A Practical Logic of Cognitive
Systems
Volume 2
The Reach of Abduction:
Insight and Trial
D RAFT O NLY
Not for circulation or quotation, please.
Comments much appreciated
Dov M. Gabbay
John Woods
Department of Computer Science
The Abductive Systems Group
King’s College
University of Lethbridge
Strand, London
4401 University Drive
ENGLAND
Lethbridge, Alberta
WC2R 2LS
CANADA, T1K 3M4
e-mail: [email protected]
email: [email protected]
and
and
The von Humboldt National Laureate
The Charles S. Peirce
in Computation Studies
Professor of Logic
Republic of Germany
Department of Computer Science
King’s College London
e-mail: ???????
e-mail: [email protected]
Intermediate Draft
December 2001
ii
c Dov M. Gabbay and John Woods
Contents
Acknowledgements
ix
I A Practical Logic of Cognitive Systems
1
1 Introduction
1
2 First Thoughts on a Logic of Abduction
2.1 Practical Logics of Cognitive Systems (PLCS) . .
2.1.1 A Hierarchy of Agents . . . . . . . . . .
2.1.2 Normativity . . . . . . . . . . . . . . . .
2.1.3 A Codicil on Validity . . . . . . . . . . .
2.1.4 Scarce-resource Compensation Strategies
2.1.5 Consciousness . . . . . . . . . . . . . .
2.1.6 Connectionist Logic . . . . . . . . . . .
2.1.7 Fallacies . . . . . . . . . . . . . . . . .
2.2 Introductory Remark on Abduction . . . . . . . .
2.3 The Basic Structure of An Abductive Logic . . .
2.4 A Schema for Abduction . . . . . . . . . . . . .
2.5 An Expanded Schema for Abduction . . . . . . .
2.5.1 The Cut-Down Problem . . . . . . . . .
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II Conceptual Models of Abduction
3 Background Developments
3.1 Brief Introductory Remarks . . . .
3.2 Peirce . . . . . . . . . . . . . . .
3.3 Explanationism . . . . . . . . . .
3.3.1 Enrichment . . . . . . . .
3.3.2 The Covering Law Model
3.3.3 The Rational Model . . .
3.3.4 Teleological Explanation .
3.4 Characteristicness . . . . . . . . .
3.5 Hanson . . . . . . . . . . . . . .
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CONTENTS
iv
3.6
3.7
Darden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fodor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Non-explanationist Accounts
4.1 Brief Introductory Remarks
4.2 Newton . . . . . . . . . .
4.3 Planck . . . . . . . . . . .
4.4 Russell and Gödel . . . . .
4.5 Popper . . . . . . . . . . .
4.6 Lakatos . . . . . . . . . .
4.7 Hintikka . . . . . . . . . .
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5 A Sampler of Developments in AI
5.1 Explanationist Diagnostics . . . . . . . . . . . . . . .
5.2 Another Example of Explanationist Diagnostics . . . .
5.3 Coherentism and Probabilism . . . . . . . . . . . . . .
5.3.1 The Rivalry of Explanationism and Probabilism
5.3.2 Explanatory Coherence . . . . . . . . . . . . .
5.3.3 Probabilistic Networks . . . . . . . . . . . . .
5.4 Pearl Networks for ECHO . . . . . . . . . . . . . . .
5.4.1 Mechanizing Abduction . . . . . . . . . . . .
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6 Further Cases
6.1 The Burglary . . . . . . . . . . . . . . . . . . . . .
6.1.1 Setting the Problem: The Logic of Discovery
6.2 The Piccadilly Line . . . . . . . . . . . . . . . . . .
6.3 The Yellow Car . . . . . . . . . . . . . . . . . . . .
6.4 A Proof from Euclid . . . . . . . . . . . . . . . . .
6.5 Neuropharmacological Intervention . . . . . . . . .
6.6 The Economy of Research . . . . . . . . . . . . . .
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7 Plausibility
7.1 Historical Note . . . . . . . . . .
7.2 Characteristicness . . . . . . . . .
7.3 A Cautionary Note on Plausibility
7.4 Rescher’s Plausibility Logic . . .
7.5 The Rescher Axioms . . . . . . .
7.6 Plausibility and Presumption . . .
7.7 Relishing the Odd . . . . . . . . .
7.8 The Filtration Structure Again . .
7.8.1 Some Suggestions . . . .
7.9 A Last Word . . . . . . . . . . . .
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CONTENTS
v
8 Relevance and Analogy
8.1 Agenda Relevance . . . . . . . . .
8.2 The Role of Relevance in Abduction
8.3 Analogical Argument . . . . . . . .
8.4 The Meta Approach . . . . . . . . .
8.5 APPENDIX . . . . . . . . . . . . .
8.5.1 Introduction . . . . . . . . .
8.5.2 The Simple Agenda Model .
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9 Refutation in Hypothesis-Testing
9.1 Semantic Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Dialectical Refutation . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Dialectical Contexts . . . . . . . . . . . . . . . . . . . . . .
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10 Interpretation Abduction
10.1 Hermeneutics . . . . . . . . . . . .
10.1.1 Abductive Inarticulacy . . .
10.1.2 Achilles and the Tortoise . .
10.1.3 Sextus Empiricus . . . . . .
10.1.4 Some Virtual Guidelines . .
10.2 Charity . . . . . . . . . . . . . . .
10.2.1 Indeterminacy of Translation
10.2.2 Dialogical Charity . . . . .
10.2.3 Radical Charity Again . . .
10.3 Semiotics . . . . . . . . . . . . . .
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185
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11 Relevance
11.1 Agenda Relevance . . . . . .
11.2 Alternative Accounts . . . . .
11.2.1 Relevant Logic . . . .
11.2.2 Probabilistic Relevance
11.2.3 Dialectical Relevance .
11.2.4 Contextual Effects . .
11.2.5 Topical Relevance . .
11.3 Agenda Relevance Again . . .
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215
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12 Limitation Issues
12.1 The Problem of Abduction . . . .
12.1.1 Justifying Induction . . .
12.1.2 Justifying Abduction . . .
12.1.3 Objections . . . . . . . .
12.1.4 Skepticism . . . . . . . .
12.2 More On Skepticism . . . . . . .
12.3 Voltaire’s Riposte . . . . . . . . .
12.4 Probativity . . . . . . . . . . . . .
12.4.1 Transcendental Arguments
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237
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CONTENTS
vi
III
Formal Models of Abduction
13 A Glimpse of Formality
13.1 Some Schematic Remarks . . . . . . . . . . . . . . . . . . . . . . .
13.2 Case Study: Defeasible Logic . . . . . . . . . . . . . . . . . . . . .
13.3 Case Study: Circumscription . . . . . . . . . . . . . . . . . . . . . .
Bibliography
257
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293
It is sometimes said that the highest philosophical gift is to invent important new philosophical problems. If so, Peirce is a major star on the
firmament of philosophy. By thrusting the notion of abduction to the forefront of philosophers’ consciousness he created a problem which — I will
argue — is the central one in contemporary epistemology.
Jaakko Hintikka, “What is Abduction? The Fundamental Problem of
Contemporary Epistemology”, 1998.
viii
CONTENTS
Acknowledgements
The present work arises from conversations the authors had at the 1999 meeting of the
Society for Exact Philosophy in Lethbridge. Early results were presented to a seminar
on abduction in June 1999 in the Department of Computing and Electrical Engineering,
Heriot-Watt University, organized by Fairouz Kamareddine; and to the Conference on
the Frontiers of Logic at King’s College, London in November 1999, organized by
Dov Gabbay. Additional work formed the basis of presentations in May 2000 to the
conference on Modelling at the University of Lethbridge in June 2000, organized by
Paul Viminitz. Further results were worked up at the University of British Columbia
in the Spring of 2000, where Gabbay was Visiting Fellow of the Wall Institute for
Advanced Study and Woods a visitor in the Noted Scholar Programme. Arrangements
in Vancouver were organized by Mohan Matthen.
A first draft of the book served as the text for a course the authors taught at the
European Summer School on Logic, Language and Information at the University of
Birmingham in August 2000. Achim Jung was responsible for these arrangements.
Subsequent results were discussed at the Conference on Formal and Applied Practical Reasoning, Imperial College, London in September 2000, organized by Jim Cunningham and Johan van Benthem, to meetings of the Ontario Society for Study of
Argumentation at the University of Windsor in May 2001, organized by Hans Hansen
and Christopher Tindale, and to the University of Groningen where Woods was the
Vonhoff Professor in the Spring Term of 2001. Arrangements in Groningen were made
by Erik Krabbe and Theo Kuipers. Further revisions were presented to the Conference
on the Logic of Rational Agency, in Dagstühl in January 2002. Mike Woolridge and
Wiebe van der Hoek were in charge of arrangements. Woods also acknowledges the
support of the Vakgroep Taalbeheersing in the University of Amsterdam where he was
Visiting Professor in the Spring and Summer 1999, and in most years since 1987. Frans
H. van Eemeren and the late Rob Grootendorst are especially to be thanked.
Alex Boston was Woods’ research assistant at UBC, as were Dawn Collins and
Brian Hepburn at the University of Lethbridge. Brian Hepburn also formatted the early
drafts of the book in LATEX. The project was supported in London by Jane Spurr,
Publications Administrator to Dov Gabbay and [...]. The authors express their warm
thanks to all.
Woods research is supported by grants from the Social Science and Humanities
Research Council of Canada, the University of Lethbridge Research Fund, the Engineering and Physical Sciences Research Council of the United Kingdom, of which he
was Visiting Fellow in the calendar year 2002, the Department of Computer Science at
ix
x
Acknowledgements
King’s College London, where he is the Charles S. Peirce Visiting Professor of Logic
and the former Dean of Arts and Science of the University of Lethbridge, Professor
Bhagwan Dua, to whom Woods owes a particular debt of gratitude.
Gabbay’s research for this book was partially supported by the European Community’s PLATO PROJECT , a joint project with Edinburgh university. The objectives
of this project is to develop lopgical tools and a goal directed database for modelling
some of the arguments of Plato. The mechanism of Abduction plays a central role
in such models. Further support is acknowledged from the Alexander von Humboldt
Foundations which provided a generouls fellowship for work in general on the logics
of practical reasoning. We are also grateful to King’s College London for a Sabbatical
leave allowing for the research on practical reasoning mechanisms, and finally both authors are grateful for the EPSRC fellowship supporting John Woods’s visits to King’s
college during the coming year.
In addition to those already mentioned, we are grateful for their support of our
work to Atocha Aliseda, Chris Reed, Jeanne Peijnenberg, David Atkinson, Jan-Willem
Romeyn, Jan Albert van Laar, Menno Rol, Alexander van den Bosch, David Makiinson, Victor Rodych, Gabriella Pigozzi, Odinalda Rodriques, Krystia Broda, Jim Cunningham, Johan van Benthem, Peter McBurney, Peter Bruza, Andrew Irvine, Bryson
Brown, Joe Wells, and Frans van Eemeren.
Part I
A Practical Logic of Cognitive
Systems
1
0
Chapter 1
Introduction
The closest thing to God is an original thought.
Abraham Isaac Kook
The surprising fact is observed. But if were true, would be a matter
of course. Hence there is reason to suspect that is true.
Peirce [1931–1958, 5.189]
This volume is the second in a series of monographs under the collective title,
The Practical Logic of Cognitive Systems. The other volumes are Agenda Relevance:
An Essay in Formal Pragmatics [Gabbay and Woods, 2002a]; and, in progress, Seductions and Shortcuts: Fallacies in the Cognitive Economy; Bobbing and Weaving:
Non-cooperation in Dialogue Logics and Belief Revision in Executable Temporal Logics. Our aim in these works is ecumenical. To the extent possible we want our own
accounts to absorb, extend and refine the virtues of alternative approaches, while eliminating features that we, and others, judge to be unsatisfactory.
Abduction is our subject here. We meet it in a state of heightened theoretical activity. It is part of the contemporary research programmes of logic, cognitive science, AI,
logic programming, and the philosophy of science. This is a welcome development. It
gives us multiple places to look for instruction and guidance.
The approach that we take in this book is broadly logical. Any fears that this will be
an over-narrow orientation is allayed by our decision to define logic as the disciplined
description of the behaviour of real-life logical agents. In this we command a theme
that has played since antiquity: that logic is an account of how thinking agents reason
and argue. We propose a rapproachment between logic and psychology, with special
emphasis on developments in cognitive science. It would be foolish in the extreme
to suggest that the hugely profitable theoretical attainments of modern mathematical
logic have no place in an agent-based, psychologically realistic account of abduction.
The rich yield in the four main sectors of post-Fregean logic — proof theory, set theory, model theory and recursion theory — are bench marks of intellectual achievement.
1
2
CHAPTER 1. INTRODUCTION
But because these results bear on properties of propositional structures independently
of the involvement of an agent or (as in proof theory) in ways involving only highly
abstract intimations of agency, the theorems of standard mathematical logic tend not
to have much in the way of direct application to what thinking agents do when they
reason. Indirect relevance is another matter. Part of the story of an agent’s discharge of
his cognitive agenda will involve his taking into account (perhaps only tacitly at times)
properties of propositional structures, such as consistency and consequence. Accordingly, we propose to appropriate the logics of propositional structures into a more capacious logic of agency. In so doing, we take special note of various important ways
in which to ‘agentify’ logics of propositional structures, whether by way of natural deduction protocols, relevance constraints, Prolog with restart and diminishing resources,
N-Prolog with bounded restart for intuitionistic logic, action and time modalities (e.g.,
‘obligatory’ and ‘now’), labelled instructions to presumed agents in proof theoretic environments (e.g., labelled deductive systems), softening the consequence relation (e.g.,
non-monotonic, default and defeasible logics) enriching semantic models (e.g., fibred
semantics), applications to fallacious reasoning (e.g., the analysis of petito principii by
way of intuitionistic modal logic); and so on.
It is these developments collectively to which the name of the new logic has been
applied [Gabbay and Woods, 2001d]. It is a conception of logic especially suited to
our purposes.
In seeking a rapproachment with psychology, we risk the opprobrium of those who,
like Frege, abjure psychologism in logic. In this we are entirely at one with Frege. If
logic is confined to an examination of propositional structures independently of considerations of agency and contexts of use, then psychology has no place in logic. If,
moreover, psychologism were to require that logic be thought of as just another experimental science, then with Frege, we would part company with psychologism. If,
however, it is legitimate to regard a logic as furnishing formal models of certain aspects
of the cognitive behaviour of logical agents, then not only do psychological considerations have a defensible place, they cannot reasonably be excluded.
If, as we think, the distinction between logic and psychology is neither sharp nor
exclusive, it is necessary to say what a logic, as opposed to a psychology, of abduction,
or of any inferential practice of a logical agent, would be. In this we see ourselves in the
historical mainstream of logic. From its inception logic has been in the first instance
an account of target properties of propositional structures, which are then adapted for
use in the larger precincts of reasoning and arguing. Aristotle invented syllogistic logic
to serve as the theoretical core of a wholly general and psychologically real theory
of argument and inference. Doing so led him to ‘agentify’ the defining conditions
on the syllogistic logic of propositional structures in ways that made it, in effect, the
first intuitionistic, linear, relevant, non-monotonic, paraconsistent logic [Woods, 2001].
Aristotle adapted the syllogistic constraints in ways that took explicit recognition of
agency, action and interaction. The father of logic, therefore, ran the broad programme
of logic by adapting the logic of propositional structures to the more comprehensive
logic of inference and argument. What made it appropriate to consider the larger project
as logic was that it was a principled extension and adaptation of the theory of those core
propositional structures. It is something in this same sense that ours too is a work of
logic. We seek a principled account of what abductive agents do, in part by extension
3
and adaptation accounts of propositional structures, which themselves have been set
up in ways that facilitate the adaptation and anticipate the approximate psychological
reality of our formal models. Ironically, then, what we are calling the new logic is in
its general methodological outlines not only the old logic, but logic as it has been with
few exceptions throughout its history.
However, for us there is a point of departure. Although we concede that key logical properties of propositional structures play a central role in the broader logic of
argument and inference, it is also our view that this broader logic must also take into
account the fact, attested to by science and intuition alike, that real-life reasoning in
real-time is not always a matter of conscious symbolic processing.
Cognitive science engages our logical interests in its two main branches. Like cognitive neuropsychology, the new logic has an interest in modelling natural cognition.
Like artificial intelligence, the new logic seeks an understanding of model of artificial
intelligence with extensions to robotics and artificial life. In both domains it remains
a characteristic contribution of cognitive science to conceive of cognition as computational; so its emphasis is on computer modelling of cognition whether artificial or
biological.
Much of the impetus for theoretical work on abduction comes from the philosophy
of science. Philosophy of science was, in effect, invented by the Vienna Circle and
the Berlin Group in the first third of the century just past. For the thirty or forty years
after the mid-thirties, philosophy of science was a foundational enterprise. It proposed
a logic of science as giving the logical syntax of the language of (all) science. It was
an approach that disclaimed as excessively psychological the context of discovery (as
Reichenbach called it), and concentrated on the context of justification (also Reichenbach). It was an imperious rather than case-based orientation, in which the flow of
enquiry is top-down, from logic to the philosophy of science, and thence to science
itself. And its analytical method was that of the explication of concepts.
In the past decades, philosophy of science has evolved from its original stance to
its present non-foundational or neoclassical orientation, as Kuipers calls it [Kuipers,
2000]. In it a place is provided for the contest of discovery and there is much less
attention paid to the (presumed) logical syntax of science and more on the construction
of mathematical models of the conceptual products of natural scientific theorising. Accordingly, the flow of enquiry is bottom-up, from science, to the philosophy of science,
to the logic of science.
The logic of science, as conceived of in the neoclassical orientation, provides formal idealised models of the most general and central feature of the conceptual content
of the process of scientific reasoning. If we remove the qualification ‘scientific’, the
logic which the new classical logic of science exemplifies resembles our own more
general conception of logic, all the more so when the modelling process is allowed to
range over thinking processes themselves, rather than just the conceptual products of
such processes.
We have remarked that a distinguishing feature of present-day cognitive science is
its tendency to regard cognition as computational. It is not however an essential condition on cognitive science that it eschew formal modelling for computational. This is
a latitude that we intend to make something of. Although we shall enquire into computer implementation of accounts of abductive reasoning, our main technical thrust
4
CHAPTER 1. INTRODUCTION
will be by way of formal models. In this respect we propose to retain a distinctive and
methodologically powerful feature of mathematical logic, even though the reach of our
models extends beyond sets of properties of agent-independent propositional systems,
as we have said. This bears on a second feature of our work, namely, the necessity to
specify inputs to our formalising devices. In a rough and ready way, these can only
be what is known informally about how reasoning agents reason. In a well-entrenched
analytical tradition these inputs are made accessible by the theorist’s intuitions about
reasoning agents, which stand as his data for a theory of reasoning. Intuitions are the
conceptual data for theory, the inputs to the theory’s formal models. We ourselves
are disposed to take this same approach to inputs, but we do so only after taking to
heart Bacon’s admonition against the use of undisciplined observations in science, and
similar advice from Suppes, who counsels the importance of presenting one’s formalising devices with models of the data [Suppes, 1962] and [Starmans, 1996, chapter 4].
Models of the data constitute the conceptual content of our theory of abduction, which
serves in turn as inputs to formalisation. To achieve the requisite conceptualisation,
we employ a methodological tool which was characteristic of the old philosophy of
science, and which served the broader purposes of analytic philosophy. This is the
methodology of concept explication or conceptual analysis. Given that our project is a
logic of abduction, concepts for explication or conceptual analysis include the obvious
ones: abduction itself, explanation, relevance, plausibility, analogy and so on. In treating of these notions, the explication should leave the explicated concept recognisably
present; it should preserve clear cases; it should be precisely specified; it should be as
simple as the actual complexities allow; and where possible it should stimulate new
questions and motive new research. The process of concept explication produces an
informal theory of abduction. It stands midway between the raw data of which they are
a conceptual model in the manner of Suppes and Starmans and aggressively specified
formalizations.
Formal modelling is an intrinsic part of our project. Models call for a certain caution. It is not too far off the mark to say that formalised models are empirically false
representations of theoretical targets. Not only do models ‘lose’ information; they
exchange accuracy and detail for precision and systematicity. The trade-off is economically sound when an agent’s actual behaviour can fairly be called an approximation to
the deviances sanctioned in the model (it’s norms, so to speak). But not every difference between fact and idealised norm is an approximation; so care needs to be taken.
Some ideal modelers are of the view that a model’s norms attain their legitimacy
and their authority because they achieve requisite standards of prescriptiveness. Thus
norms tell us how we should reason, and actual practice reveals how we do reason,
and therefore how close actual reasoning is to correct reasoning. We ourselves have
little confidence in this approach. We are not of the view that prescriptive legitimacy is
self-announcing. Accordingly we favour a more circumspect approach by way of what
we call the
Actually Happens Rule: To see what agents should do, look first to what
they actually do. Then repair the account if there is particular reason to do
so.
In espousing the Actually Happens Rule we find ourselves somewhat in the com-
5
pany of cognitive scientists who take the so-called panglossian approach to the analysis
of human rationality [Stanovich, 1999]. (We return to this point in section 2.2 below.)
It is said that the neo-classical approach to the philosophy of science is neutral on
the realism-instrumentation question [Kuipers, 2000]. The ambiguity of the claim calls
for a clarification. If it means that neo-classical philosophy of science doesn’t adjudicate the rivalry between realism and instrumentalism in science, then the claim is
correct. If it means that neo-classical philosophy of science doesn’t acknowledge realist and/or instrumentalist manoeuvres discernible in actual scientific practice, then it is
a claim from which we demur. Our own position is that the distinction between purporting to be a tenable instrumentalist move and being one is easier to take theoretical
command of than the distinction between purporting to be a tenable realist move and
being one. We do not claim to have the conceptual or epistemological wherewithal to
control this second distinction in a principled way. We take the scientist and the everyday researcher as they come. We grant that human reasoning in general is embedded in
realist presumptions, and yet we also acknowledge episodes of self-proclaimed instrumentalist reasoning. (The early Gell-Mann was an utter instrumentalist about quarks.
Later on, he lightened up considerably, encouraged by empirical developments! See
here [Johnson, 1999].) The logic we seek to construct is a logic for the human reasoner
as he presents himself to the world. We leave anti-skeptical validation of that stance to
the epistemological experts.
Speaking of matters epistemological, we ourselves favour a non-foundationalist
perspective, which can be seen as echoing the non-foundationalism of neo-classical
philosophy of science. Ours is a generically reliabilist approach, in what passes for
knowledge is belief induced by the appropriate cognitive devices when functioning
properly. It is not a view for which we here argue; others (e.g., [Millikan, 1984]) have
done so before us, and in ways adequate for our purposes here. (But see also Wouters
[1999].)
6
CHAPTER 1. INTRODUCTION
Chapter 2
First Thoughts on a Logic of
Abduction
[T]he limit on human intelligence up to now has been set by the size of
the brain that will pass through the birth canal .... But within the next few
years, I expect we will be able to grow babies outside the human body, so
this limitation will be removed. Ultimately, however, increases in the size
of the human brain through genetic engineering will come up against the
problem that the body’s chemical messengers responsible for our mental
activity are relatively slow-moving. This means that further increases in
the complexity of the brain will be at the expense of speed. We can be
quick-witted or very intelligent, but not both.
Stephen Hawking, The Daily Telegraph, 2001.
2.1
Practical Logics of Cognitive Systems (PLCS)
The theory of abduction that we develop in this volume seeks to meet two conditions.
One is that it show how abduction plays within a practical logic of cognitive systems
(PLCS). The other that it serve as an adequate stand-alone characterization of abduction. In the first instance we try to get the logic of cognitive systems right, with specific
attention to the operation of abduction itself. In the second instance, we try to get abduction right, and we postulate that our chances of so doing improve when the logic of
abduction is lodged in the more capacious PLCS.
We open this chapter with a brief discussion of what we take a PLCS to be. Readers
who wish a detailed discussion can consult chapters 2 and 3 of our Agenda Relevance:
An Essay on Formal Pragmatics.
7
8
2.1.1
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
A Hierarchy of Agents
A cognitive system is a triple of a cognitive agent C, cognitive resources R, and cognitive target performed in real time. Correspondingly, a logic of
a CS is a principled description of conditions under which Cs deploy Rs in order to
perform Ts. Such is a practical logical when the C it describes is a practical agent.
How agents perform is constrained in three crucial ways: In what agents are disposed towards doing or have it in mind to do (i.e., their agendas); in what they are
capable of doing (i.e., their competence); and in the means to convert competence
into performance (i.e., their resources). Loosely speaking, agendas are programmes of
action, exemplified by belief-revision and belief-update, decision-making and various
kinds of case-making and criticism transacted by argument. For ease of exposition we
classify this motley of practices under the generic heading “cognitive”, and we extend
the term to those agents whose practices these are.
An account of cognitive practice should include an account of the type of cognitive
agent involved.
Agency-type is set by the degree of command of resources an agent needs to advance or close his (or its) agendas. For cognitive agendas, three types of resources are
especially important. They are (1) information, (2) time, and (3) computational capacity. Seen this way, agency-types form a hierarchy H partially ordered by the relation C
of commanding greater resources than. H is a poset (a partially ordered set) fixed by
the ordered pair C, A of the relation C on the unordered set of agents.
Human agency ranks low in H. If we impose a decision not to consider the question
of membership in H of non-human primates, we could say that in the H-space humans
are the lowest of the low. For in the general case the cognitive resources of information,
time and computational capacity are for human agents comparatively scarce resources.
For large classes of cases, humans perform their cognitive tasks on the basis of less
information and less time than they otherwise might like to have, and under limitations
on the processing and manipulating of complexity.
Institutional entities contrast with human agents in all these respects. A research
group has more information to work with than any individual, and more time at their
disposal; and if the team has access to the appropriate computer networks, more firepower than most individuals even with good PCs. The same is true, only more so, for
agents placed higher in the hierarchy — for corporate agents such as the NASA, and
collective endeavours such as quantum physics since 1970.
These are vital differences. Agencies of higher rank can afford to approximate to
conditions of optimization (they can wait long enough to have a serious shot at total
information, and they can run the calculations that close their agendas that are both
powerful and precise). Individual agents stand conspicuously apart. For most tasks,
the human cognitive agent is a satisficer. He must do his business with the information
at hand, and, very often, sooner rather than later. Making do in a timely way with what
he knows now is not just the only chance of achieving whatever degree of cognitive
success is open to him as regards a given agenda; it may also be what is needed in
order to avert unwelcome disutilities, or even death. (We do not, when seized by an
onrushing tiger phenomenon, wait, before fleeing, for refutation of skepticism about the
external world or a demonstration that the approaching beast is not an hallucination.)
2.1. PRACTICAL LOGICS OF COGNITIVE SYSTEMS (PLCS)
9
Given the humbleness of his place in H, the human individual is faced with the
need to practise cognitive economies. He must turn scarcity to advantage. That is, he
must (1) deal with his resource-limitations and in so doing (2) he must do his best not
to kill himself. There is a tension in this dyad. The scarcities with which the individual
is chronically faced are the natural enemy of getting things right, of producing accurate
and justified answers to the questions posed by his agenda. And yet not only do human
beings contrive to get most of what they do right enough not to be killed by it, they also
in varying degrees prosper and flourish.
This being so, we postulate for the individual agent scarce-resource compensation strategies with which he subdues the cognitive disadvantages that inhere in the
scarcities presently in view. We do so in the spirit of Simon [1957] and an ensuing
literature in psychology and economics. At the heart of this approach is the wellevidenced fact that “fast and frugal” is almost as good as full optimization, and at
much lower cost [Gigerenzer and Selten, 2001]. We shall not take time to detail at
length the various conditions under which individuals extract outcome-advantage from
resource-disadvantage, but the following examples will give some idea of how these
strategies work. For a fuller discussion, see also [Woods et al., 2001] and [Gabbay and
Woods, 2002a] cf. [Sperber and Wilson, 1986]. We note that the hierarchical approach
to agency gives us a principled hold on the distinction between practical and theoretical agents and, correspondingly, between practical and theoretical reasoning. Practical
reasoning is reasoning done by a practical agent. An agent is a practical agent to the
extent that he ranks low in H, i.e., commands comparatively few cognitive resources.
Theoretical reasoning is reasoning done by a theoretical agent. An agent is theoretical
to the extent that it stands high in H, i.e., commands comparatively much in the way of
cognitive resources.
In some respects our interpretation of the practical-theoretical dichotomy may strike
the reader as nonstandard, if not eccentric. We ourselves are prepared to put up with
the nonstandardness in return for conceptual yield. Beyond that, there is a point of considerable intuitiveness between our notion of theoretical agent and the standard notion
of theory. We could almost say that an agent is a theoretical agent to the extent that
he discharges his or its cognitive agendas by constructing or adapting theories — e.g.,
by driving the engines of quantum physics or molecular biology. Not only do theories
have characteristic semantic and epistemic structures (see, e.g., [Suppe, 1977]), they
also exhibit particular economic facets. One is the indispensability of science-to-date
to what Kuhn has called normal science. (Even revolutionary science is highly contextualized.) Another is that even among individual scientists, science is prosecuted in
terms and under conditions that greatly reduce the press of time, while making technologically sophisticated provision for the efficient use of large and complex databases.
What factors such as these emphasize is the institutional character of theories; and it is
precisely this feature that characterizes our conception of theoretical agency.
2.1.2 Normativity
Our Actually Happens Rule raises the question of the extent to which a PLCS sets
normative standards for rational cognitive practice. We have already said that this rule
bears some affinity to a position in cognitive science called panglossism. We should
10
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
say something more about this. Contemporary cognitive science marks a distinction
among
different models of cognitive performance. They are the normative model
whichthree
specify sets standards of optional rational performance, irrespective of the
costs — especially the computational costs — of compliance. The prescriptive model
attenuates these standards so as to make them computationally executable by beings
like us. The descriptive model gives a law-governed account of how beings like
us actually perform. Following Stanovich [1999], it is possible
to discern three quite
different positions concerning how the three models, , , and , are linked. The
principle of linkage is nearness to the goal of good reasoning. On the panglossian
approach, there is little or nothing to distinguish , , and in relation to the good
reasoning goal. At the opposite extreme is the “apologist” position in which meets
the goal and both and fail it, and do so both seriously
next-to-equally.
meetsand
fails it, The
“meliorist” position takes up the middle position.
the goal.
but
not so badly as to preclude approximate realization. , on the other hand, fails it
significantly.
It is not our intention to deal with the panglossian-meliorist-apologist rivalries at
length. If we were forced to choose among the three, we would opt for the panglossian
position. On the other hand, we find ourselves attracted to a fourth position, which
the panglossian position somewhat resembles, but which is also importantly different.
Baldly stated, we are disposed to reject the normative model, and to reject its prescriptive submodel. Thus our own position is vacuously panglossian: merely reflects
good reasoning rather well, and no other model reflects it better
(since
there are none).
What so inclines us is the failure of those who espouse the , , trichotomy to
demonstrate
that
provides objectively correct standards for optimal rationality and
that provides objectively correct standards for computationally realizable optimal
rationality. (Sometimes called “optimization under constraint”. See [Stigler, 1961].)
We will not debate this issue here (but see, e.g., [Woods, 2002b, Ch. 8]). But perhaps
an example would help explain our reluctance. It is widely held that a system of logic
is a normative model of good reasoning, because it contains provably sound rules for
the valid derivation of conclusions from premisses (or databases). This presupposes
that the hitting of validity-targets is invariably an instance of good reasoning. The truth
is that in lots of situations valid deduction from premisses or a database is not the way
in which good reasoning proceeds. Consider the case in which John comes home and
sees smoke pouring from an open door ( , for short) John then reasons as follows:
“Since , then or 2+2=4; since or 2+2=4, then or 2+2=4 or Copenhagen is the
capital of Denmark”. Meanwhile John’s house burns to the ground. This does not mean
that on occasion valid deduction is part of what the good reasoner should aim for. For
such cases, it is nothing but good to have a reliable account of what it is that is being
aimed for. A system of logic may fill the bill by providing a good analysis of validity.
Welcome though such an analysis would be, the logic that provided it would not be a
plausible normative model for good reasoning.
But what if the reasoner’s task were such as to require the use of deduction rules?
Wouldn’t a system of logic whose deduction rules were provably sound and complete
be a convincing model of that particular sort of good reasoning? No. Good reasoning
is always good in relation to a goal or an agenda (which may be tacit). Reasoning of
the sort in question is good if it meets its goal or closes its agenda using only valid
2.1. PRACTICAL LOGICS OF COGNITIVE SYSTEMS (PLCS)
11
deduction rules. Reasoning validly is never itself a goal of good reasoning; otherwise
one could always achieve it simply by repeating a premiss or by uttering an arbitrary
necessary truth, such as “ ”.
Suppose, finally, that the would-be deductive reasoner had ready to hand a sound
and complete set of deduction rules and a set of heuristic principles that, for any goal
attainable through deductive reasoning, guided the reasoner in the selection of particular rules at each step of the deduction process. Wouldn’t those deduction rules together
with those heuristic rules serve as a normative model of the sort of reasoning our reasoner has set out to do? And, given that no such heuristics would work if they weren’t
actually used by real-life deducers, isn’t there reason to think that we have here a case
in which a normative and a descriptive model converge in a panglossian way?
Yes. But we propose a clarification and a caveat. The clarification is that we do
not eschew the idea of normative cognitive performance. Our view is that there is
no reliable way to capture this normativity save by attending to what beings like us
actually do. Thus our normativity is descriptively immanent, rather than descriptively
transcendent. The caveat that we would make is that no such set of deductions rules
supplemented by the requisite heuristic rules suffice for the rationality of the goal that
our reasoner may wish to achieve on any given occasion. For, again, suppose John, on
seeing the smoke pouring from his house, sets out with the help of additional premisses
to establish deductively that the house from which the smoke is pouring is not in a
suburb of Copenhagen. Equipped in the way we have been supposing, John might
achieve his deductive goal perfectly. Meanwhile his house will still have burned to the
ground.
Before leaving this matter, it would be well to take note of two prominent arguments
on behalf of the existence of normative models (and, by extension, of prescriptive models) of human cognitive performance. Let be a kind of cognitive performance, and
let ! #" $ % % % $ #& ' be the set of standards for that are sanctioned in a normative
model ( . According to those who favour the normative model approach, there are two
reasons for supposing that the standards *) are normative for us ground-zero reasoners.
1. The analyticity rationale.
The #) are normative for us because they are analytically true descriptions of what it is for to be rational.
2. The reflective equilibrium rationale.
The #) are normative for us because they are in reflective equilibrium
with what is taken to be rational -practice.
We reject both these rationales. The analyticity rationale stands or falls with the theoretical utility of the concept of analyticity (or truth solely by virtue of the meaning of
constituent terms). There is a large literature, with which it is appropriate to associate
the name of Quine, that counsels the rejection of analyticity. (See, e.g., [Quine, 1953];
cf. [Woods, 1998].) These criticisms will be familiar to many readers of this book,
and we will not repeat them here. (We are not so much attempting to prove a case
against the normative models approach as to indicate to the reader why it is that we do
not adopt it.) Still, here is an instructive example. Until 1903 it was widely held that
the axioms of intuitive set theory were analytic of the concept of set. Then the Russell
paradox put an end to any such notion.
12
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
The reflective equilibrium rationale can briefly be characterized as a balancing
act between conservatism and innovation. Consider a proposed new + -principle. It
should be rejected if it contradicts received opinion about + -rationality. Equally, consider a piece of + -behaviour. It should be rejected if it contradicts the established
+ -principles. But a new principle can be adopted if it attracts the requisite change in
accepted + -behaviour.
The doctrine of reflective equilibrium pivots on the fact that the + -theorist, like
everyone else, begins in medias res. He cannot proceed except on the basis of what he
already believes about + -hood, and he cannot proceed unless what he initially believes
about + -hood is also believed by others who are relevantly situated. The qualification
“relevantly situated” is important. If + -theory is a relatively technical matter, then the
relevantly situated others could be the community of researchers into + . Their beliefs,
and our + -theorist’s starting point, are a blend of individual + -judgements and sets of
+ -principles that are now in reflective equilibrium.
We accept this account as a descriptively adequate representation of how a + theorist actually proceeds. We concede that at a practical level there is no other way for
the + -theorist to proceed. These ways of proceedings are an indispensable heuristic for
the would-be + -theorist. But it is a mistake to think that, because this is the way that he
must proceed it must follow that the reflective equilibrium from which he does proceed
is epistemically or normatively privileged. It is a mistake of a sort that we call the
Heuristic Fallacy. This is the fallacy of supposing that any belief that is indispensable
to the thinking up of a theory is a belief that must be formally sanctioned by the theory
itself.
Finally, it is necessary to say a word about ideal models. In the final Part of this
book, we develop formal models of abduction. A formal model is an idealized description of what an abductive agent does. As such, it reflects some degree of departure
from
accuracy. Thus an ideal model , is distinct from a descriptive model
- . Butempirical
isn’t this tantamount to a capitulation to normativity? It is not. Ideality is not
normativity in our use. Ideality is simplification in support of systemacity. The laws
of frictionless surfaces are idealizations of the the slipperiness of real life. The laws
that hold in the ideal model are approximated to by the pre-game ice at Maple Leaf
Gardens. But no one thinks on that account that there is something defective about
the Gardens’ ice-conditions. We do not want to minimize the difficulty in getting the
right ideal models of + -behaviour. All that we say now is that it is no condition on the
selection of ideal + -models that they hit pre-set standards tricked out in a presumed
normative model . .
2.1.3
A Codicil on Validity
It is very much part of our intellectual tradition to value truth-preservation in what we
think and say. It was not always so. Aristotle invented the property of syllogisity, as
we might call it. Syllogisity is a restriction of the property of validity. It is validity
subject to further constraints. One is that for a valid argument to be a syllogism it must
have no redundant premisses. Another is that the conclusion not repeat a premiss. A
third is that there be no multiple conclusions. Yet another (a derived condition) is that
premiss-sets be consistent.
2.1. PRACTICAL LOGICS OF COGNITIVE SYSTEMS (PLCS)
13
Truth-preservation is enabled
2 by argument-structures that are monotonic. Let /10
be any valid2 argument, with the conclusion and / a (possibly empty) set of premisses.2 /0
is valid just in case any valuation that makes every 35476 true also
makes true. Suppose now that we supplement / by arbitrarily selected propositions
arbitrarily many times. Call this
all
2 new set /!896 . Let : be any valuation on which
members of / are true only if is true. Let : * be any valuation making every
member
2 is valid,
of /8;6 true. Then : is properly
included in : *. Since : provides that /0
2
: * provides that /<8=6>0 is valid.
Validity therefore is a natural suppressor of enquiry. Once an argument has been
determined to be valid there is nothing to be said for any further investigation from the
point of view of truth-preservation. From that perspective, the distinction between admissible and inadmissible new information is lost, just as there is nothing to be gained
by belief-updating. And given that truth-preservation is your cognitive target, there is
little to be said for belief-revision either.
What justifies our (supposed) enthusiasm for truth-preservation? Part of the answer
lies in the high levels of risk aversion implicit in truth-preserving manoeuvres. Truthpreserving manoeuvres are mother’s milk to error-avoidance, though this is not quite
the word for it. What is intended is that valid reasoning is guaranteed to present you
with conclusions that are no worse than the worst that can be said
2 of the premisssets that imply them. If / is an inconsistent premiss-set, deducing from it will not
in general make for more inconsistency; and if the premiss-set contains a falsehood,
2
deducing from it won’t in general land you with a new falsehood.1 But here, too,
the empirical record suggests that human beings are not routinely committed to erroravoidance.
A third feature of truth-preservation should be mentioned. Truth-preservation is
valuable only if there are truths to preserve. At the level of common sense, truths
are liberally available. But no one with the slightest interest in the history of human
cognitive performance will accept the congruence of what is true and what is taken
for true by common sense. Validity does not preserve what-is-taken-for-true. What
it preserves is truth. The value of truth-preservation is parasitic upon the existence of
truth. So determining whether a piece of truth-preservation is valuable is no easier
than determining whether a proposition you think is true is true. Where, then, do these
points leave us? If we are happy with vacuous truth-preservation, validity is a standing
invitation to admit new information indiscriminately and without adjustment. It is an
attainable goal, but no one will think it a cognitive virtue.
If, on the other hand, non-vacuous truth-preservation is what is wanted, it is as hard
to pin-down as are the strict truths that are the necessary inputs to the preservation
mechanism. In vast ranges of cases these truths are precisely those whose digging
up requires the larger resources of institutional agencies. But in general individuals
lack sufficient command of such resources to make these targets possible, never mind
economically attainable.
This is not to say that, a knowledge of classical consequence is not of use to the
2
?
1 Of course, if you allow for schematic deduction of an arbitrary proposition from an inconsistent , you
do get more inconsistency if your deductive apparatus is classical. (This is because negation-inconsistency is
classically equivalent to absolute inconsistency.) But, given the equivalence, it is far from clear that absolute
inconsistency is worse than negation-inconsistency, pace logicians of paraconsistent bent.
14
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
human reasoner. It helps knowing what follows from what when it comes to database
updates and revisions. Logical consequence is the heart and soul of certain varieties
of hypothetical reasoning. It is indispensable to a form of counter-argument which
Aristotle called “refutation.” And so on. But where the consequence relation is classical
consequence, then knowledge of its provenance will, in virtually any given case, leave
the performance of these tasks underdetermined [Harman, 1986, ch. 1 and 2].
Perhaps the single most daunting point to make about truth-preservation is how
slight a role it plays in our cognitive lives. In and of itself valid deduction plays no
role in ampliative reasoning, and ampliative reasoning is practically all the reasoning
that a human agent has time for. Such is the rapidity and extent of the new information
that plays over him second by second, and the abiding imperative of adjusting to new
circumstances with genuinely new appreciations of the passing scene. In general, individual agents lack the time and information to lodge these new beliefs in combinations
of old and new inputs that constitute a deductive demonstration of it.
2.1.4
Scarce-resource Compensation Strategies
Hasty Generalization
Individual cognitive agents are hasty generalizers. In standard approaches to fallacy
theory and theories of statistical inference, hasty generalization is a fallacy; it is a
sampling error. This is the correct assessment if the agent’s objective is to find a sample
that is guaranteed to raise the inductive probability of the generalization. Such is an
admirable goal for agents who have the time and know-how to construct samples that
underwrite such guarantees. But as Mill was shrewd in seeing, human individuals
usually lack the wherewithal for constructing these samples. The business of sampleto-generalization induction for the most part exceed the resources of individuals and is
better left to institutions.
Generic Inference
Part of what is involved in a human reasoner’s disposition towards the one-off generalization is his tendency to eschew generalizations in the form of universally quantified
conditional propositions. When he generalizes hastily the individual agent is likely
making a generic inference. In contrast to universally quantified conditional propositions, a generic claim is a claim about what is characteristically the case. “For all x, if x
is a tiger, then x is four-legged” is one thing; “Tigers are four-legged” is quite another
thing [Carlson and Pelletier, 1995]. The first is felled by any true negative instance,
and thus is fragile. The second can withstand multiples of true negative instances, and
thus is robust. There are significant economies in this. A true generic claim can have
up to lots of true negative instances. So it is true that tigers are four-legged, but there
are up to lots of tigers that aren’t four-legged. The economy of the set-up is evident:
With generic claims, it is unnecessary to pay for every exception.
2.1. PRACTICAL LOGICS OF COGNITIVE SYSTEMS (PLCS)
15
Natural Kinds
Our affection for generic inference and hasty generalization is connected to our ability
to recognize natural kinds. Natural kinds have been the object of much metaphysical
skepticism of late [Quine, 1969], but it is a distinction that appeals to various empirical
theorists. The basic idea is evident in concepts such as frame [Minsky, 1975], prototype
[Smith and Medin, 1981], script [Schank and Abelson, 1977] and exemplar [Rosch,
1978]. It is possible, of course, that such are not a matter of metaphysical unity but
rather of perceptual and conceptual organization.
Defaults
Generic inference tolerates exceptions, but it is not infallible. The cognitive economy
of individual agency is a highly fallibilist one. It is an economy characterized by defaults. A default is something taken as true in the absence of indications to the contrary
[Reiter, 1980]. It is characterized by a process of reasoning known as “negation-asfailure” [Geffner, 1992]. For example, Harry checks the departure times for a direct
flight from Vancouver to London early Saturday afternoon. Finding none posted, he
concludes that there is no such flight at that time on Saturday.
Default reasoning is inherently conservative and inherently defeasible, which is
the cognitive price one pays for conservatism. Conservatism is, among other things,
a method for collecting defaults D. One of the principles of collection is:“D is what
people have thought up to now, and still do” or “ D is common knowledge.” The
economies achieved are the off-set costs of fresh-thinking. (Desartes’ epistemological project would be costly beyond price for an individual to execute.) Defaults are
intimately tied to factors of genericity. (See below, section 7.2.)
Discourse Economies
Further economies are evident in regularities affecting conversation. One such has been
called
The Reason Rule: One party’s expressed beliefs and wants are a prima facie reason for another party to come to have those beliefs and wants and,
thereby, for those beliefs and wants to structure the range of appropriate
utterances that party can contribute to the conversation. If a speaker expresses belief X, and the hearer neither believes nor disbelieves X, then
the speaker’s expressed belief in X is reason for the hearer to believe X
and to make his or her contributions conform to that belief. ([Jacobs and
Jackson, 1983, p. 57] and [Jackson, 1996, p. 103]).2
The reason rule records a default. Like all defaults, it is defeasible. Like most defaults,
it conserves crimped resources. Like defaults in general, it appears to do little cognitive
harm.
A corollary to the reason rule is the ad ignorantiam rule:
2 The reason rule reports a de facto regularity between real-life discussants. When the rule states that a
person’s acceptance of a proposition is reason for a second party to accept it, “reason” means “is taken as
reason” by the second party.
16
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
Ad Ignorantiam Rule: Human agents tend to accept without challenge the
utterances and arguments of others except where they know or think they
know or suspect that something is amiss.
Here, too, factors that trigger the ad ignorantiam rule are dominantly economic. Individuals lack the time to fashion challenges whenever someone asserts something or
advances a conclusion without reasons that are transparent to the addressee. Even when
reasons are advanced, social psychologists report that addressees tend not to appraise
them before accepting the conclusions they purport to underwrite. Addressees tend
to do one or other of two different things before submitting such reasons to critical
scrutiny. They tend to accept person’s conclusion if they find it plausible. They also
tend to accept the other party’s conclusion if it seems to them that this is a conclusion
which is within that person’s competence to make; that is, if he is judged to be in a
position to know what he is talking about, or if he is taken as having the appropriate expertise or authority. (See, e.g., [Petty and Cacioppo, 1986], [Eagly and Chaiken,
1993], [Petty et al., 1981], [Axsom et al., 1987], [O’Keefe, 1990], and the classic paper
on the so-called atmosphere effect, [Woodworth and Sells, 1935]. But see also [Jacobs
et al., 1985].)
2.1.5
Consciousness
A further important respect in which individual agency stands apart from institutional
agency is that human agents are conscious. (The consciousness of institutions, such as
may be figuratively speaking, supervenes on the consciousness of the individual agents
who constitute them.) Consciousness is both a resource and a limitation. Consciousness has a narrow bandwidth. This makes most of the information that is active in a
human system at a time consciously unprocessible at that time. In what the mediaevals
called the sensorium (the collective of the five senses operating concurrently), there
exist something in excess of 10 million bits of information per second; but fewer than
40 bits filter into consciousness at those times. Linguistic agency involves even greater
informational entropy. Conversation has a bandwidth of about 16 bits per second [Zimmermann, 1989].
Conscious experience is lineating experience. Human agents are adept at being
in multiples of conscious states at a time. And time flows. These two facts suggest
an operational definition of the linearity of consciousness. Linearity operates in the
cognitive economy in much the way that tight money operates in the actual economy.
It slows the flow of information and simplifies it. Linearity is a subduer of complexity
and therewith an inhibitor of information.
Psychological investigations indicate that most of our waking experience is less
than fully conscious. This carries interesting consequences for conversational phenomena. It is widely held that when someone says that, e.g., it’s snowing, he is giving
voice to a mental state — the belief that it’s snowing — which is an object of his attention then and there. His assertion is sincere if the mental state actually obtains and true
if the belief that the mental state induces is a true belief. On this view, conversation in
general is a sequence of exchanges of transparent propositional contents coded in the
requisite speech acts.
2.1. PRACTICAL LOGICS OF COGNITIVE SYSTEMS (PLCS)
17
If the psychological studies are right, the received view is wrong. Conversation
cannot in general be the exchange of the propositional contents of conscious mental
states.
The narrow bandwidth of consciousness bears on the need for cognitive economy.
It helps elucidate what the scarcity of information consists in. We see it explained
that at any given time the human agent has only slight information by the fact that
if it is consciously held information there is a bandwidth constraint which regulates
its quantity. There are also devices that regulate consciously processible information
as to type. A case in point is informational relevance. When H.P. Grice issued the
injunction, “Be relevant”, he left it undiscussed whether such an imperative could in
fact be honoured or ignored by a conscious act of will. There is evidence that the
answer to this question is No; that, in lot’s of cases, the mechanisms that steer us
relevantly in the transaction of our cognitive tasks, especially those that enable us to
discount or evade irrelevance, are automatic and pre-linguistic [Gabbay and Woods,
2002a]. If there is marginal capacity in us to heed Grice’s maxim by consciously
sorting out relevant from irrelevant information, it is likely that these informational
relevancies are less conducive to the closing of cognitive agendas than the relevancies
that operate “down below”. Thus vitally relevant information often can’t be processed
consciously, and much of what can is not especially vital.
Consciousness can claim the distinction of being one of the toughest problems, and
correspondingly, one of the most contentious issues in the cognitive sciences. Since
the agency-approach to logic subsumes psychological factors, it is an issue to which
the present authors fall heir, like it or not. Many researchers accept the idea that information carries negative entropy, that it tends to impose order on chaos.3 If true, this
makes consciousness a thermodynamically expensive state to be in, for consciousness
is a radical suppressor of information. Against this are critics who abjure so latitudinarian a conception of information [Hamlyn, 1990] and who remind us that talk about
entropy is most assured scientifically for closed systems (and that ordinary individual
agents are hardly that).
The grudge against promiscuous “informationalism” in which even physics goes
digital [Wolfram, 1984] is that it fails to explain the distinction between energy-toenergy transductions and energy-to-information transformations [Tallis, 1999, p. 94].
Also targeted for criticism is the view that consciousness arises from or inheres in neural processes. If so, “[h]ow does the energy impinging on the nervous system become
transformed into consciousness?” [Tallis, 1999, p. 94].
In the interests of economy we decline to join the metaphysical fray over consciousness. The remarks we have made about consciousness are intended not as advancing
the metaphysical project but rather as helping characterize the economic limitations
under which individual cognitive agents are required to perform.
Conscousness is tied to a family of cognitively significant issues. This is reflected
3 See here Colin Cherry: “In a descriptive sense, entropy is often referred to as a ’measure of disorder’
and the Second Law of Thermodynamics as stating that ‘systems can only proceed to a state of increased
disorder; as time passes, entropy can never decrease.’ The properties of a gas can change only in such a way
that our knowledge of the positions and energies of the particles lessens; randomness always increases. In a
similar descriptive way, information is contrasted, as bringing order out of chaos. Information, then is said
to be ‘like’ negative energy.” [Cherry, 1966, p. 215]
18
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
in the less than perfect concurrence among the following pairs of contrasts.
1. conscious v unconscious processing
2. controlled v automatic processing
3. attentive v inattentive processing
4. voluntary v involuntary processing
5. linguistic v nonlinguistic processing
6. semantic v nonsemantic processing
7. surface v depth processing
What is striking about this septet of contrasts is not that they admit of large intersections
on each side, but rather that their concurrence is approximate at best. For one thing,
“tasks are never wholly automatic or attentive, and are always accomplished by mixtures of automatic and attentive processes” [Shiffrin, 1997, p. 50]. For another, “depth
of processing does not provide a promising vehicle for distinguishing consciousness
from unconsciousness (just as depth of processing should not be used as a criterial
attribute for distinguishing automatic processes . . . ” [Shiffrin, 1997, p. 58]. Indeed
“[s]ometimes parallel processing produces an advantage for automatic processing, but
not always . . . . Thoughts high in consciousness often seem serial, probably because
they are associated with language, but at other times consciousness seems parallel . . . ”
[Shiffrin, 1997, p. 62].
2.1.6
Connectionist Logic
There is a large literature — if not a large consensus on various aspects of subconscious, prelinguistic cognition. If there is anything odd about our approach, it can only
be the proposal to include such matters in the ambit of logic. Most, if not all, of what
people don’t like about so liberal a conception of logic is already present in the standard
objections to psychologism, which we have already discussed. Strictly speaking, there
is room for the view that, while psychologism is not intrinsically hostile to logic, psychologism about the unconscious and the prelinguistic simply stretches logic further
than it can go, and should therefore be resisted.
This is an admonition that we respect but do not intend to honour. In this we draw
encouragement from work by Churchland and others ([Churchland, 1989] and [1995])
on subconscious abductive processes. As Churchland observes, “... one understands at
a glance why one end of the kitchen is filled with smoke: the toast is burning!” [1989, p.
199]. Churchland proposes that in matters of perceptional understanding, we possess
“... an organized library’ of internal representations of various perceptual situations,
situations to which prototypical behaviors are, the computed output of the well-trained
network” [1989, p. 207]. Like Peirce [1931–1958, p. 5.181], Churchland sees perception as a limit of explanation, and he suggests that all types of explanation can be
modelled as prototype activation by way of “... vector coding and vector-to-vector
transformation” rather than linguistic representation and standardly logical reasoning.
On this approach the knowledge that comes from experience is modelled in the patterning of weights in the subject’s neural network, where it is seen as a disposition of
2.1. PRACTICAL LOGICS OF COGNITIVE SYSTEMS (PLCS)
19
the system to assume various activation configurations in the face of various inputs.
Thus, as Robert Burton nicely puts it, Churchland is drawn to the view that “inference
to the best explanation is simply activation of the most appropriate available prototype
vector” [Burton, 1999, p. 261].
The suggestion that abduction has (or has in part) a connectionist logic is attractive
in two particular ways. One is that, unlike every other logic of explanation, connectionist explanation has a stab at being psychologically real. The other, relatedly, is
that a connectionist logic is no enemy of the subconscious and prelinguistic sectors of
cognitive practice. It is no panacea, either. (See the section just above.) There is nothing in the connectionist’s prototype-library that solves the problem of the deployment
of wholly new hypotheses, as in the case of Planck’s postulation of quanta. On the
other hand, the same is true of computer systems such as PI [Thagard, 1988], which
mimic simple, existential, rule-forming and analogical genres of abduction. (See here
[Burton, 1999, p. 264]). We return to systems such as PI in chapter 5 below. For, again,
Beyond that, we should not want to say that serial processing requires consciousness:
Thoughts high in consciousness often seem serial, probably because they
are associated with language, but at other times consciousness seems parallel, as when we attend to the visual scene before us. So the distinction
between parallel and serial processing does not seem to map well onto the
distinction between the conscious and the unconscious [Shiffrin, 1997, p.
62].
2.1.7 Fallacies
Before leaving the issue of an individual’s cognitive economics we touch briefly on
objections that might be brought. On the account sketched here, the individual is
an inveterate fallacy-monger, whether of hasty generalization, ad verecundiam or ad
ignorantiam. In fuller accounts of the cognitive economy of individuals, the appearance of inveterate fallaciousness is even more widely evident [Woods et al., 2001;
Gabbay and Woods, 2001a]. It is not impossible that the human agent is an intrinsic a fallacy-monger, but we ourselves are disinclined to say so. The charge may be
answered in one of two ways.
(1) The practice would be a fallacy if interpreted in a certain way. But
under more realistic construal, the practice in question doesn’t fit with the
fallacy in question.
(2) The practice in question even under realistic interpretation qualifies as
a fallacy by the lights of a certain standard, does not qualify as a fallacy
under a lesser standard, and it is the lesser standard that has the more
justified application in the context in question.
If we go back to the example of hasty generalization, we see that if the generalization
is inference to a universally quantified proposition, then it cannot be a truth-preserving
practice. But if the hasty generalizer is taken to be making a generic inference, there
20
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
is no prospect of guaranteed truth-preservation; so the practice cannot be convicted of
failing to attain what it does not aim at.
The individual agent also economizes by unreflective acceptance of anything an
interlocuter says or argues for, short of particular reasons to do otherwise. This outrages
the usual ban on the ad verecundiam, according to which the reasoner accepts his
source’s assurances because he thinks that the source has good reasons for them. (The
fallacy, then, would be the failure to note that the source is not suitably situated to
have good reasons for his assurances.) Empirical findings indicate that this is not the
standard which real-life individuals aim at. They conform their responses to a weaker
constraint: If you have reason to believe that your source lacks good reasons for his
assurances, then do not accept his assurances. The default position of ad verecundiam
reasoners is that what people tell one another is such that incorporating it into one’s
own database or acting on it then and there is not in the general case going to badly
damage one’s cognitive agendas, to say nothing of wrecking one’s life. We see in
this a (virtual) strategy of cooperative acceptance, tentative though it is and must be,
rather than a strategy for error-avoidance or error-minimization. Judged by the requisite
standard, such trust is in general neither misplaced nor fallacious. A fallacy is always
a fallacy in relation to a contextually appropriate standard.
Ad ignorantiam is our final example. In its most basic form it is an inference in the
form
1. It is not known that @
2. So, not-@ .4
In that form there is not much to be said for it. But no one argues just by way of
argument forms. In requisitely incarnate arrangements we sometimes get rather good
arguments, such as negation-as-failure arguments. In their turn, negation-as-failure
arguments are variations of autoepistemic arguments, such as:
1. If there were a Department meeting today, I would know about it.
2. But I don’t,
3. So there isn’t.
Or, as in the departure-announcement example,
1. If there were a direct flight from Vancouver to London early Saturday
afternoon, the schedule would make that known to me.
2. But it doesn’t
3. So there isn’t.
Autoepistemic inferences are inferences to a default. My default position is that there
is no such meeting and is no such flight. Such inferences are non-monotonic. New
information might override these defaults. Here, too, there are fallacious cases of the
ad ignorantiam depending on what the relevant standard is. Nobody thinks that the ad
4 We are discussing the modern form of the ad ignorantiam, not Locke’s conception, which in turn is a
variant of Aristotle’s ignoratio elenchi. [Woods, 1999] and [Woods, 2002a]
2.2. INTRODUCTORY REMARK ON ABDUCTION
21
ignorantiam is truth-preserving. 5 For agents who are constitutionally and circumstantially bound to transact their cognitive agendas on the cheap (fast and frugal), who will
say that the standards of default reasoning are inappropriate?
Let us say in passing that the variabilities that inhere in the hierarchy of agencytypes suggest a general policy for the fallacy-attribution. It is roughly this. A fallacy
is a mistake made by an agent. It is a mistake that may not seem to be a mistake.
Moreover, it is a mistake that is naturally made, commonly made, and not easy to repair (i.e., to avoid repeating). [Woods, 2002a, ch. 1] An inference or a move in a
dialogue, or whatever else, is a fallacy relative to the type of agent in question and the
resources available to agents of that type; and to the performance standards appropriate thereto. Given that individuals operate with scarce resources, given the economic
imperatives that those scarcities impose, what may have the look of fallacious practice
lacks the cognitive targets and the performance standards against which fairly to judge
inferences or moves in fulfillment of such targets as fallacious. On the other hand, for
agencies of a type that occurs higher up in H — NASA, for example — cognitive targets are different (and more expensive) resources are abundant, and standards for the
assessment of performance are correspondingly higher. Relative to those targets and
those standards, cognitive practices having this appearance of fallaciousness are much
more likely to be fallacious.
This helps motivate the traditional idea of a mistake that seems not to be a mistake.
At the appropriate level, a cognitive practice is a mistake and may not appear to be a
mistake, because at lower levels of the hierarchy it is not a mistake. Similarly, at least at
the level of individual agency, we have an unforced explanation of why practices, which
higher up would be fallacies, are lower down natural common and hard to change. It
is because they are evolutionarily and experientially the best ways for individuals to
manage their resource-strapped cognitive economies.6
In our way of proceeding, conceptual accounts are inputs to formalization. Outputs
are formal models of this conceptual content, in which the goals of greater precision
and systematicity are realized. Conceptual content inputs are sometimes called intuitive
theories. So we begin with an intuitive pragmatics for abduction. 7
2.2
Introductory Remark on Abduction
The term ‘abduction’ was introduced into logical theory by Charles Peirce in the late
19th century. The introduction was not wholly original, since ‘abduction’ is a passable
5 An
exception:
1. If I had a throbbing headache I would know it.
2. But I don’t
3. So I haven’t.
6 This resource-based approach to fallacies can only be lightly sketched here. The fuller story may be
found in [Gabbay and Woods, 2001b]and [Gabbay and Woods, 2001a].
7 The idea of cognitive economics is also the subject of important research in such disciplines as political
science and marketing. See, for example, [Simon, 1982; Lilien et al., 1992; Stigler, 1961; Shugan, 1980].
Still, there are important differences between Simon’s approach to scarce-resource (or bounded) rationality
and that of, say, Stigler.
22
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
translation of Aristotle’s apagogB A , which is also translated as ‘reduction’ and was given
the Latin rendering abductio by Julius Pacius. For Aristotle, an abduction is a syllogism
from a major premiss which is certain and a minor premiss which is merely probable,
to a merely probable conclusion (Prior Analytics 2.25 69 C 20–36). In his early attempts
to characterize abduction, Peirce also takes a syllogistic approach:
Conceived of in Peirce’s way, abduction is a form of reasoning in which a new
hypothesis is accepted on the grounds that it explains or otherwise yields the available
data. More broadly, the logic of abduction is a general theory of hypothesis formation.
It extends its reach to the logic of discovery and the economics of research. It interacts
with or gives the appearance of interacting sublogics of relevance, plausibility, and
analogy.
It is necessary to note at the outset some significant ambiguities in the concept of
abduction. In its most general sense, abduction is a process of justifying an assumption, hypothesis or conjecture for its role in producing something in which the abducer
has declared an interest. Even within this category various distinctions press for recognition. For example, the hypothesis might help explain a given set of data or some
phenomenon; or it might facilitate the generation of observationally valid predictions;
or it might permit the elimination of other hypotheses, thus providing the theorist with
a simpler and more compact account. Here, then, are three distinct reasons which an
abducer might offer as a justification for using a given hypothesis or for making a given
conjecture. Abductions of this sort have an unmistakably pragmatic character. They are
justifications of use without necessarily being arguments for the truth of the hypotheses
in question.8
2.3
The Basic Structure of An Abductive Logic
In its use here, “abduction” is the generic name of a threefold cognitive process: (1) the
process of generating hypotheses; (2) the process of selecting hypotheses from rival hypothesis; and (3) the process of locking hypotheses in. In most writings on abduction,
a cruder distinction is proposed between thinking hypotheses up and adopting them for
use in a theory or in a piece of reasoning. Answering to this twofold contrast is the distinction between a logic of discovery and a logic of justification. We propose a more
textured approach to the logic of abduction, one that reflects the present trichotomy.
Accordingly, we will speak of a logic of hypothesis-generation, which is the logic of
process (1), a logic of hypothesis-engagement, which is the logic of process (2), and a
logic of hypothesis-discharge, which is the logic of process (3). If H is an hypothesis
selected in accordance with the logic of process (2), then at state (3) H is discharged
when it is asserted rather than merely assumed. The propositional content of the H of
stage (3) remains the same as that of H at stage (2). What changes is the manner in
which it is forwarded. What was entertained at stage (1) is assumed at stage (2), and
is asserted at stage (3). Thus a stage (3) H loses not its propositional content but rather
its prior hypothetical character.
8 Newton, for example, accepted the action-at-a-distance theorem, but he was firm in thinking it unbelievable. (See below, section 2.4)
2.3. THE BASIC STRUCTURE OF AN ABDUCTIVE LOGIC
23
Nicholas Rescher [1964] attributes to Peirce [1931–1958 ] the decision to reserve
the word “abduction” for our stages (1) and (2) combined, and “retroduction” for our
process (3). Without pressing the question strict historical accuracy, we see retroduction as covering hypothesis-engagement and hypothesis-discharge (i.e., our processes
(2) and (3)). The combined logics of these two processes we would call retroductive
logic. What we call the logic of hypothesis-generation is also called the logic of discovery. By our lights, this could be said to be abductive logic in the narrow sense.
The truth is that our trichotomy doesn’t jibe very naturally with prior classification
and distributions, whether the “logic of discovery”, “abduction” v “retroduction” or
“abduction in the narrow sense” v “abduction in the broad sense”. We are conservatives
about inventing nomenclature, which we think should honour a variant of Ockam’s
Razor. In the present case, we find that our trichotomy more nearly cuts the nature of
abduction at the joints, and shall accordingly persist with it, reserving “abduction” for
the three sublogics combined.
Our third sublogic is that part of a logic of abduction in which the hypothetical character of a previously engaged hypothesis D is discharged. It is the de-hypothesization
of a proposition by dint of categorial commitment. A little reflection shows that this
process makes a twofold provision for the target H. Not only does it categorialize H,
but it also deploys it: that is, it gives it the standing of a premiss for further inferences
in the domain of which H is now a paid-up member. Categorialization and deployment
often go hand in hand, but it is not necessary that they do. It is possible that an H be
deployed short of categorialization. This is done when, e.g., H is taken as a working
assumption in a police investigation. It is characteristic of Popper’s understanding of
scientific methodology that the deployment of an H should never be accompanied by its
categorialization. On this view, competent science is intrinsically noncategorical, even
when its conjectures are given huge load-bearing roles to play. Popper’s conservatism
is also apparent in Peirce’s, “Provisionally, we may suppose S to be of the nature of
M, [Peirce, 1992, p. 140]. In an alternative formulation of the conclusion of a piece
of abductive reasoning, Peirce writes, “Hence there is reason to support that A is true.”
[Peirce, 1931–1958, p. 5.189], which may give a lesser tentativeness than required by
Popper.
It is possible that Popper (and perhaps Peirce too) has confused fallibility with
hypotheticality, and correspondingly the utterance of defaults with the deployment of
assumptions. The whole issue turns on whether it is possible to give categorical expression to a default, e.g., that Pussy the tiger is four-legged. Certainly it is not in general
the case that, when one asserts something about which one might be mistaken, one
also asserts that one might be mistaken. For our purposes it is unnecessary to bring
this question to decisive settlement. Suffice it to say that Popper holds that the closest that an H can come to categorial issuance is by way of deployment as a working
assumption.
Abductive logic, then, is a triple of the sublogics of generation, engagement and
discharge, both Popperian and full-bore. 9 By some lights, no such logic is possible, a
9 Peirce also calls for a tripartite conception of logic. One of the sublogics is semiotics, which is a theory
of possible forms. The second sublogic is a logical theory of argument. The third is a theory of abduction,
deduction and induction as constituents in a self-adjusting dynamic, which is expressed in methods that
inform both science and ordinary life alike. It is clear that Peirce’s trichotomy is not ours, but there are
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
24
theme that replays the usual run of criticisms of logics of discovery. It is also possible
to press the reverse complaint, according to which abductive logic, while possible, is
nevertheless not necessary. Such is the view of [Kapitan, 1992] who holds that
1. what is genuinely logic about the logic of abduction is reducible to
deductive or inductive logic; and
2. what is not thus reducible isn’t logic, but is rather a merely practical
component having to do with “the conduct of inquiry and, consequently, with procedures for evaluating inferences to practical directives” [Kapitan, 1992, p. 3].
There are conceptions of logic which lend plausibility to Kapitan’s thesis of the reducibility of abductive logic to the logics of deduction and induction. However, earlier
in this chapter we tried to motivate a conception of logic which denies Kapitan’s thesis
much of this plausibility.
The question of the irreducibility or otherwise of abductive logic is an important
question. It is possible that it might be settled solely by virtue of these most abstractly
general features of abductive practice. We ourselves admit to a certain skepticism.
Better, we say, to develop the logic of abduction in as much detail as possible before
turning to the metalogical task of ascertaining its degree of independence from other
logics.
It bears on this question that in one use of the word “inductive”, abduction straightaway is part of induction, though not necessarily reducible to induction’s other parts.
On this approach, induction is non-demonstrative reasoning of any kind. We have no
interest in terminological squabbles. We see induction as non-truth-preserving inference from samples to generalizations and to predictions, and abduction, therefore, as
something apart. In the end, it matters little for the task of getting the logic of abduction right that we agree on how broadly to throw the nominative nets of “abduction”.
Once it is up and running, abductive logic will be part of inductive logic, or will, for
a different conception of it, be a separate thing from the logic of induction. The main
thing is to get the logic of abduction right.
2.4
A Schema for Abduction
At a certain level of abstraction, a solved abduction problem has the following structure
[Aliseda-LLera, 1997, pp. 26–32]. Let EGFIH JK L M M M L JN O be a database of some
kind. It could be a theory or an inventory of beliefs, for example. Let P be a yielding
relation, or, in the widest possible sense, a consequence relation. Let Q be a given
wff representing, e.g., a fact, a true proposition, known state of affairs, etc. And let
JN RS L TFVU L M M M L W be wffs. Then X EL PYL QZL JN RS [ is an abductive resolution only if
the following conditions hold.
1.
2.
E<\=] J N *R S ^PZQ
E<\=] J N *R S ^ is a consistent set
points of similarity which we shall note in due course.
2.4. A SCHEMA FOR ABDUCTION
3.
4.
_`>a
b cd e*f gY`>a
25
.
The generality of this schema allows for variable interpretations of h . In standard AI
approaches to abduction there is a tendency to treat h an classical deductive consequence. But, as we have seen, this is unrealistically restrictive. h can also be treated
as nonclassical (e.g., non-monotonic) consequence, as statistical consequence, causal
consequence, and so on. At this level of generality the abduction schema subsumes
both Hempel’s deductive-nomological and his inductive-statistical models of explanation [Aliseda-LLera, 1997, p. 2]. However if, as Peirce insisted, it is a condition on
b cd ef g that it be introduced as a conjecture, Hempel’s models cease being abductive
(see [Hintikka, 1999, p. 93]).
h can be treated as a relation which gives with respect to a whatever property the
investigator (the abducer) is interested in a ’s having, and which is not delivered by _
alone or by b cd ef g alone.
Conditions (3) and (4) provide that the consequence relation that delivers the desired result has the structure of what informal logicians call linked reasoning. The
concept of linked reasoning also plays a role in the analysis of relevance [Gabbay and
Woods, 2002a].
An abduction problem arises when, for some _ and some state of affairs a , there
is an agent who desires that some property of a be delivered by adjustment of _ , in
ways that bear on a , the sentence that once derived, delivers the goods for a . In the
abduction schema, this manipulation is always by way of a consistent extension of _ ,
and is a second respect in which the schema is not sufficiently general. We want to
leave it an open resolution option to erase one or more ci in _ and replace with zero
or more new wffs. In other words, where a is a sentence that contradicts _ , there is an
abduction problem for a _ -theorist. It is the problem of determining which wffs or wffs
in _ are refuted by the presence of a . a is the property of being a falsifier of _ . What
the _ -theorists wants to know is the particular wff(s) in _ that endows a with this
property. Then, too, it is desirable to recognize a class of abduction problems which
arise from the fact that _Vha and a is ‘unwanted’, in reductio ad falsum arguments,
for example.
The standard triggers [Aliseda-LLera, 1997, p.3] of an abduction problem are novelty and anomaly. Novelty-triggers are an instance of belief-update problems, and
anomaly-triggers are a kind of belief-revision problem. In the first case, a is new information uncontradicted by _ . Accordingly a novelty-trigger requires that in addition
to
_`>a
it also could be the case that _
_`>jYa
and a
are consistent or that
.
Novelty-triggers can be denoted by ‘ ka ’. Anomaly-triggers, on the other hand, require
that
_`>a
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
26
and that l
and m
lnZoYm
are inconsistent or that
.
and are denoted by ‘ pqm ’. (We note in passing that information that qualifies as a
novelty-triggers does not conform to the more colloquial notion of ‘new information’
which can also be an anomaly-trigger.)
As we see, the n -relation plays a central role in abduction problems. At a certain level of abstraction n is a consequence relation that doesn’t deliver the goods (or
does deliver the ‘bads’) from resources at hand. Computer scientists have fallen into
the habit of describing such situations as those in which what is wanted is not proved,
and what is not wanted is proved. As here used the notion of proof imbibes the same
abstract character, reflected in the habit of thinking of n as consequence. So we shall
speak, accordingly, of the proof-theoretic character of abduction problems. The present
class of cases exemplifies a proof-theoretic conception of abduction. Of course, wanting a body of evidence to yield an explanation of something is also a target; so explanation abductions are a subcase. The proof-theoretic approach to abduction dominates
in the AI tradition, although diagnostic abduction is an important exception. Explanationist abduction is perhaps more typical of the way in which abduction is approached
in the philosophy of science. We intend to make something of this contrast between the
explanationist and proof theoretic conceptions, in as much as most of what is important
in the literature, historical as well as current, seems to fit the dichotomy rather easily.
Whether thought of in explanationist or proof-theoretic terms, the variability already noted in the abduction schema is preserved. Indeed it is enlarged. An hypothesis
may be wanted for any number of reasons, i.e., for its potential contribution to any
number of different targets. Its contribution to a given target exposes it to a variety of
judgements which can be taken, with various degrees of confidence, to be justified by
that contribution. We remark in passing that not all abduction problems are problems
of symbol manipulation. The house-painter who wants a certain shade of yellow may
consider mixing a bit more white and a dollop of brown. Stirring carefully, he then
applies some of the new mix. (And smiles.)
Inference to the best explanation is a case of what AI-theorists call backward chaining. The locus classicus of backward chaining arguments are reductio ad falsum arguments. It is not easy to distinguish in a purely semantico-syntactic way between reductio arguments and a subclass of valid but unsound arguments. Reductio arguments are
instances of these latter harnassed to an arguer’s intention. They have an irreducibly
pragmatic character. When an arguer constructs a reductio he doesn’t try for argumentsoundness (and fails), but rather he tries for argument unsoundness (and does not fail).
If his unsound arguments is a successful reductio, its conclusion is not that of the argument he constructs for reductio, but rather it is that one or more of the premisses of that
argument is untrue. His is a backward chaining manoeuvre because his conclusion is
not the conclusion of the deduction he constructs but rather is a claim involving a prior
line in it. It is thus an inference from a property of the conclusion of the deduction
to a property of one or more of the deduction’s premisses. It is the conclusion of this
inference, rather than the original deduction, that the reductio-maker seeks to draw;
and it is this that makes it a case of backward chaining.
2.5. AN EXPANDED SCHEMA FOR ABDUCTION
27
Refutation and other kinds of ad hominem reasoning (in Locke’s sense) are also
classic examples of backward chaining. We discuss these matters in chapter 9.
If we examine our tripartite logic of abduction, it will be easy to see that the
metaphor of backward chaining gets a plausible grip only in the sublogic of hypothesisgeneration. Once hypotheses are generated, all else having to do with them is full-speed
ahead, through the sequential filters that shrink large spaces of possible hypotheses to
(ideally) unit sets of the tried and the true. So one shouldn’t overdo the idea of backward chaining.
Abductions also have to do with what Reichenbach called the ‘context of discovery’, which he contrasted with the ‘context of justification’ [Reichenbach, 1938]. Abductions in this sense are said to be the business of the logic of discovery. Reichenbach
was not alone in thinking it possible to have a logic of scientific justification, which,
as he supposed, is precisely what an inductive logic is designed to be. But a logic of
discovery, or — in this second sense of abduction — an abductive logic, Reichenbach
regarded as a mistake in principle, because it confuses psychological considerations
with logical considerations. Reichenbach’s skepticism was shared by most logical positivists, but as early as 1958, [Hanson, 1958] worked up a contrast between reasons
for accepting a given hypothesis and reasons that suggest that the correct hypothesis
will be one of a particular kind or description. A theory which analyzes reasons of
this second kind Hanson called a logic of abductive reasoning. In Hanson’s account,
a logic of abduction resembles a logic of analogical reasoning. Hanson’s efforts were
criticized—even pilloried. This has as much to do with their novelty as with their deficiencies. Even so, the very idea of a logic of discovery trails some important questions
in its wash. One deals with the extent to which the supposed contrast between contexts
of justification and contexts of discovery catches a hard and fast distinction. Another
question is whether the contrast between a psychological account of hypothesis formation and a logical theory of the same thing will hold up. A related issue is the extent
to which a heuristics for a set of reasoning problems can be distinguished in a principled way from a logic for such problems. A further matter — also closely connected
with the others — is whether a theory of hypothesis formation is able to retain a sharp
distinction between descriptive adequacy and normative soundness.
It is commonly supposed that the class of discovery abductions is disjoint from
the class of justification abductions. For this to be strictly true, it would seems to be
the case that an agent is given no guidance in thinking a conjecture up from how the
conjecture would fare in a justificatory sense. One of the virtues of Hanson’s pioneering
work is the pressure it puts on any assumption of strict disjunction between discovery
abductions and justification abductions. It is our own view that this is an issue on which
to keep an open mind.
2.5
An Expanded Schema for Abduction
We now show that abduction problems have a substantially more general and more
complex structure than we have lately been supposing.
Let
a = an agent
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
28
r
s
= a database of a, a set of wffs
generic
binary relation on wffs, read yields or proves.
v r;w –state
t ==aau s7
of affairs
i = an interest of a.
s
Since
is a generic relation of yielding it here serves
as a variable. We exemplify its
s7v r;w –states
of affairs, or t s.
values in the following non-exhaustive list of u
r
s
For some , and x a member of the counter-domain of (i.e., is that to which the
s -relation
is borne),
r
r
2.
r
3.
r
4.
r
5.
6.
7.
r ys x )
z
r{ x )
does not predict x (
r{ x )
does not motivate x (
r{ x )
is not evidence for x (
r{ x
is not strong evidence for x (
r s x)
proves x (
r s x)
is evidence for x (
does not explain x (
1.
r
r
)
Etc.
An agent a has an abductive problem with respect to a t if
a.
a’s interest i is such that he seeks to find the means to reverse the t in
s
question
by replacing its -relation with its converse, e.g., replacing
r ’s not explaining
x with something’s explaining x ; or replacing r ’s
proving x with something’s blocking the proof that x . When this
condition is met, we shall say that t is a trigger for a.
rZ|
b. a undertakes to achieve this reversal by changing
in one of three
ways:
r such that r; u Hw s x or r
Expansion:
By finding a wff }=~
€
w
{
u}
, depending on what a’s trigger is. We shall say
r for short
that when the requisite H is found, then it together with “gives the
desired yield”.
r
r
w
Contraction: By finding an }=~ such that - u H gives the desired
yield.
r
r such that
Contraction
expansion:
finding a }=~ and a }= ‚
ru 5ƒ u } w w„and
 u }=
w gives theBydesired
yield.
rZ| s …
The desired yield is a’s target … . … is the wff such that
t represented
delivers the goods for or corrects the trigger-situation
r
s
by the fact that
state of affairs x either
x or, as the
rfor
{ x some
case may be,
. Thus a solved abduction problem for a always
v ˆ v t v …‰ .
involves a quadruple †‡
The quadruple
tions are met.
†‡ v ˆ v t v …Š‰
solves its abduction problem only if further condi-
2.5. AN EXPANDED SCHEMA FOR ABDUCTION
1. For no H, H‹ is it the case that H, H‹ or Œ H, H‹
yield.

29
delivers the desired
2. It is essential that agent a have no prior knowledge of H, H‹ or prior
evidence sufficient to justify their assertion. On the contrary H, H‹
must be introduced by conjecture. That is to say, H, H‹ must be
hypotheses.
Variations: read ‘theories’ for ‘wffs’,
read ‘rules’ for ‘wffs’.
Example Let Ž be the evidence presently on hand in a police enquiry. Assume
that Ž produces weak evidence for the proposition that Spike committed the crime.
Suppose that the police extend the investigation in ways that produces new evidence.
Suppose, finally, that when this new evidence is taken in conjunction with the original
evidence Ž , the case against Spike strengthens to the point of indictability. Then this
constitutes for the police a solution of an abduction problem, provided that the new
evidence alone is not sufficient to indict the suspect.
Example Let Ž be made up of a wff  9‘’Y“ , the proof rules of simplification,
addition, disjunctive syllogism, and the wffs ”  •–“ ”  ’Y“ ” – , which follow from
predecessors in accordance with those rules. Then the Œ ŽZ” — -situation ˜ that an
abducer is faced with is that a contradiction 9‘;’™ proves arbitrary – . If the abducer
is not minded to accept this result, then his abduction problem is triggered by that
fact. The problem is solved if, by manipulation of Ž , the undesired consequence no
longer obtains. Let us suppose that, upon reflection, the abducer deletes the disjunctive
syllogism rule from Ž , declaring it now to be admissible at best, but not valid.
We note in passing that it is a condition on abduction solutions that, from the alteration ŽZš of Ž , the desired consequence situation Œ Ž;š9—œ› be construable as
something like a linked argument, that is to say, an argument in which for no proper
subset  of ŽZš do we have it that 5—G› . The idea of linked arguments originates
with Aristotle’s premiss-irrelevancy condition on syllogisms. Contemporary versions
are advanced by argumentation theorists (e.g., Thomas [1977, p. 36]), and linear logicians (e.g., Girard [1987]).
The abductive constraints on Œ Ž;š—I› do not, however, expressly rule out that
some proper subset of Ž š might yield › . But it must preclude that Œ ž9 yield › where
ŽZšqŸ7Ž> ;Œ ž9 for new evidence ž , or that where Ž;šŸ<Œ Ž7¡=Œ ž9 # ;Œ ’™ž9 ” Œ ’™ž9
yield › .
We repeat that propositional contents of adjustments to an original database in
an abductions must be introduced as hypotheses, rather than as premisses expressing
known facts, or purported facts. So, in our second example, the removal of disjunctive
syllogism counts as abductive only if it is done conjecturally and in ways subject to
subsequent tests.
Historical Note In their early note on a complete formulation of propositional
logic Anderson and Belnap [1959], produced the system and the noted en passant
that it couldn’t consistently permit the validity of disjunctive syllogism. Instead of
faulting their formulation, they performed an abduction. They conjectured that the best
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
30
explanation of the fact that their formulation excluded disjunctive syllogism was that
disjunctive syllogism was an invalid proof rule. ¢
It is well to emphasize that £<¤¥ ¦ ¥ §*¥ ¨V© is the solution of an abduction problem
only within the generation sublogic. In nothing we have said so far is there the slightest
reason to suppose that the H, Hª associated with any given £<¤¥ ¦ ¥ §¥ ¨1© are the best
candidates for what is required to be assumed to hit the abducer’s target ¨ . We shall
say, then, that a solved problem £¤¥ ¦ ¥ §*¥ ¨© is the minimal condition on the solution
of a full abduction problem. Delivering the desired yield hypothetically is the least of
what is required to consummate a full abduction, for which there are many different,
incompatible and implausible ways of doing this. Something further is needed. Atocha
Aliseda speaks of “integrity constraints”, and proposes the following. Let «;¬ be the
alteration of « in a solved abduction problem £<¤¥ ¦ ¥ §*¥ ¨V© . Then «Z¬ should satisfy
the following constraints.
a.
preservation of consistency
b. deductive closure
c.
minimal sufficient change
d. preservation of embeddedness
e.
other? [Aliseda-LLera, 1997, sec. 2.6]
We shall in due course expose these suggestions to considerations which, we think, cast
them in some doubt. Suffice it to say here that even if these constraints were honoured,
it would remain true (though not as true as before) that far too many H, Hª pass through
the £¤¥ ¦ ¥ §¥ ¨5© –filter. Let us say instead that any wffs ­ that perform the role of H or
Hª in an £¤¥ ¦ ¥ §¥ ¨© are hypotheses generated for ¨ by that very fact. Accordingly
we shall abbreviate our quadruple as Gen. The task of the sublogic of hypotheses
generation is to produce every Gen for any ¨ that has one. Given the structure of
Gens, whether a given target ¨ for a is realized by an H or by an H, Hª pair, it does so
independently of whether a is aware of it. If, however, we want our generation sublogic
to be a logic of how the abductive agent reasons, then we must qualify our Gens as
those who embedded hypotheses are known to a to facilitate delivery of his desired
yield. There are no known mechanical rules for finding such hypotheses in arbitrarily
selected situations. Even so, there is a strong heuristic available to real-life abducers:
Subject to the constraints already in place, try to perform the requisite ® -manoeuvres;
i.e., try to generate ¯ or to block ¯ by drawing the requisite ® -consequence from «Z¬ . If
doing so hits the target, the hypothesis or hypotheses in question have been generated,
and a knows it.
2.5.1
The Cut-Down Problem
It is possible to speculate that what the abductive agent aims for in the generation
sublogic is specification of a space ° of possible hypotheses. Let ­ be a wff in ° . Then
­ plays the role of an H in a Gen. And it is this fact that makes ­ a possible candidate
for actual deployment. We see in this the necessity to characterize with due care the
notion of hypothesis and conjecture as they operate in the generation logic. Rather than
2.5. AN EXPANDED SCHEMA FOR ABDUCTION
31
say that when ± meets the present conditions that it is conjectured outright as a fullyfledged hypothesis by the abductive agent, better to say that the agent has identified as
a candidate for such deployment, i.e., as a possible hypothesis for possible conjecture.
The space ² of such possibilities can be seen, on the present assumption, as input
to the next step in the abduction process, as input to the engagement-sublogic. Engagement is a way of activating (some of) the possibilities in ² . Here we might suppose that
the activation condition is one of relevance. Relevance we can take as a filter that takes
² into a proper subset ³ of relevant possibilities. Accordingly the Gens of the generation sublogic can be seen as having substructures, Engages, which are Gens whose
embedded H, H´ meet a relevance condition. Thus Engages are Gens that specify the
relevant possible hypotheses. Thus the relevance filter seems to cut down the space ²
to the smaller space ³ .
The relevance filter plays a role similar to Harman’s Clutter Avoidance Principle,
which bids the reasoner not to clutter up his mind with irrelevances [Harman, 1986, p.
12]. In its role here, the relevance filter enables the abducer to concentrate on possibilities that are fewer than those found in ² , and make a better claim for activation. ³ , the
space of relevant possibilities, advances subsets of ² closer to the point of activation.
A second principlal task is to bring hypothesis-activation off, to replace possible
hypotheses that are conjectural possibilities with hypotheses that the abducer actually
deploys by actual conjecture. This, too, involves the passing of larger sets through
a contraction filter. The relevance filter took possibilities into relevant possibilities.
What is wanted now is a filter that will take relevant possibilities into plausibilities
(a cut down of ³ to µ ). The plausibility screen accordingly shrinks ³ to subset µ , µ
presents the abducer with three activation options. Assuming µ to be nonempty,
1. If µ is a unit set of ³ , then engage the hypothesis in µ .
2. If µ is a pair set of ³ or larger, then engage all the hypotheses in µ .
3. If µ is a pair set of
pothesis in µ .
³
or larger, then engage the most plausible hy-
The final phase of the logic of abduction in the discharge logic, the logic of categoricalization or de-hypothesization. Logicians are already familiar with the natural
deduction distinction between a hypothesis and a premiss. A similar distinction awaits
us here. Unlike the case of indirect proofs, in which lines introduced by hypothesis must be dropped before the proof terminates, in the discharge sublogic hypothesis
must be transformed into premisses, and correspondingly conjecture must give way to
assertion.
Hypothesis-discharge comes when the items in the plausibility space µ are passed
through a further filter. More accurately, they are passed through one or other or both
of two filters.
1. The filter of independent confirmation.
2. The filter of theoretical fruitfulness.
Either way, discharge is triggered by the successful negotiation of a test. Discharge
is the reward for passing muster. But there is muster and muster, with (1) comparatively strong in most variations and (2) rather weaker in most variations. Collectively
32
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
we might speak of these modes of trial as the test filter. Plausibilities that successfully
pass through the test filter form the (possibly empty) set of retroductions (in Rescher’s
understanding of Peirce). Correspondingly we have structures Dis that are substructures of Engages, in the way that Engages are subsets of Gens, i.e., by restrictions of
the associated hypotheses to those that pass the requisite tests. Accordingly, a solution
to a full abduction problem is the structure ¶ (Gen, Engage, Dis) = · Dis ¸ .
It hardly needs saying that there are lots of abductive targets that have only partial
solutions, that is solutions Gen but not Engage, or solutions Engage but not Dis.
Hypothesis-discharge is an essential part of abduction. It is the categoricaliazation
of an hypothesis. Hypotheses must not be confused with what logicians call hypothetical propositions. A hypothetical proposition is a proposition of the form “If A then B”,
hence is a conditional proposition. Like all propositions, a conditional statement can be
assumed, i.e., forwarded as an hypothesis, or asserted categorically. But no proposition
conditional or otherwise, can concurrently be entertained hypothetically and forwarded
categorically.
Categoricalizing an hypothesis H is a twofold process. It is finding the means to
dock H, and it is doing so to some desired effect. H is docked when together with other
data it delivers a target ¹ and does so with the desired effect. If someone’s target is
some wff, say Q, which cannot be derived from his database º , the target can be got
by adding Q to º as an hypothesis. Doing so does not however result in a docking of
Q, since although Q now follows from º!»=· ¼¸ , it does not do so with desired effect.
It is a derivation, as Russell said about another matter, that has all the virtues of theft
over honest toil. When the requisite conditions are met, full-bore retroduction docks
an hypothesis fully. the docking is acknowledged in a variety of ways, one of which is
its categoricalization.
Hypotheses are docked, hence categoricalized, in ways consonant with the particularities of the abducer’s problem. It is a long list: predicative accuracy, unifying power,
explanatory force, simplicity, conformity with other theories, and so on.
If docking is a good metaphor for full-bore discharge quasi-or Popperian discharge
might be likened to a ship’s weighing anchor in the outer harbour. Affairs ashore can
be abetted by both docked and more distantly anchored vessels, but some services are
unperformable short of a close tie-up at quayside. Skippers who anchor their craft at a
distance have a particular interest in the quick getaway. Captains of docked vessels are
not so risk-averse.
As an aid to exposition we can now use the letters ½™¾ ¿ as names of the presumed
filters and as the names of the resultant sets. The same is true of À . It names either
the order on ½ or the resultant set. The triple Á Â#¾ ¿¾ ½YÃ represents a filtration structure
on an initial space of possibilities in which the succeeding spaces are cutdowns on
their predecessors. It is possible that there also exists a further filter Ä on the space
of plausibilities to a unit subset, of (intuitively) the most plausible candidate for engagement/discharge. It is important to emphasize that Á Â*¾ ¿¾ ½YÃ and, where it exists,
Á Â*¾ ¿¾ ½„¾ · À>¸ à exists independently of whether any abductive agent has ever thought
of it. The filters that cut sets down to smaller sets at each stage do so independently
of whether anyone has actually tried to deploy those filters. In the speculation that
we have been entertaining these past few pages we have supposed that in reaching his
desired Å , the abductive reasoner proceeds by solving the cutdown problem by con-
2.5. AN EXPANDED SCHEMA FOR ABDUCTION
33
structing (or reconstructing) the requisite filtration structure. In so supposing we have
it that the would-be engager of Æ proceeds in a top-down fashion, first by entertaining
mere possibilities, then cutting to relevant possibilities and finally to the most plausible
of these.
Everything that is known of actual human agency suggests that this is not the structure of the abducer’s reasoning; for one thing it would involve the searches of spaces
that are computationally beyond the reach of such agents. This can hardly be a surprising finding, given what is known of the fast and frugal character of practical reasoning.
So, while it is perfectly reasonable to suppose that hypothesis-engagement involves
considerations of relevant plausibility; how it does is not all that clear.
Let us be clear about the present point. Ç is a structure of propositions. We already
have it that Ç satisfies the consequence constraint; i.e., each element in Ç is part of the
domain of the È -relation with regard to some abductive target É . We assume that there
is a theory of relevance that cuts Ç down to Ê , a proper subset of those propositions.
We suppose further that there is a plausibility logic that cuts Ê down to Ë , a proper
subset of plausible and relevant propositions. And we suppose, finally, that there is an
order on Ë that, at least sometimes, picks Ì , the most plausible candidate in Ë .
There is reason to think that for any given Ç , such a filtration structure exists. If
so, for any Ç there is at worst a subset of most plausible members of it. Sometimes,
this is a unit set Ì . Given that what an abductive agent seeks to do is to choose the
most plausible hypothesis Æ , there is reason to think that for any such Æ there exists a
filtration structure whose component Ë contains Æ , and whose component Ì , if there
is one, just is Í Æ9Î .
On the current assumptions, if the Æ for which the abducer seeks exists, then there
exists a filtration structure in which Æ exists and in which it has at worst an approximately unique occurrence. This it owes to the relevant logic, the plausibility logic and
the partial order that take Ç to Ì . These are the devices that locate Æ in the filtration
structure. So why not suppose that since the abducer’s task is also to locate Æ , he does
so by deploying on Ç successively the relevant logic, the plausibility logic and the partial order? If this were a tenable supposition, then the tasks of our first two sublogics
of abduction — the generation and engagement sublogics — would be discharged by
the logics of relevance and plausibility, supplemented by the partial order; in short by
constructing the requisite filtration structure.
Yes. But, again, there is no empirical evidence that this is how actual abducers
proceed. This is not to say that relevance and plausibility play no role in a tenable
account of abduction. It is only to say that they do not play the top-down roles that they
play in the construction of filtration structures. (We return to the factors of relevance
and plausibility in later chapters.)
34
CHAPTER 2. FIRST THOUGHTS ON A LOGIC OF ABDUCTION
Part II
Conceptual Models of
Abduction
35
Chapter 3
Background Developments
As I see it, to have good reason for holding a theory is ipso facto to have
good reason for holding that the entities postulated by the theory exist.
Wilfrid Sellars, Between Science and Philosophy, 1968.
3.1
Brief Introductory Remarks
In the previous chapter, we said that we would try to make some headway with the three
problems of complexity, approximation and consequence. In the present chapter, we
take up the cudgels of conceptual analysis. Our targets include, but are not exhausted
by, explanation, relevance, plausibility and analogy. We begin with explanation, concerning which there is an unstoppably huge literature. Our aim is to expose enough of
the structure of present-day debates about the philosophical analysis of explanation to
bring some detailed clarity to explanationist accounts of abduction.
In this chapter we review some of the historical background of abduction in the
modern era.1 Our purpose is not to make a complete survey, but rather to specify some
issues of importance to our project. In particular, we wish to emphasize some salient
contrasts. One is that abduction needn’t be an inherently explanatory exercise, as we
have already indicated. Another is that there are ways of providing explanations other
than by way of abduction. A third point of importance, which we owe to Peirce, is
that when an abducer makes a conjecture there is no general requirement that he will
believe it to be true or even that he will find it plausible, although there are also, of
course, contrary cases.
In so saying, it might be supposed that we have contradicted a claim in chapter 2
about the engagement sublogic of abduction. We said there that part of the process of
selecting a hypothesis for use in an abduction solution involved subjecting it to a plausibility test. But there is plausibility and plausibility. A proposition may be plausible
in what it says. A proposition may be implausible in what it says, but nevertheless
1 Some scholars discern its presence in antiquity. A case can be made, for example, that Aristotle gave an
abductive account of the first principles of the various scientific disciplines. See Woods [2001, Ch. 8].
37
CHAPTER 3. BACKGROUND DEVELOPMENTS
38
be a plausible candidate for conjecture. (We take this latter to be Peirce’s view of the
matter.)
These are points that bear with considerable force on the question of whether there
is a core structure to abduction, one which involves only contingently factors of explanation, plausibility and the like. It is well at this juncture to take note of the possibility
that what a stripped-down abductive structure will look like will vary contextually, depending on the subject matter of the abduction problem and the intentions or goals of
the abducer, as well as the sort of resources on which the abducer is able to draw in
working through his abductive agenda. It is with these things in mind that we briefly
examine in section 3.3 an important controversy in the philosophy of science over the
status of historical laws and historical explanation.
3.2
Peirce
The history of logic has not accorded Charles Peirce the attention that his work demands, although there are signs that repair of this omission is now underway. (See,
for example, [Peirce, 1992], [Nathan Houser and van Evera, 1997] and [Hilpinen,
forthcoming ].) Writing in the tradition of Boole, Peirce extended the algebraic approach in the direction of what would become modern proof theory. Yet his highly
original logic of relations would also play an influential role in the efforts of Schröder,
Lowënheim, Skolem and Tarski to bring forth a mature model theory.
In 1898 Peirce gave a series of eight lectures at Harvard.2 Delivered ‘off-campus’,
these powerful and original pieces were presented at William James’ behest, in yet another gesture of generosity towards his beleaguered friend. Peirce’s range was striking.
He had novel and prescient things to say about probability, likelihood and randomness;
about quantification (of which, with Frege, he was independent co-developer); about
causality, space and time, and cosmology. Of particular note is Peirce’s emphasis on
abduction (or retroduction, as he would also call it). Peirce had been writing on abduction since the 1860s. The Harvard lectures reflect what [Kruijff, 1998] calls the
‘late theory of abduction’. (See also [Burks, 1946; Fann, 1970] and [Thagard, 1988,
p. 53])3 As Peirce came to believe, abduction is the dominant method in science, in
fact, the only purely scientific method beyond brute observation. For Peirce, scientific
reasoning stands in sharp contrast to practical reasoning or, as he also says, to reasoning about ‘matters of Vital Importance’ [Peirce, 1992, p. 110]. In this contrast he was
hardly alone, although it may be mentioned in passing that it was a matter on which he
disagreed fundamentally with Mill, for whom ‘a complete logic of the sciences would
also be a complete logic of practical business and common life’ [Mill, 1974, Bk. III,
Ch. i, Sec. 1].
Peirce’s writings on abduction are important in a number of ways, but especially
for its emphatic rejection of the possibility of a practical logic, hence of a logic of
abductive reasoning in practical contexts. As should by now be clear, this is not a view
which the present authors are disposed to share.
2 Recently
reprinted as Peirce [1992].
identifies 1865–1875 as the early period, followed by a transition period and then, in 1890–1914,
the late period.
3 Fann
3.2. PEIRCE
39
A key distinction in Peirce’s logical writings is that between corollarial and theoremic reasoning. Corollarial reasoning simply draws deductive conclusions from sets
of premisses, whereas theoremic reasoning analyzes some ‘foreign idea’ which, although it may be discharged in the final conclusion, is needed for the conclusion to
be derived. Appropriation of certain set theoretic ideas in mathematical reasoning is
an example of theoremic reasoning. Indirect proof is a second example. Peirce also
likened abductive reasoning to theoremic reasoning.4
Peirce saw abduction as hypothetico-deductive reasoning under conditions of trial.
Thus the by now familiar pattern:
The surprising fact Ï is observed
But if Ð were true, Ï would be a matter of course.
Hence there is reason to suspect that Ð is true [Peirce, 1931–1958, p.
5.189].5
Abduction, says Peirce,
although it is little hampered by logical rules, nevertheless is logical inference, asserting its conclusion only problematically or conjecturally, but
nevertheless having a perfectly definite logical form [Peirce, 1931–1958,
p. 5.188].
Abduction is, or subsumes, a process or processes of inference. This is what Kapitan calls the Inferential Thesis.6 It is also a strategy for making refutable conjectures.
This is what Kapitan calls the Thesis of Purpose. The purpose of scientific abduction
is twofold. It produces new hypotheses and it chooses hypotheses for further scrutiny
[1931–1958, p. 6.525]. The purpose of abduction is therefore to ‘recommend a course
of action’ (p. MS637:5). 7 Abducted conclusions are not matters for belief or for probability. In abduction ‘not only is there no definite probability to the conclusion, but no
definite probability attaches even to the mode of inference. We can only say that ... we
should at a given stage of our inquiry try a given hypothesis, and we are to hold to it
provisionally as long as the facts will permit. There is no probability about it. It is a
mere suggestion which we tentatively adopt’.8
We now turn to a sketch of Peirce’s fledgling positive theory, beginning with the
Comprehension Thesis. Scientific abduction subsumes all procedures and devices
4 Reasoning of this form is commonplace in theoretical computer science. (There is also a connection
with Interpolation Theorems).
5 The factor of surprise persists in subsequent accounts. See [Hanson, 1958] and [Thagard, 1988, p. 52].
6 Kapitan [1997, pp. 477–478], cf. [Peirce, 1931–1958, pp. 5.188–189, 7.202].
7 Kapitan [1997, pp. 477–478] . MS citations are to the Peirce MSS, in care of the Peirce Edition Project.
‘MS 637’ denotes manuscript 637; (5) denotes page 5.
8 [Kapitan, 1997, p. 142] Peirce distinguishes (1) induction, or the establishment of frequencies in populations by sampling and (2) probable inference, or the inference from a known frequency in a randomly
selected sample. He anticipates Neymann-Pearson statistical sampling theory with its notion of likelihood.
By requiring that inductions have a premiss to the effect that sampling is random, Peirce thought that all
inductions turn on prior discernment of law like statements. That a method of sampling is random requires
recognition of the equality of certain frequencies, and so is a kind of law-like knowledge; that is, knowledge
of generals. Of course, Peirce didn’t get randomness right. No one did or could until the development of
recursion theory well into the last century. (Putnam is good here. See his ‘Comments on the Lectures’ in
Peirce [1992, pp. 61 and 68. Lecture Two, pp. 123–142].)
CHAPTER 3. BACKGROUND DEVELOPMENTS
40
whereby theories are produced ([Kapitan, 1997]; cf. [Peirce, 1992, p. 5.146]). Yet
‘a retroductive inference is not a matter for belief at all,. ..[an] example is that of the
translations of the cuneiform inscriptions which began in mere guesses, in which their
authors could have had no real confidence’ [Peirce, 1992, p. 176]. In this Peirce agrees
with Popper. There can be no logic of hypothesis-discharge [1992, p. 5.146] since
in truly scientific retroduction, hypotheses are not discharged; they are only assumed.
In practical matters, hypotheses may well be asserted rather than merely assumed, but
Peirce insists that there can be no logic of such practices.
[We see, then, that Science] takes an entirely different attitude towards
facts from that which Practice takes [Peirce, 1992, p. 177].
Practice involves belief ineradicably, ‘for belief is the willingness to risk a great deal
upon a proposition. But this belief is no concern of science ... .’ Belief contrasts with
acceptance, as can be gathered from this passage, which might well have been penned
nearly a century later by the philosopher and inductive logician, L.J. Cohen:9
...whether the word truth has two meanings or not, I certainly do think
that holding for true is of two kinds; the one is that practical holding for
true which alone is entitled to the name of Belief, while the other is that
acceptance of a proposition which in the intention of pure science remains
always provisional [Peirce, 1992, p. 178].
Hence, I hold that what is properly and usually called belief, that is, the
adoption of a proposition as [a possession for all time] Ñ Ñ Ñ has no place in
science at all. We believe the proposition we are ready to act upon. Full
belief is willingness to act upon the proposition in vital crises, opinion
is willingness to act upon it in relatively insignificant affairs. But pure
science has nothing at all to do with action. The proposition it accepts, it
merely writes in the list of premises it proposes to use.
And since
[n]othing is vital for science [and] nothing can be Ñ Ñ Ñ , [t]here is Ñ Ñ Ñ no
proposition at all in science which answers to the conception of belief
[Peirce, 1992, p. 178].
Whereupon,
Ñ Ñ Ñ the two masters, theory and practice, you cannot serve [Peirce, 1992,
p. 178].
Peirce anticipates Quine.
My proposition is that logic, in the strict sense of the term, has nothing
to do with how you think Ñ Ñ Ñ . Logic in the narrower sense is that science which concerns itself primarily with distinguishing reasonings into
Ò
Ò
9 ‘...to accept that
is to have or adopt a policy of deeming, positing, or postulating that —i.e. of
including that proposition or rule among one’s premisses for deciding what to do or think in a particular
context, whether or not one feels it to be true that ’ [Cohen, 1992, p. 4].
Ò
3.2. PEIRCE
41
good and bad reasonings, and with distinguishing probable reasonings into
strong and weak reasonings. Secondarily, logic concerns itself with all that
it must study in order to draw those distinctions about reasoning, and with
nothing else [Peirce, 1992, p. 143].
About these things, ‘it is plain, that the question of whether any deductive argument,
be it necessary or probable, is sound is simply a question of the mathematical relation
between Ó Ó Ó one hypothesis and Ó Ó Ó another hypothesis’ [Peirce, 1992, p. 144].
Concerned as it is with the presence or absence of that mathematical relation between propositions,
[i]t is true that propositions must be expressed somehow; and for this reason formal logic, in order to disentangle itself completely from linguistic,
or psychical, considerations, invents an artificial language of its own, of
perfectly regular formation, and declines to consider any proposition under any other form of statement than in that artificial language. Ó Ó Ó As for
the business of translating from ordinary speech into precise forms, Ó Ó Ó that
is a matter of applied logic if you will Ó Ó Ó [Peirce, 1992, pp. 144–145 ].
But applied logic stands to logic much as white chocolate stands to chocolate, for it
is ‘the logical part of the science of the particular language in which the expressions
analyzed [or translated] occur’ [Peirce, 1992, p. 145]. Peirce, like Mill, is somewhat
ambivalent about the primacy of induction. ‘As for Ó Ó Ó Induction and Retroduction, I
have shown that they are nothing but apogogical transformations of deduction and by
that method the question of the value of any such reasoning is at once reduced to the
question of the accuracy of Deduction’ [Peirce, 1992, p. 168]. On the other hand, the
‘marvelous self-correcting property of Reason, which Hegel made so much of, belongs
to every sort of science, although it appears as essential, intrinsic, and inevitable only
in the highest type of reasoning which is induction’ [Peirce, 1992, p. 145]. Of Peirce’s
transformations we have the space to say only that he regarded induction, probabilistic
inference and retroduction as distinct forms of reasoning of which Aristotle’s first three
syllogistic figures were limiting cases. (See Lecture Two [Peirce, 1992, pp. 131–141 ].)
The distinctness claim is challenged, however, by the high degree of entanglement, in
actual examples of reasoning, of deductive, inductive and abductive patterns of reasoning, never mind what their status is considered as ideal types of reasoning. (See
chapters by [Crombie, 1997] and [Kapitan, 1997] in Houser et al. [1997].) Kapitan
goes so far as to attribute to Peirce an Autonomy Thesis, according to which abduction
is, or subsumes, reasoning that is distinct from and irreducible to either deduction or induction (cf. [Peirce, 1992, p. 5.146]). That is to say, it is a part of empirical linguistics,
the natural science of natural languages.
Logic, then, has nothing to do with how we think. Still less has it to do with how
we think about vital affairs.
Once you become inflated with [what good deducers, i.e. mathematicians,
are up to,] vital importance seems to be a very low kind of importance
indeed.
But such ideas [i.e. the procedures of deductive thinkers] are only suitable
to regulate another life than this. Here we are in this workaday world, little
42
CHAPTER 3. BACKGROUND DEVELOPMENTS
creatures, mere cells in a social organism itself a poor little thing enough,
and we must look to see what little and definite task circumstances have set
before our little strength to do. The performance of that task will require
us to draw upon all our powers, reason included. And in the doing of it we
should chiefly depend not upon that department of the soul which is most
superficial and fallible — I mean our reason — but upon that department
that is deep and sure — which is instinct [Peirce, 1992, p. 121 (emphasis
added)].
Concerning the claim of abductive innateness, Thagard observes that it is not at all
clear that we have any special faculty for guessing right when it comes, say, to theoretical physics [Thagard, 1992, p. 71]. This is a view to which the present authors are
sympathetic (but see 2.9 below). Thagard goes on to remark that in “current subatomic
physics, many theorists are investigating the properties of spaces with ten or more dimensions, and it is hard to see how their speculations might be at all constrained by
biologically evolved preferences for certain kinds of hypotheses . . . [T]here is no current reason to adopt Peirce’s view that abduction to scientific hypotheses is innately
constrained” [Thagard, 1992, p. 71].
Peirce thinks that all forms of reasoning, even deduction, require observation. What
is observation? It is
the enforced element in the history of our lives. It is that which we are
constrained to be conscious of by an occult force residing in an object
which we contemplate. The act of observation is the deliberate yielding of
ourselves to that force majeure, an early surrender at discretion, due to our
forseeing that we must, whatever we do, be borne down by that power, at
last. Now the surrender which we make in Retroduction, is a surrender to
the Insistence of an Idea. The hypothesis, as The Frenchman says, c’est
plus fort que moi.10
This is a view that has attracted some experimental support. (See, e.g., [Gregory, 1970;
Rock, 1983] and [Churchland, 1989] and [Churchland, 1995], concerning which see
section 2.1.6 above.)
The passage is striking and at seeming odds with Peirce’s insistence that is not
useful in ordinary affairs and that it has nothing to do with belief or probability. (After
all, for Peirce the most common fallacy of abduction is Bayesianism, that is, to ‘choose
the most probable hypothesis’ [Putnam, 1992, p. 78].) But belief is like observation.
It presses in on us and demands our surrender. And what is so insistent about the
Insistence of an Idea if it does not somehow call for and eventuate in belief?
We begin to see the irreducibly practical aspect of abduction, contrary to what
Peirce wants to think. It shows itself in two ways, in the Insistence of an Idea and in
the inductions that let loose our sentiment, configure our habit and culminate in belief.
This last aspect is a scientific, an all too human acquiescence in a proposition by which
our vital affairs might be guided. But the first is purely scientific, an intrinsic part of
10 Peirce also thinks that observation is itself abductive. Perceptual claims are ‘critical abductions, an
extreme case of abductive inference’ [Peirce, 1992, p. 5.181]. Cf. [Thagard, 1992, p. 170].
3.2. PEIRCE
43
abduction itself, and a seeming contradiction of Peirce’s settled and repeated views.
Let us see.
Abduction is two things for Peirce:
(a) surrender to an Idea, to a force majeure; and
(b) a method of testing its consequences.
In surrendering to an idea, say the existence of quarks, or the promise of quantum
electrodynamics, we do not come to a belief in quarks or to a belief that QED is true,
but rather to a belief that quarks are a good thing to suppose and a good bet for testing,
or that QED has a good explanatory record or at least the promise of such. In this,
abduction resembles practical reasoning (and observation, too). At the heart of this
resemblance is the concept of a ‘vital affair’. Here, too, there are two conceptions to
ponder.
Ô Õ*Ö
A vital affair is made so by its subject matter, i.e. its vitality is topic-intrinsic (e.g.,
ethics possibly).
[× ] A vital affair is made so by its methods of enquiry irrespective of its subject matter.
It seems to be Peirce’s view that the vitality of an affair is entirely a matter of the
manner in which we conduct its enquiry. There may be matters that especially conduce
to such methods, but there seems to be nothing intrinsic to the methods that preclude
them from application to any subject matter.
In the case of abduction, these methods are applied to the important business of
making conjectures for subsequent test. Essential to this is the conviction that a hypothesis H is a good enough conjecture to warrant our making it. That is a vital affair
for scientist and laymen alike. What this shows is that belief is indispensable for abduction, but it does not show an inconsistency in Peirce’s account. For here the object
of belief is not hypothesis H, but rather is some proposition of which H is the subject,
viz., the proposition that H is a good conjectural bet (or, in some variations, that H is
a good explanation of some set of target phenomena). Equally, a test of Ø is no good
if the experimenter is in any doubt — to say nothing of states of indifference — about
whether these things are consequences of Ø or about whether this other thing implies
the negation of a consequence of Ø , and so on.
We have been suggesting that abduction has been revealed to have an irreducibly
practical component. If that were true, it would matter for of a theory of abduction. And
it would matter greatly if there could be such a thing as a theory of practical reasoning.
We note here for subsequent consideration the possibility that while abduction may
have been shown to have a practical aspect, it has not been shown to be irreducible in
the requisite sense of being an aspect that any competent theory of abduction must take
note of and accommodate within its principled disclosures.
For the present we merely mark Peirce’s ambivalence about the possibility of a
science or a theory of practical reasoning. Certainly there is for him little prospect
of there being a logic of practical reasoning. Standing in the way is logic’s necessary
silence on how to represent a piece of practical reasoning in the artificial language in
which the logic transacts it proper business.
CHAPTER 3. BACKGROUND DEVELOPMENTS
44
‘In everyday business’, says Peirce, ‘reasoning is tolerably successful; but I am inclined to think that it is done as well without the aid of a theory as with it’ [Peirce, 1992,
p. 109]. This suggests not the impossibility of a theory so much as its unhelpfulness,
and it leaves open an option of a type that he has already acknowledged in connection
with deductive logic. Given that deductive logic doesn’t fix the representation relation
that takes natural language arguments into a logic’s artificial language, this is a task
that is left for the science of the natural language in question. By these lights, there
is no logic of abduction, but it might well have a scientific theory for all that. If so,
the science of abduction will itself be abductive — as all genuine science is.11 And
therewith is met an interesting problem. It is the problem of specifying what it is about
our abductive practice that calls out for abductive theoretical analysis and disposition.
If we agree that abduction is basically the making of conjectures in the light of their
consequences and the testing of those consequences in the light of new information,
then what are the abductive questions to put about these practices?
We are not here attempting to prime the reader for an assault upon the Problem of
Abduction, as we might say; but we might briefly pause over the point before moving
on.12 If we ask by what means might abduction be justified, there is little chance that
we will like the suggestion that its legitimacy is the conclusion of a deductive proof or
the conclusion of a forceful abductive argument. The first doesn’t exist and the second
commits the fallacy of petitio principii. Of course,
[a]fter a while, as Science [i.e. abduction] progresses, it comes upon more
solid ground. It is now entitled to reflect, this ground has held a long time
without showing signs of yielding. I may hope that it will continue to hold
for a great while longer Ù Ù Ù For a large sample has now been drawn from
the entire collection of occasions in which the theory comes into comparison with fact, and an overwhelming proportion, in fact all the cases that
have presented themselves, have been found to bear out the theory. And so
Ù Ù Ù I can safely presume that so it will be with the great bulk of the cases in
which I shall go upon the theory, especially as they will closely resemble
those which have been well tried [Peirce, 1992, p. 177].
This is induction, and Peirce, like Popper, says that induction has no place in science.
If, on the other hand, the justification is not scientific, it is practical. This is not sufficient ground to disentitle induction, but it won’t work as a justification of abduction.
The reason is that induction itself is abductively justified (cf. [Harman, 1965, pp. 90–
91]; see also [Woods, 2002b, Ch. 8]). For what justifies our inductive practices if not
that if nature were uniform our practices would be successful, together with the observation that our practices have been successful up to now? This leaves the justification
of abduction with a circularity problem and then some. (But see chapter 11.)
Pages ago, we said that, by Peirce’s lights, abduction was inherently practical in
certain of its aspects. The abducer is always driven by a belief that ‘this proposition is
worth entertaining’, a belief held by the abducer independently of whether he or anyone else actually believes the proposition in question. Given that the abducer’s belief
11 For
example, logical rules are abduced in [Gabbay, 1994; Gabbay, 2001].
there is a problem of abduction; and we shall deal with it in chapter 9, where we discuss
limitations on abduction.
12 However,
3.2. PEIRCE
45
is always a function of the Insistence of an Idea, abduction is always practical and, as
such, ascientific. If this is right, the same will hold for any theory of abduction itself
which aspires to be abductive. It may seem to follow that there can be no theory of
abduction, but we may think that this goes too far. The fallacy of composition looms.
It is, here, the fallacy of supposing that if a theory has an ascientific component the
theory as a whole is ascientific. That thinking so would be a fallacy in the present case
is suggested by the fact that the abducer’s sole belief (hence his sole practical contribution) is his belief about what is a plausible candidate for testing. Of course, it may turn
out that the candidate tests negatively, but this wouldn’t show that it wasn’t worth testing. And so we might find ourselves ready to concede that such beliefs are ascientific
in the degree that they are untestable. But ascientific or not, such beliefs don’t discredit
the scientific enterprise. We will be led by such beliefs to no scientific theories of a
sort not already admissible under the testing provisions of abduction generally. If these
let in bad theories, it won’t have been for the ascienticity of the beliefs that set them
up as conjectures in the first place. So it cannot be supposed that abduction entire is
ascientific just on account of the ascientificity of that aspect.
As we have seen, Peirce takes abduction as having the following general form:
The surprising fact Ú is observed,
But if Û were true, Ú would be a matter of course;
Hence there is reason to suspect that Û is true [Peirce, 1931–1958, p.
5.189].
In all of Peirce’s massive writings there is nothing that qualifies as a comprehensive
working-up of this basic insight. Even so, Peirce has occasional things to say that are
as important as they fragmentary (and, in places, contentious).
It is perhaps surprising that Peirce should make the factor of surprise a condition
on abduction. We ourselves suppose Peirce to have been excessively restrictive in
imposing this condition; but it cannot be denied that often what an abductive agent
sets out to deal with he does so precisely because it does take him by surprise, that the
surprise is the occasion of an abductive problem.
A surprise is an event or a consequence which does unexpected violence to a preexistent belief. A surprise is doxastically startling. ‘A man’, writes Peirce, ‘cannot
startle himself, by jumping up with an exclamation of Boo!’
What makes a surprising event or state of affairs surprising is that it instantiates a
law or a lawlike connection of which the enquirer has been unaware. In conjecturing
that Û , the abductive agent tends to suppose that there is a lawlike connection between
it and Ú . The conjecturing of Ú furnishes the antecedent of the conditional ‘If Û then
Ú ’, and Û itself imbibes something of the conditional’s hypothetical-causal character.
Peirce also sees the postulation of Û as ‘originary’ or creative. Abduction ‘is Originary in respect to being the only kind of argument which starts a new idea’ [Peirce,
1931–1958, p. 5.171]. Conjectural creativity comes in degrees, in ascending order as
follows:
1. Showing for the first time that some element, however vaguely characterized, is
an element that must be recognized as distinct from others.
2. to show that this or that element is not needed.
CHAPTER 3. BACKGROUND DEVELOPMENTS
46
3. giving distinctness — workable, pragmatic, distinctness — to concepts already
recognized
4. constructing a system which brings truth to light
5. illuminative and original criticism of the works of others [Peirce, 1931–1958 ].
It is possible to discern a less fine-grained taxonomy in which novelty comes in just
two degrees, what J.R. Anderson calls conceptual re-arrangement and concept creation.
It is interesting that this distinction is virtually identical to one drawn by Kant in his
pre-critical writings and persisted with until his last published work [Kant, 1974]. Seen
Kant’s way, the distinction is one between analysis and synthesis. Analysis, says Kant,
is the business of making our concepts clear, whereas the business of synthesis is that
of making clear concepts. Here is Peirce on the former notion: ‘It is true that the
different elements of the hypothesis were in our minds before; but it is the idea of
putting together what we had never before dreamed of putting together which flashes
the new suggestion before our contemplation’ [Peirce, 1931–1958, p. 5.181]. The
second kind of novelty Anderson characterizes as the formation of a new concept. It is,
as Kruijff says, ‘the creation of an idea that had never been in the mind of the reasoner
before’ [1998, p. 15], as witness Planck’s innovation of the quantum.
It is arguably fundamental to the logic of discovery — if such a thing there is —
that abductive agents have the creative capacity to hit upon the correct, or nearly so,
hypothesis out of very large sets of possibilities. Peirce’s approach to this, central issue
is itself abductive.
It is a primary hypothesis underlying all abduction that the human mind is
akin to the truth in the sense that in a finite number of guesses it will light
upon the correct hypothesis [Peirce, 1931–1958, p. 7.220].
The guessing instinct Peirce conjectures is that which explains why beings like us
are able with striking frequency to venture hypotheses that are at least approximately
correct, while in effect discarding an almost infinite number of unsuitable candidates
[Peirce, 1931–1958, p. 5.172].13
The heart and soul of abductive reasoning appears to involve the reduction of large
sets of possibilities to small subsets of candidates for hypothetical endorsement (see
chapter 2 above). In selection of a hypothesis, the abductive agent must solve what
we are calling the Cut Down Problem, although, as we said at the end of the previous
chapter,
not be (cannot be) be done by reconstructing the filtration-structure
Ü Ý*Þ ßÞ à„Þ this
á âã need
ä . According
to Peirce, the Cut Down Problem is solved by largely economic considerations.
Proposals for hypotheses inundate us in an overwhelming flood while the
process of verification to which each one must be subject before it can
count as at all an item even of likely knowledge, is so very costly in time,
energy, and money, that Economy here, would override every other consideration even if there were any other serious considerations. In fact there
are no others [Peirce, 1931–1958, p. 5.602 (emphasis added)].
ååå
13 cf. Kant “
the schematism of our understanding
soul” [Kant, 1933, pp. A141–B181].
å å å is an art concealed in the depths of the human
3.2. PEIRCE
47
Notwithstanding that an inductive agent’s conjectural economies are largely a matter
of instinct, Peirce supposes that there are three ‘leading principles’ for the selection
of hypotheses [Peirce, 1931–1958, p. 7.220]. One is that the hypothesis should be
generated with a view to its explanatory potential for some surprising fact or state
of affairs [1931–1958, p. 7.220]. Another is that the hypothesis should be testable
[Peirce, 1931–1958, p. 5.599]. The third such principle is that the abducer’s choice of
hypothesis be made economically [Peirce, 1931–1958, p. 5.197], since,
If he examines all the foolish theories he might imagine, he never will
light upon the true one [Peirce, 1931–1958, p. 2.776].
æææ
This is important. It explains why we don’t specify ç by search è for the relevant
possibilities why we don’t specify é by searching ç for plausibilities, and so on. Such
searches are both uneconomic and ineffective.
The economics of abduction is driven in turn by three common factors: the cost of
testing [Peirce, 1931–1958, p. 1.120], the intrinsic appeal of the hypothesis, e.g. its
simplicity, [1931–1958, p. 5.60 and 6.532], where simplicity seems to be a matter of
naturalness [1931–1958, p. 2.740]; and consequences that a hypothesis might have for
future research, especially if the hypothesis proposed were to break down [1931–1958,
p. 7.220].
There remain two further matters to mention. One is the caution with which Peirce
proposes hypotheses should be selected. Caution is largely a matter of cardinality. In
general, it is better to venture fewer hypotheses than more. ‘The secret of the business
lies in the caution which breaks a hypothesis up into its smallest logical components
and only risks one at a time’ [1931–1958, p. 7.220]. On the other hand, Peirce favours
the use of broad hypotheses over narrow, since it is the business of an abductive explanation to ‘embrace the manifold of observed facts in one statement, and other things
being equal that the theory best fulfills its function which brings the most facts under a
simple formula’ ([Peirce, 1931–1958, p. 7.410]; see also [1931–1958, p. 5.598–599 ]).
In other words, the abducer is bidden to strive for the fewest conjectures that will cover
the largest set of data. (See also [Peng and Reggia, 1990] and [Thagard, 1993]. As
Thagard has it, preferred hypotheses explain many facts without co-hypotheses [1993,
p. 56].)
It is easy to interpret Peirce’s leading principles and the other criteria of economy
as instructions for the free use of the abducer, as rules which the abductive agent may
follow or break at will. It is difficult however, to reconcile such a view with Peirce’s
repeated insistence that hypothesis selection is largely instinctual. We shall return to
this question. It is an issue which bears in a central way on the question whether a logic
of discovery is possible.
In sum, according to Peirce abduction problems are set off by surprising events
ê . The
abducer’s task is to find an hypothesis that generates an explanation of ê , i.e.,
the fact that for some state of affairs ë and some available set ì of wff, ì does not
explain ë . Such hypothesis are somehow “selected” from indefinitely large sets of
possibilities. In some cases, the abductive process is “originary”; it produces wholly
novel conjectures. The Cut Down Problem is handled instinctually, by the abducer’s
innate flair for thinking of, for picking and being moved by (force majeure) the right
hypothesis. It is all a matter of guessing, but the guesses have a remarkable track
CHAPTER 3. BACKGROUND DEVELOPMENTS
48
record for fruitfulness, if not strict accuracy. Even so, at the tail-end of abduction, no
hypothesis ever scientifically justifies discharge, or the displacement of hypotheticality
by categorical utterance. The best that a hypothesis can aspire to is quasi-discharge
(or anachronistically) Popperian discharge. Notwithstanding that hypothese are forced
upon us by an innate capacity for making the appropriate guesses, there are conditions
which favour the exercise of the guessing instinct, each having to do with hypotheses
under consideration, so to speak. The innate guesser will favour hypotheses that have
greater rather than less explanatory yield, that are testable, low-cost, few in number,
and broad in their reach. Explanatory yield, in turn, turns on factors of simplicity and
relevance for further research. Quite a lot to be encompassed in the innate capacity for
guessing right!
3.3
Explanationism
There is a large class of abduction problems that call for conjectures that would explain
some given fact, phenomenon or feature. An account of the abductive structure of such
problems and of the solutions that they admit require that the concept of explanation
have or load-bearing role in such accounts. This in turn requires that the idea of explanation be well understood. Such is a job for the conceptual analysis component of a
logic of abduction.
We have already noted the tension between philosophical and AI approaches to
what might be called the very idea of abduction. The dominant philosophical idea,
both in epistemology and studies in the foundations of science, is that abduction is inherently explanationist. What this comes down to in particular is the proposition that
within the logic of discharge, the degree to which an í is legitimately dischargeable
is strictly a matter of its explanatory contribution with respect to some target phenomenon or state of affairs. In AI approaches, including logic programing and logics
influenced by computational considerations more generally, the dominant (though not
by any means exclusive) idea is not explanatory but proof-theoretic. On this view, an
í is legitimately dischargeable to the extent to which it makes it possible to prove
from a database a formula not provable from it as it is currently structured. This makes
it natural to think of AI-abduction in terms of belief-revision theory, of which beliefrevision according to explanatory force is only a part. In the preceding chapter, we
pledged ourselves to the idea that it is preferable to have a logic of abduction that absorbs both conceptions, an approach that derives some encouragement from the fact
that the AI-conception already subsumes the philosophical conception.
We recur to this matter here, not to re-argue the issue but to point out that the
proposed ecumenism receives additional support from conceptions of explanation that
smudge the intuitive distinction between proof-theoretic and explanationist conceptions of abduction. A case in point is the Deductive-Nomological (D-N) theory of
explanation, to which we turn in the following section. As we will see, it is possible to have D-N explanations of a phenomenon without having an iota of explanatory
command of it in the intuitive sense. But since D-N explanations always have a prooftheoretic and explanatory component, it may be said that any supposed rockbound
rivalry between proof-theoretic and explanatory conceptions is misbegotten. For here
3.3. EXPLANATIONISM
49
is one form of what is called explanationism which delivers the goods for a prooftheoretic conception of abduction as much as for an intuitively explanationist one.
In its broadest sense, explanationism (a term coined by James Cornman), holds that
what justifies nondeductive inference is the contribution it makes, or would make, to
the explanatory coherence of the agent’s total field of beliefs. Aristotle, Copernicus,
Whewell [1840 and 1967], Peirce, Dewey, Quine and Sellars, are explanationists of
one stripe or another; but Harman was the first to formulate it as ‘the inference to the
best explanation’, and to argue for it as an indispensable tool of ampliative reasoning
[1965]. More recent defences of the principle can be found in Thagard [1992; 1993],
Lycan [1988], Josephson and Josephson [1994] and Shrager and Langley [1990]. In
contrast to explanationist approaches are probabalisitic accounts. A part from the inventors of probability theory (Pascal, Bernouilli, and others), a probabilistic approach
to induction is evident in the work of Laplace and Jevons. It is also at work epistemologically in Keynes [1921], Carnap [1950], Jeffrey [1983], Levi [1980], Kyburg [1983]
and Kaplan [1996]. Lempert [1986] and Cohen [1977] bring a probabilistic perspective to legal reasoning, and concerning how juries reach decisions, Allen [1994] and
Pennington and Hastie [1986] combine explanationism and probabalism. (This issue
is also discussed below in chapter 5 and in [Thagard, 2000].) In the philosophy of science, there is a continuing rivalry between probabilist theories of scientific reasoning
(Achinstein [1991], Hesse [1974], Horwich [1982], Howson and Urbach [1989], Maher [1993]) and explanationist accounts (Eliasmith and Thagard [1997], Lipton [1991],
Thagard [1988; 1992].)
Inference to the best explanation is perhaps the dominant current expression of
explanationism [Harman, 1965]. On this view an inference to the best explanation is
one that concludes that something is so on the grounds that it best explains something
else the reasoner also takes to be so.
Peirce holds that ‘the new hypothesis that abduction yields should explain the available data’ [Peirce, 1931–1958, MS 315:15–16 ]. But there is also reason to believe that
Peirce may not think that inference to the best explanation is intrinsic to abduction. Inference to the best explanation is an inference to something which explains something
already known to the enquirer, or as Peirce says, to ‘the available data’. But it is also
Peirce’s view [Peirce, 1931–1958, p. 6.525], that one hypothesis could be preferable
to another because it is not based on prior knowledge. What Peirce appears to mean
at [1931–1958, MS315:15-16 ] is that where data are available and the abduction is run
in relation to those data, then the new hypothesis should explain them. (See Hintikka
[1999, p. 96].)
Irrespective of what Peirce may have meant, it does seem apparent that bestexplanation abductions are a proper subclass of abductions. As Hintikka notes,
Even so, the discovery of the general theory of relativity was not an inference to the best explanation of the perihelion movement of the planet
Mercury and of the curvature of light rays in a gravitational field during a
solar eclipse, even though they were the first two new phenomena explainable by the general theory [Hintikka, 1999, p. 95].
The special theory also provides an interesting kind of exception to best-explanation
models. The goal of the special theory of relativity was to integrate Maxwell’s electro-
CHAPTER 3. BACKGROUND DEVELOPMENTS
50
magnetic theory with Newton’s mechanics. In terms of the abductive schema sketched
in section 2.5, î;ï is Newton’s theory, îð is Maxwell’s and î;ñ is the special theory.
î;ñ is such that every law of î;ï and every law of îð are covered by some law or laws
derivable in îñ and no such derivation is valid in î;ï or in îð or in their extensional
union òZó îï*ô îðõ . This gives an abductive trigger ö . The target ÷ is to specify a
set of laws underivable in îï and in î;ð but which cover the laws derivable in îð .
The targe was hit with the construction of îñ . But it is stretching things unreasonably
to suppose that in achieving this objective, Einstein was explaining either Maxwell’s
theory or Newton’s. (See also [Hintikka, 1999, p. 96].)
We note in passing that this example justifies our decision to abandon the condition
of the abductive schema of early in the previous chapter, which provides that abduction
problems always be solved by update, contraction or revision of î .
Lycan distinguishes three different grades of inference to the best explanation. He
dubs them ‘weak’, ‘sturdy’ and ‘ferocious’. Here, too, let î be a theory, H an hypothesis such that H î and ø a trigger. Unlike its role in Newtonian explanations ø here is
not merely the fact that îŠù9úû#ü ü ü ô ú , but rather there is some phenomenon or set
ï
of phenomena ý which is derivable (or fails to be derivable) in î , but which need not
be restricted to observational data. Then
Weak IBE: The statement that þ , if true, would explain ý
vide inferential justification for concluding that þ .
and could pro-
On the other hand,
Strong IBE: The sturdy principle of inference to the best explanation satisfies Weak IBE, and is itself a justificatorily independent principle. That is,
it need not be derived from, or justified in terms of, more basic principles
of ampliative reasoning such as probability theory.
Finally,
Ferocious IBE: The ferocious principle of inference to the best explanation
satisfies Weak IBE, and is the sole justified method for ampliative reasoning.
Van Fraassen demurs from Weak IBE (see below, 2.10); in this he is joined by Nancy
Cartwright [1983] and Ian Hacking [1983]. Cornman [1980] and Keith Lehrer [1974],
support Weak IBE, but part company with Strong IBE. Concerning Ferocious IBE,
Peirce appears to have been, apart from the young Harman, the sole supporter of note;
although, as we have seen, he is not always constant in his ferocious inclinations. Harman himself defended the ferocious version in [1965], but has moderated views in his
later writings (e.g., [1986]).
3.3.1
Enrichment
In these pages our intent is dominantly ecumenical. We seek a theory of abduction that
preserves what is good in alternative accounts. In pursuing this account of ecumenical
abduction, we filter our speculations through a more comprehensive practical logic of
3.3. EXPLANATIONISM
51
cognitive systems, or PLCS. In constructing PLCS and the account of abduction that
it implicitly harbours, we turn the principle of cooperation to our own advantage. We
build on achievements already to hand in logical theory and in the disciplines cognate
to a PLCS, notably cognitive science and philosophy of science. We believe in paying
for our appropriation. This is done under the guidance of what, in Agenda Relevance,
we called the Principle of Enrichment [Gabbay and Woods, 2002a]. In its most informal and intuitive sense, Enrichment bids us reciprocate our borrowings in the following
way. Just as our borrowings may guide us to insights into abduction which otherwise
would have been impossible or difficult to arrive at, so too should the theory of abduction facilitate developments in those cognate disciplines and subdisciplines which
otherwise would have been impossible and difficult to get clear about. A case in point
is relativity theory. Relativity owes a substantial debt to Newton’s and Maxwell’s theories, but it sanctions derivations underivable in either borrowed theory and provides
correction for them. The Enrichment Principle is closely related to something we call
the Can Do Principle. Informally put, the Can Do Principle sanctions the following
arrangement. If a theorist is investigating an issue in a domain of enquiry ÿ , then it is
obvious that he has work to do in ÿ . Sometimes, the work that must be done in ÿ can
be facilitated by doing further work in another discipline ÿ *. The principle asserts that
if the facilitation condition can be met then the ÿ -researcher is justified in pursuing,
in part, his ÿ -objectives in ÿ *. It is not uncommon however, that working in ÿ * is
easier than working in ÿ . This is a standing occasion to abuse the Can Do Principle,
as when the ÿ -theorist purports without showing that the easier work in ÿ * does indeed facilitate the work required in ÿ . According to bounded-rationality theorists in
the manner of Gigerenzer and Selten [2001], the use of the probability calculus, in the
analysis of human rationality is nearly always, in effect, unjustified subscription to the
Can Do Principle.
In the three sections to follow, we pursue an obvious task for any account of explanationist abduction. It is the task of saying, as best we can what exploration is.
As it happens there are rival and incompatible answers to this question. It would be a
welcome thing if the theory of abduction were to help to adjudicate this rivalry.
3.3.2 The Covering Law Model
In the covering law tradition of the 1940s onwards, something similar to Hume’s idea
of a causal law survives in the notion of a general scientific law as developed by Frank
Ramsey [1931]. On this view, laws are true generalizations that are a ‘consequence
of those propositions which we would take as axioms if we knew everything and organized it as simply as possible in a deductive system’ [Ramsey, 1931]. Mere correlations
on the other hand are true generalizations which would not pass the Ramsey-test.
Let us quickly note that a number of prominent philosophers of science take an
anti-Humean position on causality. In the 1970s D.M. Armstrong [1973], Fred Dretske
[1988] and Michael Tooley [1987] independently developed accounts of causality which
reinstate necessary connections and devise empirical tests for their correlation. Perhaps
the best-known of these is the so-called randomized experimental design test (see, e.g.,
Giere [1991, Ch 8]). Against this, critics of the non-Humean approach doubt that the
test of necessity meets Hume’s own requirement of observability.
CHAPTER 3. BACKGROUND DEVELOPMENTS
52
It is important that the randomized experimental design test is a statistical procedure. The best case for non-Humeans comes from the sister discipline of probability.
The difficulty for Humeans is that not even a perfect constancy of conjunction of s
and s to date is a necessary condition — to say nothing of a sufficient condition — for
the truth of a probabilistic law covering s and s. So it is clear that probabilistic laws
involve something stronger than proportionalities. (See, e.g., Papineau [1986]).
We owe the term ‘covering law model of explanation’ to William Dray [1957].
Also known as Hempel’s Model, the Hempel–Oppenheim Model, the Popper–Hempel
Model and the Deductive-Nomological model, it is a view anticipated by John Stuart
Mill [1974]. It involves two main factors — general laws and the subsumption of the
explanandum thereunder.
An individual fact is said to be explained by pointing out its cause, that
is, by stating the law or laws of causation, of which its production is an
instance [1974].
Furthermore,
a law or uniformity in nature is said to be explained when another law or
laws are pointed out, of which that law itself is but a case, and from which
it could be deduced.
In 1948 Carl Hempel and Paul Oppenheim expanded on this basic idea, schematizing
Deductive-Nomological (D-N) explanation as follows:
, , , (statements of initial conditions)
,
, ,
(general laws or lawlike sentences)
E
description of empirical data to be explained.
may be either a particular event, calling for a singular explanation, or a uniformity,
calling for an explanation of laws. Explanation is a deduction offered in answer to the
question, ‘Why ?’ The question
is answered (and the intended
is ren is deducible
andexplanation
dered)
when
it
is
shown
that
from
general
laws
initial
conditions
. If the deduction is valid it is assumed that is nomically forseeable from those
laws and those conditions. A good account of the logic of the covering law model is
Aliseda-LLera [1997, sec. 4.2].
In strictly logical terms, a D-N argument is indistinguishable from predictive and
postdictive demonstrations. Even so, explanation is contextually and pragmatically
distinguishable from the others by the assumption that E is already known to the investigator. A further condition on acceptable D-N explanations is that the explanans
must be true and showable as such by experiment or observation. The strength of this
requirement, together with the requirement of subsuming laws, suffices to distinguish
D-N explanations from reason-giving explanations. Reason-giving explanations
are
answers to the question, ‘What reasons are there for believing that ?’ If is ‘The
reals are indenumerable’, it is a reason to believe that Professor Mightily Important
said that Cantor proved this theorem. But short of the Professor’s universal reliability,
3.3. EXPLANATIONISM
53
this argument fails both the general laws and the deductive consequence conditions of
D-N explanation.
If the requirement that the explanans be true is dropped and yet the requirement of
experimental testability or observability is retained, then an argument in the form of a
D-N argument is said to afford a potential explanation of (cf. Aliseda-LLera [1997,
p. 136].
The D-N account of the covering law model admits of various extensions. The
explanans may admit statistical or probabalistic laws, and the consequence relation
on the explanans, explanandum pair could be inductive. This produces four different
types of covering law explanations: (1) deductive-general; (2) deductive-probabalistic;
(3) inductive-general; and (4) inductive-probabalistic [Aliseda-LLera, 1997, p. 121].
It is possible to soften the Why-question in the form ‘How possibly ?’ It is
also possible to produce corrective explanations in answer to both hard and soft Whyquestions. Here the answer indicates that sentence is not entirely true. Such explanations occur in approximate explanations in which, for example, Newton’s theory
implies corrected versions of Galileo’s and Kepler’s laws.
The Hempel-Oppenheim version of the covering law model is in several respects
too weak. It permits auto-explanations such as ‘Water is wet because water is wet.’
It suffers from monotonic latitude, permitting both, ‘Henry’s nose is bleeding because
Bud punched Harry or Copenhagen is in Denmark’, and ‘This powder pickles cucumbers because it is salt mined under the Detroit River.’ Also missing from the HempelOppenheim model is the essential directedness from causes to effects. This is shown
by Sylvain Bromburger, who pointed out [van Fraassen, 1980] that the length of a
flagpole’s shadow is explained by the length of the flagpole, not the other way around.
It is debatable whether D-N explanations are canonical for all science. But there
is no debate as to whether they are canonical for all explanations. We take up this
claim in the section to follow. We note in passing, however, partial conformity to the
Enrichment Principle. If abduction were dominantly explanationist and if the D-N
model were canonical for explanation, then for practical agents abduction would be a
rarity, not a commonplace, as would appear to be the case.
We have noted that answers to reason-questions, e.g., ‘What reasons are there for
believing that ?’, are not in general satisfactory answers to D-N Why-questions, such
as ‘What brought E about?’ The distinction between the two sorts of questions —
reason-questions and cause-questions — lies at the heart of a conception of explanation
that re-emerged in the first half of the past century among philosophers of history, and
which has flourished ever since. Its earlier proponents included the neo-Kantians and
thinkers such as Dilthey, who argued against the scientific model of explanation, insisting that in some areas of enquiry, history for example—and the Geisteswissenschaften
more generally—explanation is a matter of irreducible factors specific to the enquiry in
question. To give an historical explanation is not to deduce the event in question from
general laws but rather is a matter of specifying attendant conditions which enable us
to understand the event.
54
3.3.3
CHAPTER 3. BACKGROUND DEVELOPMENTS
The Rational Model
This conflict was revived in the mid-twentieth century by R.G. Collingwood [1946]
and William Dray [1957], who challenged the applicability of the covering law model
to the study of history. The anti-covering law position invests heavily in the irreducible
character of the distinction between reasons and causes. It holds that it is possible to
know the cause of any event without understanding the event caused by that cause.
They hold with equal force that a satisfactory explanation of an event in terms of the
reasons behind it is always a contribution to the event’s intelligibility. There is little
chance of the cause-reason distinction lining up so tightly with the distinction between
explanations that needn’t and explanations that must increase the intelligibility of their
explanans. However, these two distinctions show better promise of concurrence when
reasons are understood in a certain way. As Collingwood saw it, to explain an historical event is to find the reasons for its occurrence, and to do that is to re-enact the
circumstances and states of mind of the actual agents whose actions brought about the
event in question. The historian gives ‘the’ reasons for that event when he gives ‘their’
reasons for it. Dray calls this the rational model of explanation. In Collingwood’s version of it, a high premium is put on the historian’s empathy with his historical subjects.
There was not then a satisfactory epistemological theory of empathy. This occasioned
stiff criticism against Collingwood’s re-enactment theory. In Dray’s hands, the rational
model retains some Collingwoodian insights, while downplaying express reliance upon
empathy. According to Dray, historical explanations are typically better explanations
when they conform to the rational model. In that model, the historical investigator
seeks to reconstruct the reasons that the historical subject’s deliberate actions, in the
light of their known, desired or expected consequences, appear justified or defensible to the agent himself, rather than deducing those actions from true or inductively
well-supported uniformities.
It is one thing to say that an event is explainable independently of the uniformities
it falls under. It is another thing to deny the very existence of such uniformities. Part of
the rational modelist’s disenchantment with covering law explanantions in history, and
the Geisteswissenschaften in general, is the tendency to think that the domain of deliberative, purposive human action is not one governed by laws in the sense required by
the D-N model. Supporters of the rational model of historical explanation tend to doubt
that there are in this sense genuine laws of psychology, sociology, political science, economics and so on. In the forty-five years since Laws and Explanation in History first
appeared, skepticism about laws of human performance has abated considerably—a not
surprising development given the inroads made by materialism during those four and a
half decades. Another respect in which the Dravian account may strike the present-day
reader as over-circumspect has to do with the role of empathy in ascriptions to others.
In recent developments, empathy has emerged as a load-bearing factor in Quine’s later
reflections on translation [Quine, 1995], and even moreso has a central place in the
important work of Holyoak and Thagard [1995] on analogical reasoning.
It is evident that the rational model is rooted in the story — the most primitive
form of explanation-giving. Stories drive the cosmogonic tradition of making sense of
events big and small. Cosmogony is in the tradition of narrative, one of two forces that
shape Western cultural and intellectual sensibility. Cosmogony derives from Jerusalem,
3.3. EXPLANATIONISM
55
whereas the second tradition, logos, derives from Athens. These two inheritances competed with one another for standing as the better way of accounting for the world and its
significant events. To this day this competition survives not only in the rivalry between
the rational and covering law models, but also — and more broadly — in the contrast
between narratives which strike us as satisfying and plausible, and accounts which lay
a strong claim on truth or high probability, even at the cost of plausibility and — in extremis — intelligibility. Philosophers of history also plump for the narrative approach
with notable frequency. As they see it, understanding human conduct in general, and
past behaviour in particular, is a matter of weaving a coherent narrative about them.
On this view, history is a species of the genus story (see [Gallie, 1964]). It is a view
that attracts its share of objections. Perhaps the most serious criticism is that effective
and satisfying narratives about events do not in general coincide with accounts of those
same events that are arguably true. Plausibility is one thing, and truth is another. So
it is questionable whether history considered as story qualifies as a bona fide mode
of truth-seeking and truth-fixing enquiry. Philosophers of history have responded to
this criticism in two basic ways. On the one hand, Louis Mink [1969] has argued that
historical enquiry stands to the duties of truth-fixing enquiry as theory construction in
science stands to this same objective. Mink’s suggestion will not still the disquiet of
every critic, needless to say; but it performs the decidedly valuable task of emphasizing
the importance of distinguishing in a principled way the theories a scientist constructs
beyond the sway of directly observable evidence, and the narratives constructed by
the historian—also beyond the sway of directly observable evidence. Mink’s implied
question for the scientific theorist is ‘Are you not also telling stories?’
A second reaction to the criticism that historical narratives make a weak claim
on probativity, is that of writers such as Hayden White [1968], who emphasize the
close structural fit of historical narratives with fiction, as well as their susceptibility to
fictional taxonomies, such as tragedy, comedy, romance and satire. White concedes
that the fictive character of historical explanation calls into question its capacity as a
reliable representation of the reality of the past.
In recent decades, historians have altered their practice in ways that makes it more
difficult to reconcile the narrative model to what historians actually do. These days historians place less emphasis on the role of individuals, and stress instead the importance
of social, cultural and economic forces, as witness Fernard Braudel’s likening of the
doings of kings, generals and religious leaders to mere surface irritations on patterns
of events constituted by these more slowly moving impersonal historical forces. Even
so, the narrative model is far from defunct, and is not without its significant supporters
(e.g. [Ricoeur, 1977]).
We must ask again, how are we doing as regards Enrichment? It seems rather
clear that if explanationist abduction is a commonplace form of reasoning for practical
agents, then, since the rational model is consistent with this presumption, that alone is
(abductive) reason to take the rational model as canonical for practical reasoning. It is
not, of course, a proof that this is so; but it does give the nod to the rational rather than
covering law model of explanation in the domain of practical reasoning.
56
3.3.4
CHAPTER 3. BACKGROUND DEVELOPMENTS
Teleological Explanation
According to rational model theorists, the DN-model of explanation fails to fit the
structure of explanations in disciplines such as history. According to functional or
teleological theorists, the D-N model also fails to fit the explanatory structure of various
sciences, e.g., biology.
To the best of our knowledge it was John Canfield who first distinguished functional
explanation from those that fit the covering law model [1964]. As Canfield sees it,
functional explanations do not place a phenomenon or a feature in the ambit of a general
law; rather they specify what that phenomenon or feature does that is useful to the
organism that possesses it. (It has proved a seminal insight, giving rise to a whole
class of scientific explanations called explanation by specification, concerning which
see Kuipers [2001, Ch 4].) Canfield writes,
The latter [explanations of the covering law model] attempt to account for
something’s being present or having occurred, by subsuming it under a
general law, and by citing appropriate ‘antecedent conditions’. Teleological explanations in biology... do no such thing. They merely state what the
thing in question does that is useful to the organisms that have it [1964, p.
295].
As with those who distinguish the rational model from the covering law model, philosophers such as Canfield distinguish the functional model from the covering law model
on the ground that the covering law model doesn’t answer the type of question that a
functional explanation does answer. Here is Ruth Millikan on the same point:
[I]f you want to know why current species members have [trait] T the
answer is very simply, because T has the function F [1989, p. 174].
A second reason for separating them from D-N explanations is that
although teleological explanations have the same logical form as explanations in terms of efficient causes, it would be highly misleading to call
them “causal” [Pap, 1962, p. 361]. (See also [Nagel, 1977, pp. 300–301 ].)
This criticism has brought to the fore a further distinction, which some covering law
theorists find attractive, between causal and inferential explanations. This distinction
is especially well appreciated by Wesley Salmon [1978], notwithstanding that Salmon
[1984] is the classic statement of the causal theory of explanation. As Salmon suggests [1978, pp. 15–20], the inferential conception pivots on a concept of nomic expectability, according to which the explanadum is something that would have been
expected in virtue of the laws of nature and antecedent conditions. (Thus inferential
explanation is the converse of a Peircean abduction trigger, which is a surprising phenomenon, i.e., one that would not be expected even given the applicable laws of nature
and the presently known antecedent conditions.) Causal explanations, on the other
hand, are those that disclose the causal mechanisms that produce the phenomenon to
be explained.
The inferential-causal pair thus gives us a distinction that invades the question of
how functional explanations are supposed to differ from D-N explanations. On the
3.3. EXPLANATIONISM
57
causal approach to explanation, the point of difference would appear to be that with
functional explanations attributed functions denote effects rather than causes. Karen
Neander is good on this point.
The general prima facie problem with the teleological explanation is often
said to be that they are ‘forward-looking’. Teleological explanations explain the means by the ends..., and so the explanans refers to something
that is an effect of the explandum, something that is forward in time relative to the thing explained.... Indeed, because teleological explanations
seem to refer to effects rather than prior causes, it looks at first sight as
though backward causation is invoked.... The prima facie problem gets
worse, if that is possible, because many... functional effects are never realized [1991, pp. 455–456 ].
Robert Cummins offers an ingenious solution of this problem [1975, pp. 745 ff.].
He proposes that functional explanations do not, and are not intended to, explain the
presence of the phenomenon or feature to which the function is attributed. Rather what
functional explanation explains is a capacity of a physical system of which the phenomenon or feature to which a function is ascribed is a part. Thus says Cummins,
the covering law model applies to transition explanations but not to property explanations, whereas functional explanations are a particular kind of property explanation.
They are explanations in which the property-explanadum is a complex capacity and
the explanans is the specification of less complex capacities in terms of which the explanadum can be analyzed.
Although Cummins restricts causal explanations to those that explain transitionexplananda, i.e., where what needs to be explained is how a physical system changed
from a given state to a successor state, Salmon uses the term more comprehensively,
including in its extension capacity-explanations. So terminological care needs to be
taken, as witness Larry Wright’s approach in which functional explanations are those
that attribute functions, but do so compatibly with the causal account [Wright, 1973;
Wright, 1976].
Functional explanations subdivide further into three categories: etiological, survivalist and causal.
Etiological approaches (Millikan [1984, pp. 17–49]; Mitchell [1989]; Brandon
[1990, pp. 184–189 ]; and Neander [1991]) propose that function ascriptions specify
effects for which a given feature has been antecedently selected. In a well-known
example, hearts have the function of distributing blood if and only if doing so caused
them to do well in past natural selection. As its name suggests, the etiological view of
functional explanation is a variant of the causal conception.
On the survivalist14 conception (Canfield [1964], Wimsatt [1972], Ruse [1973],
Bigelow and Pargetter [1987], and Horan [1989]), the function of the blood is to propagate the blood because doing so is how hearts facilitate survival and reproduction of
organisms that have hearts.
On the causal account, to say that the heart has the function of propagating the blood
14 This
is our contraction of Wouters’ “survival value approach” [1999, p. 9].
58
CHAPTER 3. BACKGROUND DEVELOPMENTS
is to say that doing so accounts for the organism’s ability to circulate the blood (Cummins [1975], Nagel [1977], Neander [1991] and Amundson and Lauder [1994]).15
The inferential and causal conceptions do not exhaust the kinds of functional explanation. A third approach is the design explanation conception of functional explanation. In it, the role of explanation is not to make a phenomenon or feature nomically
expectable; neither is it to lay bare mechanisms that produce the states, transitions or
properties that the explainer wants an account of. On the design explanation approach,
it would be an explanation of the hollowness of the heart that this is the way the heart
has to be designed if it is to be a blood-pumper. Of the three conceptions, design explanations are comparatively little-discussed in the relevant literature. So we shall tarry
with it a bit (and in so doing we follow [Wouters, 1999, pp. 14–15 and Ch 8]).
Design explanations connect the way in which an organism is constructed, how its
parts operate and the environmental conditions in which it exists to what is required or
useful for survival and reproduction. Design explanations come in two types.
1. Those that undertake to explain why it is useful to an organism to behave in a certain way; e.g., why it is useful for vertebrates to transport oxygen. Explanation
of this stripe are two-phased. In phase 1, the explainer specifies a need which is
satisfied causally by the behaviour in question. In phase 2, it is explained how
this need hooks up with other features of the organism and its environment.
2. The second kind of design explanation shows why it is useful that a given feature
or piece of behaviour has a certain character. Here, too, the explanation proceeds
in two phases. In phase 1 the feature or behaviour in question is assigned a
causal role. In phase 2, it is explained why this causal role, given the other
relevant conditions, is better formed in the way that it is performed rather than
some other way.
Wouters argues that design explanations are not causal explanations, even though the
causal idiom is used in their description. They are not causal because they do not
explain how or why a certain state of affairs is brought about.
Reflecting on the issues of this and the prior two sections, it is evident that explanation — even in explanation in science — is beset by a hefty pluralism. There are
two responses to pluralisms of this sort. One is to take differences at hand as having
rivalrous import, and then to seek for reasons to pick a (near-to-)unique winner. A
second response is to employ the strategy of ambiguation, in which each difference
marks a different (and legitimate) sense of explanation. Seen the first way, there is a
single most tenable conception of explanation, and the others at best are less tenable
than it, if not outright wrong. Seen the second way, the different conceptions are best,
if at all, in particular contexts governed by particular objectives, and with particular
kinds of access to the requisite cognitive resources. Seen this way, it is relatively easy
to see why abduction is so commonplace a practice among practical agents. Even if
explanationist abduction is not all there is to abduction, it is a considerable part of it.
Its ubiquity is explained by the availability to abducers of a multiplicity of conceptions
of explanation, each contextually and resource-sensitive. From the point of view of a
15 Canfield’s theory is non-explanationist. Ruse and Horan are inferentialists and Winmsatt, too, in a
somewhat different way. Bigelow and Pargetter hold to the causal account.
3.4. CHARACTERISTICNESS
59
theory of abduction that recognizes its commonplaceness, there is a lesson to propose
to theorists of explanation. It is to analyze explanation in the spirit of reconciliation
fostered by the ambiguation strategy.
3.4
Characteristicness
For all their considerable differences, covering law, rational, functional-causal and
functional-acausal (or design) models of explanation answer to a common idea. In
each case, one achieves an explanation of a phenomenon when one sees that it would
happen in the circumstances at hand. Suppose that Sarah has lost her temper. Even
if this is something that (generically) Sarah would never do, it is explicable that she
would do it here because the object of her wrath has vulgarly insulted Sarah’s adored
mother. There is a sense of characterictness in which Sarah’s doing this utterly uncommon thing is not out of character for her in those circumstances.
Even the covering law model could be brought into the ambit of this point if we
allowed its embedded notion of law the status of generic propositions rather than that
of the fragility of universally quantified conditional statements. This alone would go
along way toward explaining the entrenched practice of scientists not to abandon a law
in the face of a single episode of observational disconformity. It would also capture
what it is about how things function that explains how they behave or the further states
that come to be in. For it is characteristic of things that function in those ways that they
behave in those ways. If we are right to see the explicable tied thus to the generic and
the characteristic, then we have reason to think that, given the centrality of abduction of
the explicable, a like centrality could be claimed for the generic and the characteristic.
We pick up this suggestion in chapters 6 and 7.
The suggestion that the covering law model could be made more user-friendly by
the simple expedient of genericizing its laws will not strike everyone as a good idea.
There are two particularly telling strikes against it. One is that for certain domains of
enquiry universality is literally exhaustive, and thus is mathematically expressible by
classical quantification. If its laws were genericized, a theory of this sort would understate what it has the wherewithal to demonstrate. A second reservation turns on the
emphasis the theorist is prepared to give the falsifiability condition. If a theory’s generalizations are formulated as universally quantified conditionals, then not only does he
(and we) know precisely in what falsification consists, but falsification is achieved with
absolutely minimal contrary turbulence — a single true negative instance. Neither of
these clarities attaches to the falsification of generic statements.
We do not dispute these points. We have already said repeatedly that the bar of
cognitive performance is set in conformity with the type of cognitive agency invalued.
Theories typically are in the ambit of theoretical (i.e., institutional agency), where exhaustive generality is not only an attainable but also an affordable goal. In disciplines
(such as certain of the social and life sciences) in which literal exhaustiveness is not
always attainable, and in contexts in which an individual seeks to solve a pressing particular cognitive problem, generic universality is a blessing that it would be foolish to
do without. Our point of lines above was simply that in such cases the covering law
model can often be adapted in such a way that here, too, explanatoriness pivots on
CHAPTER 3. BACKGROUND DEVELOPMENTS
60
considerations of characteristicness.
3.5
Hanson
In Hanson [1958], an attempt was made to establish a logic of discovery (in our terms, a
combined logic of generation and engagement), although later on he came to have second thoughts. In 1958, the ‘received view’ of scientific theories was that they are physically interpreted deductive systems, the data of which are deducible consequences.
Hanson points out that when a scientist is trying to take the theoretical measure of a
set of data or a body of evidence, he does not search for a physical interpretation of
a deductive apparatus within which those data are deductive consequences. Rather he
seeks for theories which ‘provide patterns within which the data appear intelligible’
[1958, p. 71]. Finding such patterns enables the theorist to explain the phenomena
which they subsume.
Fundamental to Hanson’s project is pattern recognition. Theories that are constructed in Hanson’s every are not discovered by inductive generalizations from the
data in question, but rather, as Hanson says, retroductively, by inferring plausible hypotheses from ‘conceptually organized data’ [Suppe, 1977, p. 152]. As it happens,
Hanson’s whole retroductive approach pivots on the view that data are not pure but
rather ‘conceptually organized’ or ‘theory-laden’; so we should pause awhile over this
issue.
Hanson imagines Tycho Brahe and Kepler up at dawn, awaiting a certain event concerning the Sun. Tycho observes the Sun as rising, which has been the view of common
sense to the present day. Kepler sees it as progressively revealed by the downward orbital slippage of the Earth on which he stands. What did they observe? A sunrise, for
Tycho, and an Earth-slip for Kepler? Or was there some core sensory experience which
they both shared? Hanson’s reasoning is that since the latter alternative is false, the former must be true. We won’t here rehearse Hanson’s full case for the theory-ladenness
of observation. It suffices to give the gist of his argument. There is a clear sense in
which one can see an object without knowing what it is, without knowing what one
is seeing. Hanson is drawn to the view that this is not real seeing, that the concept of
seeing analytically involves the concept of seeing as. The present authors are of the
opinion that Hanson needn’t be right in this claim in order to be right about a claim that
matters considerably for his theory. The claim that is right is that bare observations—
observations such that the observer does not know what he is seeing—are of no use to
science, just as they are wholly inadequate guides to action in practical affairs generally.
We receive insufficient instruction from ‘Oh look, there’s a whatchamacallit’ or ‘Watch
out for the thing I’m presently looking at, whatever it is!’ The observations that matter
are those that carry appropriate descriptive signatures. For this to be true it needn’t
be the case that the Tycho-Kepler story is strictly true. There is plenty of taxnomic
slack with which to characterize a heavenly body as the Sun short of commitment to
its status as a planet, or otherwise. But it remains the case that useful observations
are observations of objects under descriptions, and that it is not uncommon that such
descriptions fail to be theoretically neutral. Whether they are or not will depend on
the context in which the observations in question have been made. It does not bear on
3.5. HANSON
61
the observation, ‘The Sun appeared at 5:10 A.M., whether the Sun is a planet or not, if
the context is the question whether it is presently winter in London. In other contexts,
the descriptions under which things are seen carry theoretical freight (granted that the
sunset-dipping Earth example is a trifle forced). 16
Hanson himself comes close to exposing this slighter view of the theory-ladenness
of observation when he emphasizes that terms are theory-laden, and must be so, if
they are to have any explanatory function. There may, he concedes, actually be pure
sense-datum languages where terms are entirely free of theoretical taint, but no such
language can serve the explanatory functions of science or of ordinary life [Hanson,
1958, pp. 59–61]. For our purposes in this book, the greater importance of the theoryladen account of observation is the credence it lends to the project of developing a
genuine logic of discovery. Here is Hanson on the point:
Physical theories provide patterns within which data appear intelligible.
They constitute a ‘conceptual Gestalt’. A theory is not pieced together
from observed phenomena; it is rather what makes it possible to observe
phenomena as being of a certain sort, and as related to other phenomena.
Theories put phenomena into systems. They are built up in ‘reverse’—
retroductively. A theory is a cluster of conclusions in search of a premise.
From the observed properties of phenomena the physicist reasons his way
toward a keystone idea from which the properties are explicable as a matter
of course [Hanson, 1958, p. 90].
What, then, will the preferred logic of discovery look like? (We here follow the treatment of Hanson [1961].)
Hanson acknowledges the distinction between what we have been calling justification abductions and discovery abductions. (See also [Peirce, 1931–1958, p. 1.188].)
But he does not expressly have our trichotomy of generation, engagement and discharge or retroduction. He distinguishes
(1) reasons for using and/or accepting a hypothesis discharging )
(i.e., engaging and/or
from
(2) reasons for thinking of in the first place (i.e., generating ).
The reasons subsumed by (1) are for Hanson reasons for thinking that is true (rather
than, for example, as with Newton and his action-at-a-distance theorem, reasons for
using , that there is no question of thinking it true). Reasons subsumed by (2) are
reasons for thinking that H is a plausible conjecture. (So, again, Hanson is not operating with our three sublogics.) Here, too, a further distinction is required, owing to the
ambiguity of ‘plausible’. In one sense, a plausible conjecture is one whose truth can
plausibly be supposed. In another sense, a conjecture H is plausible when it is a plausible candidate for consideration. However this distinction is drawn in fine, it pivots on
16 We note the widespread view among perception theorists that perception is always perceiving as, and
the related view that, since unconceptualized perception is impossible, perception is inherently epistemic.
(See, e.g., Hamlyn [1990, Ch. 4 and 5].)
CHAPTER 3. BACKGROUND DEVELOPMENTS
62
the essential difference between the plausibility of conjecturing something even when
there isn’t the slightest reason for judging it true, and conjecturing something precisely
because the idea that it is true is a plausible one. In the first sense, it is plausible to make
all other family members suspects in a crime of violence against a family member even
without a shred of evidence for the guilt of any of those individuals. In the second
sense, it might become plausible to concentrate on Uncle Harry as the miscreant, not
that there is any direct evidence of his guilt, but rather that, given the various facts of
the case, Harry’s involvement would not at all be out of character.
Hanson’s approach to discovery abductions is rather narrow. He wants to preserve
a logical distinction between reasons for thinking that an hypothesis, , is true and
reasons for thinking up in the first place. But reasons of this latter kind are, for
Hanson, always reasons in the form ‘ is the right kind of hypothesis for the problem
at hand, irrespective of the particular claims it may succeed in making.’ The present
authors join with Thagard in thinking that putting it this way is tantamount to giving
up on the idea of a bona finde logic of discovery [Thagard, 1993, p. 63]. Reasoning of
this kind is retroductive reasoning which Hanson schematizes as follows:
1. Some surprising, astonishing phenomena are encountered. 17
2. But would not be surprising were a hypothesis of ’s type to obtain. They would follow as a matter of course from something like and would
be explained by it.
3. Therefore there is good reason for elaborating a hypothesis of the type of
for proposing it as a possible hypothesis from whose assumption might be explained [Hanson, 1961, p. 33].
;
Hanson’s abductive schema hooks up with his theory-laden view of observation in a
straightforward way. For if it is true that observations involve a certain conceptual organization for the phenomena in question, that fact alone will preclude various possible
kinds of explanation and will encourage, or at least be open to, other kinds of possible
explanation. If an observer sees some data as having the conceptual organization that
makes Keplerian explanations the right kind to consider, then Tychoean explanations
will be of the wrong kind for serious consideration.
While our brief resumé doesn’t do full justice to Hanson’s abductive project, the
same can be said of Hanson’s own writings which, while more voluminous than the account sketched here, nevertheless present a highly programmatic research programme,
with a good deal more promised than actually delivered. In this we agree with Achinstein.
Hanson [ describes] one such pattern [i. e., actually employed in the
sciences], retroduction. This consists in arguing from the existence of
observed phenomena that are puzzling to a hypothesis, which if true would
explain these phenomena. I do not believe that Hanson describes this type
of inference in a way that is free of difficulties, but I do believe there
is an explanatory type of inference often used in science and that it can
17 Note
the Peircean cast.
3.5. HANSON
63
be correctly described. Moreover, contrary to what Hanson sometimes
suggests and to what has been stated by Gilbert Harman, I believe
that there are certain nonexplanatory, nondeductive patterns of inference
actually employed in the sciences [Achinstein, 1977b, p. 357 (emphasis
added)].
Before bringing the present section to an end, it would be well to consider an objection
which is leveled against the entire project of discovery abduction. Where, the critic
might ask, do these abductive hypotheses come from?
And is it not the case that whatever the precise answer to this question, it brings
psychological factors into the picture? No doubt, this is so. When Kepler was in process of thinking up his laws on the basis of Tycho’s data, causal factors were at work.
Perhaps Kepler would not have been able to make these discoveries had he not had
the formal education that he did in fact have or had he had different interests, and so
on. It is, even so, a grave error — indeed, a special case of the genetic fallacy — to
suppose that if how a theorist came to think up a law is subject to a causal explanation,
then it must follow that he could not, in so doing, have been involved in a reasoning
process. It remains an open — and good — question as to whether there are any systematic connections between the causal conditions under which reasoning is produced
and the logical conditions in virtue of which it counts as correct or reasonable. But
there appears to be little reason for thinking that the logic of hypothesis formation cannot proceed apace without exposing the etiology of the reasoning process that is in
question. (See also Achinstein [1977a, p. 364].)
Of course, thinking things up is often a matter of intuition (although saying it just
this way doesn’t offer much of an explanation). Still, no one doubts that having the
right intuition can be crucial for the theorist’s larger purpose. Consider, for example,
a theory in which two definitions are equivalent, but only one generalizes. There was
a point at which the physicist could have chosen either " # ! or %$ " &# ! ' . If
his intuitions were ‘reliable’, he would have opted for the second, since it is the second
that generalizes to relativity!
Hanson, the promissory-note abductive logician, is also an explanationist.
Deductive-nomological explanations are fine as far as they go; they may even be necessary conditions on explanations that do the full job. But D-N explanations do not as
such serve in the cause of ‘rationally comprehending the ‘go’ of things ’ [Hanson,
1958, p. 45].
What can be wrong with our seeking examples of scientific theory which
are capable both of explaining a( la Hempel and of providing understanding and illumination of the nature of the phenomena in question? Even
if distinguishable, the two are genuinely worthwhile objectives for scientific enquiry; they are wholly compatible. And, it may be noted, the second is unattainable without the first. So although Hempel’s [D-N] account
of scientific explanations may not be sufficient, it seems to be necessary
[Hanson, 1958, p. 45].18
18 There is a slip here. If a staisfactory Hansonian explanation implies a satisfactory D-N explanation, then
it cannot be true, as Hanson strenuously avers, that explanations are sometimes non-predictive.
CHAPTER 3. BACKGROUND DEVELOPMENTS
64
What is more—and note well:
Ontological insight, unstructured by quantitatively precise argument and
analysis is mere speculation at best, and navel-contemplatory twaddle at
worst [Hanson, 1958, p. 45].
Even in these later posthumous writings (Hanson was killed in a flying accident in
1967, at the too young age of 43), there is nothing in the way of ‘quantitatively precise
argument and analysis’ about what it is to make sense of something, to figure it out, or
make it comprehensible. The account may not be mere speculation, but it is decidedly
promissory.
We see, then, Hanson’s approach to abduction implicitly recognizes the threefold
distinction between the generation and engagement and (quasi-)discharge of hypotheses. As with Peirce, the trigger is an astonishing event or state of affairs that can’t be
explained with available resources. ) is the target, the sentence that repairs the astonishment. Hypotheses are generated when they are seen to be of a type that would
explain ) . Hanson does not explain how hypotheses of that type are recognized (so
strictly the generational sublogic is idle), but it is clear from context that he sees analogy at work here. Crudely, phenomena like the trigger are explained by hypotheses
of such-and-such type; therefore, an hypothesis of like type is a good bet to explain
) . This is all but what we call engagement (Hanson calls it elaboration). Once elaborated, it is subjected to a quasi-discharge; it is proposed “as a possible hypothesis
from whose assumption [) ] might be explained” [Hanson, 1961, p. 33]. Hanson retains Peirce’s idea that triggers must surprise or astonish, but he doesn’t accept Peirce’s
faith in the abducer’s innate capacity for guessing right. Peirce saw perception as the
limit of abduction. For Hanson, too, observations are essentially bound up with abduction, but differently. Observations always involve conceptual organization. Thus how
a phenomenon is perceived shapes what it is about it that may call out for explanation.
Central to this approach is the idea that abductive inference is inference from considerations of like-typeness. We read this as analogical inference, to which we shall return
in chapter 8.
3.6
Darden
Neither Hanson’s nor Lakatos’ (see below, section 4.6) favourable disposition to a logic
of discovery has spurred much follow-up work among philosophers of science. A
notable exception is Darden [1974]. Central to her approach is the role of analogy.
Darden proposes ‘the following general schema for a pattern of reasoning in hypothesisconstruction:
problems
posed by
fact
plausible
solution to
this problem
* * generalize
*******+
- * * * * * * * * * **
particularize
general form
of the problem
general form
of solution
to problem
* * analogize
* * * * * * * to* * +
- * * * * * * * **
construct
,
general forms of
similar problems
with solutions
general forms of
other KNOWN
solutions
3.7. FODOR
65
The pattern of reasoning is used for each of the facts in turn. The use of the same
explanatory factor in each one results in a hypothesis made up of a set of postulates all
of which involve the same explanatory factors’ [Darden, 1976, p. 142].
Darden’s approach to the logic of discovery emphasizes the use of hypotheses for
the purpose of achieving explanation-abduction, but there is no reason in principle
why, if the scheme works for explanation abductions, it can’t be generalized (in its
own spirit, so to speak) so as to apply to other forms of justification abductions, such
as those that turn on the predictive yield of the hypothesis. We meet here for the first
time a sketch of abduction in which analogy plays a central role. In chapter 11, we
develop an account of analogy which bears considerable abductive weight, in the spirit
of Darden’s scheme.
3.7
Fodor
In a widely discussed essay Fodor [1981] it is proposed that abduction is intrinsic to
conceptualization. The work in question is lengthy and highly nuanced, but its main
features can briefly be set out, beginning with the following question. “Everyone agrees
that human agents conceptualize their experience, but where do concepts come from?”
Fodor’s answer is that all concepts are innate or are computationally derivable from
those that are. Derived concepts are constructed; and constructing a concept is like
constructing a theory with which to explain certain facts. Such theories need to be
cognitively prior to the data that are to be explained in their terms. The same is true of
constructed concepts. They are put together out of existing resources for the purpose
of having some application to experience.
Some will not see it as plausible to think of highly technical and original concepts
as already ‘there”, so to speak (e.g., quark; cf. Thagard [1988, p. 71]). But Fodor
can plausibly plead that the critic has confused lexical unfamiliarity with conceptual
novelty.
It is arguable that Fodor’s is the most extreme form of nativism since Plato’s famous
example in the Meno, in which Socrates elicits from the slave-boy the Pythagorean
theorem. But it must be said that Plato is a nativist about knowledge, whereas Fodor’s
nativism extends to concepts (and, if having a concept of K is believing what it is to be
a K-thing, to such beliefs as well).
For our part, we grant that even Fodor’s nativism is counterintuitive, but we are not
prepared to infer from this its indefensibility. For our purposes here, it isn’t necessary
to settle the nativist-empiricist controversy about concept acquisition. It suffices to note
the sheer extent to which on Fodor’s account, the human agent is an abductive animal.
Fodor’s emphatic nativism brings to the fore another problem of celebrated difficulty. Given our notion of logic as the principled description of the moves of a logical
agent, it is not a matter to which we can be wholly indifferent. [More TBA]
As anyone familiar with Russell’s copious logical and philosophical writings will
know, this is a division of labour impossible to honour. Russell himself constantly
employed the softer standards of his epistemological orientation in service of the higher
standards of metaphysical objectivity. A locus classicus of this confusion is Russell’s
attempt in Principles of Mathematics [Russell, 1903] to originate a realist account of
66
CHAPTER 3. BACKGROUND DEVELOPMENTS
post-paradox set theory from nominal definitions. We shall not pursue this vexed issue
here except to say that it is not as vexed as it initially appears. (Interested readers could
consult [Woods, 2002b, ch. 1, 6, and 9].)
Chapter 4
Non-explanationist Accounts
Today I have made a discovery as important as that of Newton.
Max Planck, in conversation with his son, 1900.
In general, we look for a new law by the following process: First, we guess
it. Then we compute the consequences of the guess to see what would be
implied if this law were guessed right.
Richard Feynman, ‘Seeking New Laws of Nature’, [19–]
4.1
Brief Introductory Remarks
There is no hard and fast rivalry between explanationist and proof-theoretic accounts
of abduction. As we saw in the previous chapter, the D-N conception of explanation
is one on which the present distinction collapses. We are not minded to resist this
turn of events. Explanation is an ambiguous and context-sensitive concept. It might
be expected therefore that there exist resolutions of abduction problems in which the
target is delivered non-explanatively and yet this very fact explains or helps explain
some collateral feature of it. (We develop this point just below.) Accordingly we are
content to offer the explanationist-non-explanationist contrast for what it is, namely, as
a loose and contextually sensitive distinction.
4.2
Newton
There is an error of substantial durability as to what Newton meant by his celebrated
dismissal of hypotheses. The error is that when Newton asserted that he did not feign
hypotheses, he was rejecting the hypothetical method in physics and that he would have
no truck with inferences to the best explanation. Newton’s dismissal is not a general
indictment, but rather a response to a particular case. In Newton’s theory of gravitation,
the gravitational force acts instantaneously over arbitrary distances. Newton’s own
67
68
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
view was that this was a conceptual impossibility, and he hinged his acceptance of
it on the extraordinary accuracy of the relevant equations. ‘Hypotheses non fingo’
is a remark which Newton directed to the particular proposition that gravitation acts
instantaneously over arbitrary distances. By this he meant that he regarded this as an
inexplicable fact. He was at a loss to explain how this apparent absurdity could actually
be the case. (It is, of course, widely believed that Einstein’s field theory eliminates the
source of the inexplicability.)1
Although ‘hypotheses non fingo’ does not stand as general indictment of the hypothetical method, there is an air of general disapproval evident in Newton’s writings,
well-summarized by Duhem:
It was this . . . that Newton had in mind when in the ‘General Scholium’
which crowns his Principia, he rejected so vigorously as outside natural philosophy any hypothesis that induction did not extract from experiment; when he asserted that in a sound physics every proposition should
be drawn from phenomena and generalized by induction [Duhem, 1904–
1905, pp. 190–191 ].
Here is the relevant part of the General Scholium: 2
But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I feign no hypotheses; for whatever
is not deduced [sic] from phenomena is to be called an hypothesis; and
hypotheses, whether metaphysical or physical, whether of occult qualities
or mechanical, have no place in experimental philosophy. [Newton, 1713,
p. 546].
That the Principia should have been formatted axiomatically bears on our issue in
an interesting way. Newton inclined the generally current view of axiomatics in supposing that axioms afforded a kind of explanation of the theorems that are derivable
from them, that the demonstrative closure of a set of scientific axioms is suffused with
the explanatory force that originates in the axioms and is preserved under and transmitted by the theory’s deductive apparatus. For Newton, explanatory force is axiomatically
secure, a view held in contrast to abductivists in the manner of Laplace [1951], for example, for whom a significant part of a science’s explanatory force must come from
hypotheses that resist efforts at refutation.
1 Tom van Flandern denies this claim in van Flandern [1999]. He argues that there are pulsars whose
times the speed
observations entail that the speed of the gravitational influence is not less than
of light. This astonishing development, which exceeds the scope of the present book, is well-discussed in
Peacock [2001].
2 Newton was an inductivist in the spirit of Bacon. Bacon condemned speculation in the absence of data,
but he was also aware that a given set of data can be made to comport with more than one explanation.
Bacon rejected the method of hypothesis — which he called Anticipation of Nature — when it was rashly
or prematurely resorted to. Bacon also thought that ‘there are still laid up in the womb of nature many
secrets of excellent use, having no affinity or parallelism with anything that is now known, but lying entirely
out of the best of the imagination, which have not yet been found out.’ [Bacon, 1905, p. 292]. But this is
nowhere close to an outright condemnation of the scientific imagination. The debate between inductivists and
conjecturalists rages to this day. Consider, for example, the sometimes heated exchanges between biologists
(e.g., [Rose and Rose, 2000]) and evolutionary psychologists (e.g., [Miller, 2000]).
/021 3 4 5
4.2. NEWTON
69
The action-at-a-distance theorem proved a methodological embarrassment for Newton. If explanatory force is axiomatically secure and preserved by a theory’s deductive
apparatus, then it ought not be the case that anything should turn out to be inexplicable
in the deductive closure of the first principles of gravitation theory. Given the unintelligibility of the action-at-a-distance theorem, then by Newton’s own lights either axioms
are not explanatory as such, or the deductive apparatus is not explanation-preserving.
The hypothetical method bears on what the Port Royal logicians would call the
method of discovery.
Hence there are two kinds of methods, one for discovering the truth, which
is known as analysis, or the method of resolution, and which can also be
called the method of discovery. The other is for making truth understood
by others when it is found. This is known as synthesis, or the method of
composition and also can be called the method of instruction [Arnauld and
Nicole, 1996, p. 232].
Arnauld and Nicole, and the other Port Royal logicians, were of the view that analysis
always precedes synthesis. They pressed this point in the context of an unsympathetic
discussion of syllogistic logic, but it is clear that they see their remarks as having a more
general application. They assert that, in as much as the challenges of real life are much
more to discover what things are true or probable than to demonstrate them, it follows
that the rules of syllogistic, which are rules of demonstration, can have only a limited
utility, and that they will soon be forgotten by young student-logicians because of their
general inapplicability to the main problems of life. We see here a considerable debt to
Descartes. Not only do Arnauld and Nicole accept the Cartesian distinction between
the logic of discovery and the logic of demonstration, they are shrewd to notice (what
others don’t) that the logic of discovery resists — even if it does not outright preclude
— detailed articulation, which is a challenge for the would-be logician of hypothesisgeneration.
That is what may be said in a general way about analysis which consists
more in judgement and mental skills than in particular rules [Arnauld and
Nicole, 1996, p. 238].
The Port Royal logic appeared in 1662 and was widely noted, not only in France;
and it had a considerable impact upon Locke. Newton’s Principia was published in
1687, by which time the method of analysis was sympathetically received throughout
Europe, and its influence was discernible in various developments in the new sciences,
if not in the formal treatment of the Principia. Hypothesis formation is a substantial
part of the method of analysis; so perhaps it is somewhat surprising not to find it in
Newton. But as the Port Royal logicians insisted, the method of analysis, abduction
included, involves the exercise of the scientific imagination; hence is more a matter of
‘judgement and mental skills’ than of ‘particular rules’. So neither is it surprising to
find little in the methodology of the new sciences ensuing from Bacon in the way of a
theoretical development of the scientific imagination.
The concept of explanation embodies a distinction which is crucial for the logic of
abduction. For ease of exposition, we label this contrast as one between explanation
70
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
in Newton’s sense, or Newtonian explanation, and explanation in Harman’s sense, or
Harmanian
Concerning Newtonian explanations, let 6 be a theory and
798:7<; 6>= ?Aexplanation.
@ B be the datum that 6DCE?AF = G G G ?H , where C is a deducibility relation
of requisite character, and the ?@ are observational or otherwise discernible consequences. Let I be an hypothesis
such that IKJL6 . In other contexts, depending on
7
his interests and goals, could be a trigger for an agent M . However, it is not this kind
of abductive stimulus that concerns us here. We put it that
7
DefNewtExp: Given 6>= N<= and ?@ as specified, I explains the fact that
7 iff 6OCP?<F = G G G ?AH and 6OQSR IUT2V CL?<F = G G G = ?H .
The distinctive
7 feature of Newtonian explanations is that the fact that I permits the
derivation of ’s discernible consequences which 6 could not furnish in its absence, is
a fact that need confer no understanding on the ?A@ themselves
(nor indeed on I ), even
7
though it does confer some understanding on why obtains, namely that IPJU6 and
6DCP?AF = G G G = 7 ?AH and 6OQSR IUT2V C ?@ = G G G = ?AH . Another way of saying this is that I
explains why obtains, i.e., why the ?@ are deducible in 6 , even though it need
7 not
explain any of the ?A@ themselves. In particular, a Newtonian explanation of need
not enhance our understanding of these observational phenomena. And let us say again
that I can figure with perfect success in a Newtonian explanation without providing a
jot of understanding of I itself.
On a point of historical accuracy, we confess to a certain liberty in invoking the
name of Newton for the present conception of explanation. Let there be no doubting
that the historical Newton is strongly pledged to a rather exclusive inductivism (the
emphatic declarations of the General Scholium were quite deliberate). Even so — and
apart from the question of how plausible inductivism is as a description of scientific
practice — the action-at-a-distance theorem has a secure place in Newton’s mature
theory, never mind that Newton thinks that it is absurd. We are less struck by this as
a deviation from inductivism (so ours is not an ad hominem move against Newton),
than as an example of scientific practice, which not only is not uncommon but which,
in Newton’s employment of it, is a dramatic deviation from a widely received view
of scientific rationality. Of course Newton’s abduction bears a dramatic resemblance
to Russell’s regressive method (see below, 4.4), which Russell thinks of as a central
part of the epistemology of mathematics and of the sciences generally. Some readers,
accordingly, may prefer the description ‘Russellian explanations’ over ‘Newtonian’,
and are, of course free to rebaptize as they choose.
Of course some people will not see Newtonian explanations as abductive. The
action-at-a-distance theorem is a consequence of the mathematics needed for celestial mechanics. It was not independently conjectured in the way, for example, that
Planck was led to think up the quantum hypothesis. It is more nearly correct to say that
action-at-a-distance was something that Newton found himself stuck with. Thinking
it a conceptual impossibility, it is natural to think of Newton’s situation as providing
8 celestial mechanics and X is the conceptual impossia 6WCYX trigger, where 6
bility. In standard such situations, the reasoner’s goal is to disable 6 so that it can no
longer deliver X . Newton considered this remedy, but decided against it. Disabling
6 would ruin celestial mechanics. So he held his nose and re-committed to X . Is
this abduction? If we hold that abduction is intrinsically ampliative or eliminative, we
4.2. NEWTON
71
may think it a stretch to classify re-affirmation either way; certainly there is no new
propositional content in Z consequent upon Newton’s decision to retain the action-ata-distance equation. On the other hand, Newton’s re-commitment to this proposition
was consequent upon some new thinking and the new thinking was induced by a [ an
abduction trigger. It strikes the present authors that the call could go either way; but
we ourselves are prepared to see this as abduction.
It is well to repeat that explanationism is a doctrine which emphasizes the intelligibility of events or states of affairs, rather than the fact that such events or states of
affairs are theoretically derivable. It is true, of course, that cases abound in which a
given H will provide an explanation in both the Newtonian and Harmanian senses of
explanation. We may have it, for some \^]UZ and a _ and a ` , that _>a bc Zed holds
and _fa ZOgSh \Ui d does not hold, and also that \ enlarges the theorist’s understanding
of ` , of which the Oj encompassed by _ are themselves a special case.
There is also a third sense of explanation, which is generalization of the Newtonian
sense. In the Newtonian conception the datum k of which \ is (part of) the explanation
is the fact that ZD[Plj m m m c ln . The theory predicts those observations, and H explains
or helps explain the fact that this is so by virtue of the joint facts that Zpoflj m m m c ln
and Zg:h \Ui<qDlAr c m m m c lAn . We may say, then, that \ explains, or helps explain,
the observational adequacy of Z as regards the lj . Observational adequacy is but
one trait — albeit an important one — of theories we regard as passing muster. Other
features indicative of good theories include among others, simplicity, conservatism,
and integratability. In each case, Newtonian explanations are possible in principle; in
any case, they are properly subject to an interest in whether they exist. When they do
exist, then a generalized Newtonian explanation is defined as follows.
DefGenNewtExp: Let s be a trait possessed by a theory Z . Let \ be an
hypothesis embedded in Z . Then s is explained by \ iff s is not true of
ZOgSh \Ui , or not true of Z to the same extent s is true of ZOgSh \Ui .
We note in passing that the generality of DefGenNewtExp extends to any property of
theories, both ‘good’ properties such as simplicity and ‘bad’ properties such as overcomplexity. By these lights, the axiom of choice3 in the ZFC theory of sets has an
explanatory role, in the Newtonian sense, in proofs of the Löwenheim-Skolem theorem, which guarantees to any true first-order theory an interpretation in the natural
numbers. Löwenheim’s original proof of 1915 turns out to be equivalent to the axiom
of choice. Skolem’s more streamlined proof of 1920 makes explicit use of the axiom.
In 1922 Skolem produced a proof which averted the necessity to assume choice. In its
generalized Newtonian sense, the choice axiom is explanatory in the 1915 and 1920
proofs, and (trivially) in the 1922 proof. Whether the choice axiom could also claim
explanatory potential in Harman’s sense might strike us as out of the question, in as
much as the Löwenheim-Skolem proofs are deductive, not ampliative arguments. In
fact, however, although the conclusion of these proofs (that first-order theories are, if
true at all, true in the natural numbers) is the result of deductive reasoning, the decision
3 The axiom of choice asserts that for any set of non-empty sets there exists a set containing exactly one
member from each. The choice axiom is equivalent, among other things, to the well-ordering theorem (i.e.
that every set can be well-ordered) and Zorn’s lemma (i.e. that if every chain in a partially ordered set has an
upper bound, then the set possesses a maximal element).
72
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
to use choice in the 1920 proof is a case of ampliative reasoning. Choice is postulated
precisely because it was thought necessary for the successful derivation of the theorem
in question. And in that proof, choice had beyond doubt the force of a Newtonian explanation. Even so, this does not yet tell us whether the role of choice in the 1920 proof
might also have had Harmanian explanatory force. If so, it enhances our understanding
not of the fact that the theorem follows from the 1920 proof and would not do so in the
absence of choice, but rather of the fact that true first-order theories are true in the natural numbers. It would seem that the descriptively adequate answer that reflects how
people are in fact affected by the proof, and by its conclusion, is that its explanatory
force ranges variously over a range of real-life reactions. At the one extreme we find
people who find choice more intelligible than the conclusion of the proof, and who find
that conclusion no more intelligible with the axiom than without it (cf. the discussion
of Russell and Gödel in section 4.4 below). At the other extreme we find those who
find the theorem richly intelligible, and who may do the same for the axiom. Given
that post-1920 developments show that choice is not needed for proving the theorem,
the question of the axiom’s explanatory force in post-1920 proofs doesn’t arise. It does
arise for the 1920 proof, however. Here, too, there is a distinction to make.
(a) Did the axiom of choice explain the conclusion of the 1920 proof?
(b) Did the axiom of choice contribute to the explanatory force that the
entire proof had with regard to its conclusion?
The answer to (a) would appear to be No. The answer to (b) is less obviously in the
negative. But it remains the case that even for someone who finds both the axiom and
the premisses of the proof richly intelligible, it might not be the case that the proof
makes the conclusion more intelligible to anyone satisfying the antecedent condition.
The issue of Harmanian explanation-potential is made somewhat more complicated
by the fact that the theorem is equivalent to the axiom. That is, a true first-order theory
is true in the natural numbers iff for any set of non-empty sets there is a set containing
one and only one member from each.
Harmanian explanations are aids to the understanding of a given phenomenon or a
given set of data or a given state of affairs. They are not aids to the understanding of
how that phenomenon or that set of data or that state of affairs came about, although it is
far from being true that there are no cases in which an inference to the best explanation
is facilitated by knowing how those things came about. This is especially the case
when the theorist knows, or thinks he knows, the things that caused those things to
come about.
Of Aristotle’s four causes — the material, formal, efficient and final4 — only the
latter two cut much sway in the logic of abduction. Concerning these two, there is
widespread, though not perfect, agreement that the natural sciences must make do with
efficient causality only, and that those disciplines or modes of enquiry that make room
for final causes or purposive causation are disciplines whose membership in the rank
of science may, to that very extent, be called into question. Such a view emanates from
4 The material cause of something is the stuff of which it is made. The formal cause is that which makes
the thing the sort of thing it is. The efficient cause is that which produced the thing. The final cause is the
end or ends for which the thing was produced.
4.3. PLANCK
73
Bacon [1905], who broke with a strong tradition in the natural sciences in proposing
the expungement of teleological explanation, i.e., ends-oriented explanation. Bacon
was drawn to the view that the true cause of something observable was the law under
which it fell, and that a law is a form correlated with the observable characteristics.
An explanation of such observables requires the specification of such a law; and the
more comprehensive the explanation furnished by the law, the more the explanation is
probative or alethic.
4.3
Planck
Planck’s discovery of the quantum arose out of his study of black body radiation, and
turned on the question of the interrelation among temperature t , frequency of radiation u , and the energy density per unit interval of wavelength vw . Experimental were
thought results to suggest two laws, the Rayleigh-Jeans Radiation law for very low
frequencies
yz u{|}
vwAx
~
(4.1)
and Wein’s Radiation Law for high frequencies,
vwAx ‚€ w
exp ƒ
„…w † tA‡ ˆ
(4.2)
(In fact, there was little by way of experimental confirmation for Wein’s Law; but let
that pass.) It is significant that (1) fails for high frequencies, and (2) for low.
Planck undertook to eliminate this kind of dichotomy. It was on its face a forbidding task, given that they represent such different functions mathematically. How was
Planck to show that each is a limiting case of some deeper mathematical equation? Of
course, in a momentous turn in the history of science, the answer lies in the postulation
of minimal separable quantities of energy, or quanta.
It is a matter of some irony that Planck abjured the very idea of them. On the one
hand, he recognized that accepting them would trigger a revolution in physics on a
scale that would displace Newton; but even so, Planck spent years trying to think up
alternatives to the postulation of quanta.
We see in this the structure of an abduction problem. ‰ are the theoretical and
experimental data concerning black body radiation. As a classically-minded physicist,
Planck places great store on the unification of physical laws. Thus he takes it that
there is a mathematical law yet to be formulated (Š ) that integrates the two highly
special laws. Accordingly, he postulates the existence of quanta which, together with
the requisite provisions of ‰ deliver the goods for Š .
Notwithstanding, some obvious similarities, Planck’s abduction deviates from the
explanationist/alethic model in a number of respects.
1. Planck did not undertake to explain the fact that the two laws didn’t integrate
under classical assumptions. He sought to prove that they did integrate under
nonclassical assumptions.
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
74
2. Planck’s conjectured the existence of quanta, but not for their explanatory power.
Rather for their role in the integration of laws.5
3. Planck’s also assumed that there was a deeper theory in which the two disparate
laws were replaced by a unified law. But this was an assumption of a kind continuous with the the whole methodological and philosophical spirit of the physics
of the day; so it is odd to call it a conjecture. Certainly it was not a conjecture on
the same scale of that of quanta.
4. The success of the postulation of quanta in unifying the physics of black body
radiation did not persuade Planck of the truth of his conjecture, but only of its
instrumental value in the cause of unification. So Planck did not think of this
wholly successful abduction as alethic.
Here is an even greater resemblance to Newton. Planck’s invention of quantum physics
resembles that later discovery by Gell-Mann of quarks. Initially, Gell-Mann saw quarks
as artifacts of the mathematics that drove his theory. Quarks were epiphenomena to
be tolerated, to be accepted because the mathematics that generated them was indispensable to the general physics. In the early days Gell-Mann was openly skeptical of
quarks, and stoutly denied their empirical reality. (When experimental confirmation
was eventually forthcoming, Gell-Mann tended to forget his earlier hostility.)
4.4
Russell and Gödel
Newton is an ardent inductionist, as the General Scholium of 1713 makes clear. ‘Hypotheses non fingo’ may have been directed at the action-at-a-distance theorem specifically, but it is clear that Newton disdained the method of hypotheses in a quite general
way. What is interesting about Newton’s decision to retain the incoherent theorem is
that it resembles a form of abduction, or a variant of it, that is not much recognized in
the literature, but which in fact plays an important role in the foundations of mathematics. For reasons that will shortly declare themselves, we will call this sort of abduction
regressive abduction, after Russell’s discussion of it in a paper delivered in Cambridge
in 1907, but which remained unpublished until 1973 [Russell, 1907]. The title of this
work is ‘The Regressive Method of Discovering the Premises of Mathematics’.
Russell’s regressive method forms an important part of his logicism, which, as correctly observed in Irvine [1989], is widely misunderstood. For our purposes here it
suffices to concentrate on one particular feature of Russell’s mathematical epistemics;
readers interested in the fuller story of Russell’s logicism could consult Irvine [1989]
with profit. The specific issue which concerns us here is the problem of determining
in what sense ‘a comparatively obscure and difficult proposition may be said to be a
premise for a comparatively obvious proposition’ [Russell, 1907, p. 272]. A further
problem is that of explaining how such obscure and difficult propositions are discovered and justified. One thing that Russell certainly did think that logicism involved is
the doctrine that ‘ ‹ ‹ ‹ all pure mathematics deals exclusively with concepts definable in
5 Thus Planck’s conjecture is a non-explanationist version of what Thagard calls existential abduction
[Thagard, 1988, pp. 56–58].
4.4. RUSSELL AND GÖDEL
75
terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles Œ Œ Œ ’
[Russell, 1903, p. xv]. However, as Russell freely admits,
There is an apparent absurdity in proceeding, as one does in the logical theory of arithmetic, through many recondite propositions of symbolic logic
to the ‘proof’ of such truisms as ŽLe: : for it is plain that the conclusion is more certain than the premises, and the supposed proof therefore
seems futile [Russell, 1907, p. 272].
Consider such a ‘proof’, in which the recondite premisses are propositions of logic and
the conclusion is an arithmetic truism such as ‘Ž><  ’. The purpose of such a proof
is not to demonstrate the truism, but rather to demonstrate that it follows from those
logical premisses. Russell then proposes that in those cases in which an antecedently
accepted conclusion can be shown to be deducible from a given logical premiss or set
of premisses, then the fact of such deducibility tends to confer some justification, not
on the antecedently accepted conclusion, but rather on the premiss or premiss-set of
the deduction in question.
Russell calls the method of reasoning by which such justifications are proposed
the regressive method, which as Irvine points out (citing Kleiner), is a ‘method similar
to Peirce’s abduction’, which, in ‘contemporary terminology Œ Œ Œ might also be termed
‘inference to the best explanation’ [Irvine, 1989, p. 322, nt. 26].6 As Russell goes on
to say,
We tend to believe the premises because we see that their consequences are
true, instead of believing the consequences because we know the premises
to be true. But the inferring of premises from consequences is the essence
of induction; thus the method in investigating the principles of mathematics is really an inductive method, and is substantially the same as the
method of discovering general laws in any other science [Russell, 1907,
pp. 273–274 ]. (Note here that Russell’s use of ‘induction’ is that of a
generic concept of non-deductive, ampliative inference of which abduction is a kind. See, e.g., the entry on ‘inference’, in Honderich [1995, p.
407].)
On Russell’s regressive account even the most fundamental of logical laws may
be only probable. This matters. If logical laws needn’t be attended by certainty or
by what Quine calls obveity, thus the logician is spared the burden of finding for his
axioms a Fregean justification in which ‘they are truths for which no proof can be given
ΠΠΠand for which no proof is needed. It follows that there are no false axioms, and that
we cannot accept a thought as an axiom if we are in doubt about its truth’ [Frege,
1914, p. 205]. (Emphasis added.)7 Since Russell thought that this burden could not be
met, especially in the wake of the antinomy that bears his name and which ended his
‘intellectual honeymoon’, he is able to plead the case for wholly non-intuitive axioms
6 However,
we ourselves demur from the suggestion that Russell’s abductions here are explanationist.
between Frege and Russell concerning the epistemology
and logic and mathematics, see Woods [2002b].
7 For a more detailed discussion of the difference
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
76
such as, notoriously, infinity, without having to concede that its truth is more than
merely probable.
In induction [sic], if ‘ is our logical [and obscure] premise and ’ our empirical [i.e., pretty clearly true] premise,8 we know that ‘ implies ’ , and
in a text-book we are apt to begin with ‘ and deduce ’ . But ‘ is only believed on account of ’ . Thus we require a greater or less probability that
’ implies ‘ , or, what comes to the same thing, that not-‘ implies not-’ .
If we can prove that not-‘ implies not-’ , i.e. that ‘ is the only hypothesis
consistent with the facts, that settles the question. But usually what we
do is to test as many alternative hypotheses as we can think of. If they all
fail, that makes it probable more or less, that any hypothesis other than ‘
will fail. But in this we are simply betting on our inventiveness: we think
it unlikely that we should not have thought of a better hypothesis if there
were one [Russell, 1907, pp. 274–275 ]. (Emphasis added in the second
instance.)
Note in passing Russell’s use of ad ignorantiam reasoning in a form which AI researchers have dubbed autoepistemic reasoning, and which is far from always being a
fallacy. In its most elementary form,
1. It is not known that ‘
2. Therefore, not-‘ ,
the ad ignorantiam is fallacious. In certain autoepistemic variations, it is not, as in:
a. If there were a Departmental meeting today, I would know it.
b. But I have no knowledge of such a meeting.
c. Therefore, the Department doesn’t meet today.
Thus,
i. If there were a better hypothesis we would know it by now.
ii. But we don’t
iii. So there isn’t.
We shan’t here press the question of how safely autoepistemic Russell’s elimination
argument is. But we note that it is of a type which often plays a central role in abductive
reasoning.
Thus it is Russell’s view that more often than not axioms are accepted abductively,
rather than by way of their self-evidence.
The reason for accepting an axiom, as for accepting any other proposition,
is always largely inductive [sic], namely that many propositions which are
nearly indubitable can be deduced from it, and that no equally plausible
8 Russell’s
use of ‘empirical’ is metalevel. He here means clearly or obviously true.
4.4. RUSSELL AND GÖDEL
77
way is known by which these propositions could be true if the axioms
were false, and nothing which is probably false can be deduced from it
[Whitehead and Russell, 1910, p. 59; p. 62 of 1st ed. (emphasis added)].
Russell is clear in seeing regressive abduction as pragmatic. He goes so far as to concede that his axiom of reducibility has a ‘purely pragmatic justification’ [Whitehead
and Russell, 1910, p. xiv]. In this, it is clear that Russell anticipates Gödel, who held
that
“ “ “ even disregarding the intrinsic necessity of some new axiom, and even
in case it has no intrinsic necessity at all, a probable decision about its
truth is possible also in another way, namely inductively by studying its
‘success’. Success here means fruitfulness in consequences, in particular, ‘verifiable’ consequences, i.e., consequences demonstrable without the
new axioms, whose proofs with the help of the new axiom, however, are
considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs [G´’odel, 1990b, pp. 476–477
(emphasis added)].9
Gödel acknowledges the Russellian lineage of his own regressive views, and he joins
with Russell in emphasizing the epistemological similarity between mathematics and
the natural sciences. ‘The analogy’, says Gödel, ‘between mathematics and a natural science is enlarged upon by Russell also in another respect “ “ “ the axioms need
not necessarily be evident in themselves, but rather their justification lies (exactly as
in physics) in the fact that they make it possible for these ‘sense perceptions’ to be
deduced “ “ “ ’ [Gödel, 1944, p. 127].
A further example of regressive abduction can be found in the Jordan Curve Theorem, which establishes that if there is a closed curve in the plane satisfying certain
conditions, then it divides the plane into ‘inside the curve’ and ‘outside the curve.’ Call
this proposition F. It is interesting to note that the proof of the Theorem was preceded
by seven years of effort and failure. When the proof was finally achieved it did not
establish the truth of proposition ” . That proposition had long been known and did not
require establishment. The seven-year lull prior to the proof was instructive. It showed
that there was something not quite right with the proof apparatus of plane geometry,
since it couldn’t deliver the obvious truth ” . In a certain way, the eventual success
of the proof resembles the situation involving Galileo’s telescope. Before its apparent
disclosures of canals on the Mars could be believed, the instrument was first trained on
a mid-distant but well-known church to verify features it was already known to have.
When the telescope ‘proved’ these features, it was considered a generally reliable optical instrument. In the same way, when geometry was finally tricked out in ways that
enabled the proof of the Jordan Curve Theorem (complex variables and topology), this
served abductively to indicate the general reliability of the theory as an instrument of
which investigates the mathematical structure of space.
•••
•••
9 Cf: ‘
probably there exist other [axioms] based on hitherto unknown principles
which more
profound understanding of the concepts underlying logic and mathematics would enable us to recognize as
implied by these concepts
[and] so abundant in their verifiable consequences
that quite irrespective
of their intrinsic necessity they would have to be assumed’ Gödel [1990a].
•••
•••
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
78
4.5
Popper
Karl Popper is unique among 20th century philosophers of science. His theories were,
and still are, taken seriously by working scientists, and to some extent they have influenced the work of some of them. Further evidence of the regard in which he was held
by scientists is the work Popper did in collaboration with them (see, e.g., [Popper and
Eccles, 1983]).
There is, apart from Popper’s own prodigious writings, a huge literature on his
views. It is not our purpose to add to that total. The basics of Popper’s approach to
science will be known to most readers of this book and, in any went, we wish here to
concentrate only on those features of it that bear on abduction. Those features are well
summed up in the title of one of his more important works, Conjecture and Refutation
[1963].
Popper rejects inductivism in science. He thinks that Hume’s Problem of Induction
has no solution save evasion. Evade it he does, declaring that scientific laws derive
none of their security from high degrees of confirmation by way of true positive instances. The security of scientific laws is provisional at best. It derives from the fact
that persistent and sophisticated attempts to refute them have so far failed. When this
condition is met, Popper is prepared to say that such a law is corroborated by its positive instances. But he emphasizes that corroboration (also a provisional attribute) has
not a jot of confirmatory significance. Thus the corroboration of a scientific law or
theory consists in its predictive success to date coupled with the failure to refute it to
date.
This aspect of Popper’s philosophy of science is especially important for the logic
of abduction. We have repeatedly made the point that part of the abduction process is
a decision to submit a hypothesis to test. But we have not yet attempted to characterize
what such a test would be. Popper does characterize it. The test of a scientific theory
is the systematic effort to refute it.
Popper is a deductivist about science. One knows a scientific theory through its
deductive consequences. Refutation imbibes the spirit of Popper’s deductivism. A scientific law or theory is refuted by as few as one true negative instance. Such a notion of
refutation flies in the face of scientific practice (see the section below), but, according
to Popper, this would show only that such scientific practice is philosophically disreputable. Even if we allow that an entire scientific theory is falsified by any given true
negative instance, there is nothing in Popper’s conception of refutation that enables us
in a principled way to localize the theory’s defect, so that comparatively minor repairs
might be considered. This raises the question of whether alternative accounts of refutation might be more hospitable to the idea of localized diagnosis. We consider this
question further in chapter 9.
There is an odd asymmetry between Popper’s conception of conjecture and his
account of refutation. Given what he takes refutation to be, scientific theories are maximally fragile and unstable. There is no latitude that can be given a theory in the face
of even a single true negative instance. Conjecture, on the other hand, is a hugely
latitudinarian notion. We have seen that Peirce thinks that conjecture is constrained
instinctually, and we ourselves have suggest the importance of plausibility. Others require that conjectured hypotheses meet various conditions of conservatism, of what
4.5. POPPER
79
Quine calls the “maxim of minimal mutilation”. Popper is not so-minded. Conjectures
need not be constrained in any of these ways. There is in the whole literature on abduction no freer rein given to the generation and engagement of hypotheses. In some
respects, Popper thinks that the wilder-the-better is the applicable standard. Thinking
so would make Planck’s quantal speculation a conjecture of the very best kind, of a
kind to which scientific revolutions are frequently indebted. Implausible, instinctually
alien, mutilating, the quantum conjecture blasted physics from its classical embrace
into the most successful physical theory in the history of human thought. This is not
to say that Popper is a total conjectural promiscuist. Hypotheses cannot be entertained
unless they are falsifiable. This is a point that returns us to a difficulty of just paragraphs ago. Let – be a hypothesis that we am considering making. If we intend to do
thing Popper’s way we must drop – if we think that it is not falsifiable. This we might
not know until we find a place for – is a complex scientific theory of a sort that we
take to be falsifiable. Suppose later that a true negative instance is unearthed. Then the
theory is false. But we still might not know whether – is false. If this is all that we
have to go on, we might not even know whether – is falsifiable.
In the logic of generation and and engagement, there is little that Popper has to
offer, except negatively.
My view of the matter . . . is that there is no such thing as a logical method
of having new ideas, or a logical reconstruction of this process [Popper,
1934, p. 31].
Indeed,
the act of conceiving or inventing a theory seems to me to be neither to
call for logical analysis nor to be susceptible of it [Popper, 1934, p. 32].
So, plausibility is not a constraint; conservatism is not a constraint; etc. And the one
positive requirement, falsifiability intuitiveness is not a constraint, is problematic. In
the logic of discharge, as we saw in section 2.5, Popper will go only half-way. Strictly
speaking, nothing about a scientific theory is categorical. All of science is on sufferance; and the best that we can hope for is that our favorite theory hasn’t yet been put out
of business. When it comes to deploying a hypothesis, the closest that one can properly
come to categorializing it is to leave in place with whatever confidence it deserves to
have on the basis of its corroboration record to date.
In section 2.5 we claimed that Popper had confused defeasibility with hypotheticality. Popper is assuredly correct to emphasize that scientific theories and the hypothesis
that they embed are forwarded defeasibly. They are forwarded in ways that recognize
the fallibility of all science. But this is no bar to categorical assertion in science it; is
not required that scientific utterance be restricted to hypothetical utterance. There is no
equivalence between
1.
—
2.
Perhaps — .
, although I may be wrong
and
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
80
When I say that Uncle Frank will live to a hundred, I may be wrong and I may say so.
This is compatible with my strong conviction that, already very old, he will achieve his
century. “Maybe he will live to a hundred” is much weaker.
Popper’s confusion of defeasible utterance with conjectural hypotheses might be
occasioned by his insistence that no conjecture is ever justified.
[T]hese conjectures . . . can neither be established as certainly true or even
as “probable” (in the sense of the probability calculus . . . . these conjectures can never be positively justified [Popper, 1963, Preface].
Even so, it strikes us that, again, insusceptibility to justification does not preclude categorical utterance.
4.6
Lakatos
It will repay us to touch briefly on some ideas loosely developed by Imre Lakatos
[1970] and [1968], which can be seen as a corrective to Popper’s conception of refutation [1959; 1965]. Popper understood a scientific theory as a set of deductions derived
from conjectures. If the deduced consequences are false, the theory that implies them
is false. (As we saw, modus tollens.) Although Popper’s general methodological orientation was something Lakatos agreed with, he finds fault with Popper’s account in
two respects. Popper erred in conceiving of the methods of source too narrowly and
too strictly. The narrow error was that of thinking that individual theories are the intended targets of scientific testing. In fact, Lakatos claimed science tests not individual
theories but families of theories, which he called ‘research programmes’. The error of
excessive strictness was the mistake of supposing that a research programme is automatically toppled if it licenses a false prediction. Lakatos writes:
The main difference from Popper’s is that in my conception criticism does
not — and must not — kill as fast as Popper imagined. Purely negative
destructive criticism, like ‘refutation’ or demonstration of an inconsistency
does not eliminate a [research] programme. Criticism of a programme is a
long and often frustrating process and one must treat budding programmes
leniently. One may, of course, shore up the degeneration of a research
programme, but it is only constructive criticism which with the help of
rival research programmes, can achieve real success [Lakatos, 1970, p.
179 (original emphases omitted in some cases)].
As Lakatos sees it, a research programme is what Kuhn should have understood a
paradigm to be [Kuhn, 1967; Kuhn, 1970]. Against Kuhn’s idea that normal science
embeds a paradigm, Lakatos holds that what ‘Kuhn calls ‘normal science’ is nothing
but a research programme that has achieved monopoly’, and ‘one must never allow a research programme to become a Weltanschauung, or a sort of scientific rigour’ [Lakatos,
1970, p. 155 (emphasis omitted)].
The idea of research programmes is central to how Lakatos conceives of a logic
of discovery. The basic concept of a logic of discovery is not the theory, but rather
families or series of theories. ‘It is a succession of theories and not one given theory
4.6. LAKATOS
81
which is approved as scientific or pseudo-scientific’ [Lakatos, 1970, p. 132]. The central issues of the logic of discovery require for their consideration the framework of the
methodology of research programmes. A research programme can be defined by way
of problem shifts. Suppose that ˜š™ , ˜š› , ˜œ ,    is a sequence of theories in which every
subsequent member results from its predecessor by addition of supplementary statements or by providing its predecessor with an inducted semantic interpretation with
which to take account of an observational anomaly. A problem shift is theoretically
progressive if each subsequent theory is empirically richer than its predecessor, if, in
other words, the successor theory predicts a new fact beyond the reach of the predecessor theory. A problem shift is empirically progressive if the new predictions of a
successor theory are empirically corroborated. A problem shift is progressive if it is
both theoretically and empirically progressive. A problem shift is degenerative if it is
not progressive [Lakatos, 1970, p. 118].
When a science is in a state of immaturity, problem shifts are a matter of trial
and error. Mature theories have heuristic power which guides the science in the more
orderly development of its research programmes. A positive heuristic gives counsel
on what research strategies to pursue. A negative heuristic guides the researcher away
from unfruitful research protocols. Lakatos sees the negative heuristic as framing the
key insight of a problem shift, its conceptual core, so to speak; and ‘we must use our
own ingenuity to articulate or even invent auxiliary hypotheses which from a protective
belt around this core’ [Lakatos, 1970, p. 133]. Thus the conceptual core is irrefutable
and is made so by the theorist’s methodological decision, an idea that calls to mind
Quine’s jape that theories are free for the thinking up, and Eddington’s that they are
put-up jobs. Concerning the matter of what enters into such decisions, to say nothing
of what makes them scientifically rational, Lakatos has nothing to say.
A science’s positive heuristic specifies its range of refutability. It identifies the
‘refutable variants’ of the research programme and makes suggestions about how to
modify the ‘refutable protective belt’ of the programmes core insights [Lakatos, 1970,
p. 135]. The positive heuristic
sets out a programme which tests a chain of ever more complicated models
simulating reality: the scientist’s attention is riveted on building his models
following instructions which are laid in the positive part of his programme;
he ignores the actual counterexamples, the available data [1970, p. 135].
A theoretically progressive problem shift arises from attempts to falsify theories and
then to save them by supplementary hypotheses or semantic reinterpretation or both.
A theory, then, can be seen as a negative heuristic together with auxiliary hypotheses
worked up as a model. The only reason to abandon a research programme is that
it is degenerate (a fact which is often enough apparent only in hindsight). The idea
that a theory should be abandoned if its fundamental insights are false or if they lack
inductive support. Lakatos thinks it likely that all core ideas of research programmes
are false, and he joins with Popper in insisting that inductivism is a wholly misbegotten
idea in the methodology of science [Lakatos, 1970, pp. 155–173, 99–103 ]. It follows,
therefore, that it is irrational to suppose that any given scientific theory is true; and
this, says Lakatos, is a powerful and correct inducement to scientific pluralism and the
proliferation of research programmes.
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
82
Hypotheses are the work of a programme’s positive heuristic. Their function is
to protect the programme’s core insight from observational discouragement or from
predictive impotence. The hypothesis is justified (or ‘rational’) to the extent that it
facilitates novel predictions that turn out to have observational corroboration, if not
immediately, then soon enough in the evolving successiveness of the research programme and before there is occasion to attribute degeneracy. Core ideas, on the other
hand, while not strictly refutable, can over time be discredited by the failure of the
programme’s positive heuristic to protect it. A core idea is in trouble if the research
programme to which it is central is degenerating.
Sketchy though it surely is, Lakatos has some conception of what a logic of justification abduction would look like. Discovery abduction is another matter. Here Lakatos
has little to say except — in ways that call Hanson to mind — that hypotheses should
be selected for their suitability to the positive heuristic of a research programme. Thus
those hypotheses should be considered which are of a type that might do well if made
part of such a heuristic.
Lakatos’ views have attracted a large number of criticisms, three of which are especially important [Suppe, 1977, pp. 667–668 ]. One is that in the research programme
defined by a series of theories žšŸ , žš , ¡ ¡ ¡ , the changes wrought by a successor theory
may not, according to Lakatos, include the deletion of non-core claims made by the
predecessor theory (cf. the ¢O£U¤ ¥¦ §¨ ©«ªS¬ constraint of our first pass at a general
schema for abduction in section 2.4). Another is that in his account of how a research
programme arises, Lakatos over-concentrates on the modification of theories within
a problem shift, and accordingly pays insufficient attention to what Dudley Shapere
[1977] calls ‘domain problems’. A domain problem is the problem of determining
with sufficient precision the target questions of an emerging research programme; it is
the question of what the programme is appropriately about. A third criticism, linked to
the first, is well developed by Toulmin [1972]. The idea of what theoretical strategies
are appropriate is seriously underdetermined by an absence of anchoring concepts of
requisite type. Lakatos admits no notion of conceptual adequacy in theory construction
save for the device of semantic reinterpretation in the production of problem shifts. But
as Toulmin points out, the history of science is dotted with ideas of such inappropriateness that no amount of semantic reinterpretation (fairly so-called) would put things
right.10
4.7
Hintikka
During the past twenty-five years and more, Jaakko Hintikka and his collaborators have
been pursuing a fundamental reform of logic. Hintikka’s view is that a psychologi10 It is interesting to note that Lakatos’ philosophy of science is a model of a form of non-cooperative
dialogue logic known as MindClosed [Gabbay and Woods, 2001a; Gabbay and Woods, 2001d]. In its pure
form MindClosed offers an impenetrable form of defence of a player’s thesis T, in which every critical move
is rejected out of hand. The reasoning underlying these rejections is that since T is true, (or obviously,
self-evidently or necessarily true), the criticisms not only must fail, but can be seen to fail without detailed
consideration of their purported weight. We join with Lakatos’ critics in thinking that he presses too far
what, in effect, is the MindClosed model. Still, it seems clear to us that a science defends its organizing
conceptualizations and core insights in ways that approximate to the MindClosed strategy.
4.7. HINTIKKA
83
cally real and technically adequate account requires us to think of logic as intrinsically
game-theoretic and erotetic. It is game-theoretic in as much as it assigns a central role
to strategic factors. It is erotetic in as much as it is a logic of interrogative inquiry
[Hintikka et al., 2001].
Hintikka’s approach to abduction is likewise game-theoretic and erotetic. It is an
approach that is neither explanationist nor probabilist, although it bears some affinity
to a class of diagnostic logics which themselves admit of probabilistic interpretation
(e.g., [Peng and Reggia, 1990]).
Hintikka finds intimations of his account in Peirce, who defines logical operators
by dialogical rules that yield decidable proof-protocols. 11
I call all such inference by the peculiar name, abduction, because it depends upon altogether different principles from those other kinds of inference [Hintikka, 1999a, p. 97 (quoted from Peirce), emphasis added in the
second instance].
These principles are summarized in Kapitan [1997, p. 479].
1. Inference is a conscious voluntary act over which the reasoner exercises control [Peirce, 1931–1958, 5.109, 2.144].
2. The aim of inference is to discover (acquire, attain) new knowledge
from a consideration of what is already known (MS 628:4 But see
section 3.2, above).
3. One who infers a conclusion C from a premiss P accepts C as a result
of both accepting P and approving a general method of reasoning according to which if any P-like proposition is true, so is the correlated
C-like proposition [Peirce, 1931–1958, 7.536, 2.44, 5.130, 2.773,
4.53–55, 7.49].
4. An inference can be either valid or invalid depending on whether it
follows a method is conductive to satisfying the aim of reasoning,
namely, the acquisition of truth [Peirce, 1931–1958, 2.153, 2.780,
7.44], Peirce MS 692:5).
Hintikka’s general logic distinguishes two types of inference rules which regulate the
moves of players in games of interrogative enquiry. Definatory rules specify moves that
are permissible. Strategic rules specify moves that are advisable. (This the ancester of
our distinction in 2.1.2 between valid deduction rules and rules it would be reasonable
to use in a given context.) Hintikka sees evidence of the distinction between definatory
and strategic rules in Peirce [Hintikka, 1999, p. 98 ff.]. He attributes to Peirce the view
that the two classes of rules have different justification conditions, and he thinks that his
own strategic rules are a good candidate for Peirce’s ‘altogether different principles.’ A
definatory rule is valid in so far as it ‘confers truth or high probability to the conclusion
of each particular application. . . ’ [Hintikka, 1999, pp. 98–99]. A valid definatory rule
must be universally truth-preserving or probability-enhancing. Strategic rules, on the
11 See also [Lorenzen and Lorenz, 1978], which extends this approach to a constructive foundation for
classical logic.
84
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
other hand, can be justified even though they may fail in individual cases. They are
justified not by their universal success but by their propensity for success. They are
game-theoretic principles. They honour game theory’s insight that for the most part
the acquisition of utilities is best achieved by way of general strategies, rather than
individual moves. Hintikka’s own deep insight is that measures for guaranteeing truthpreservation or probability-enhancement greatly underdetermine what an investigator
should do, move-by-move. Strategic rules describe a general method which pays off in
the long run even though it may land the investigator in local turbulence. Hintikka sees
his strategic rules prefigured in Peirce’s description of the contrast between induction
and abduction. Induction is a type of reasoning
which professes to pursue such a model that, being persistent in, each
special application of it ... must at least indefinitely approximate to the
truth about the subject in hand, in the long run. Abduction is reasoning,
which professes to be such that in the case there is ascertainable truth
concerning the matter in hand, the general method of this reasoning though
not necessarily each special application of it must eventually approximate
the truth [Eisele, 1985, p. 37]12
Where Hintikka parts company with Peirce is in claiming that his definatory-strategic
distinction cuts across the deduction-induction-abduction trichotomy. Thus a bit of
reasoning is not made abductive just because it involves the use of strategic rules. Every
move of a reasoner, deductive, inductive or abductive, involves the presence of both
types of rules. The definatory rules tell him whether the move is permissible. The
strategic rules tell him whether it is smart. A definatory rule must be ‘conductive to the
acquisition of truth.’ A strategic rule need only be part of a general method which may
have local failures (hence is a type of rule tailor-made for our conception of resourcestrapped individual agents).
It is Hintikka’s contention that it is only through the providence of strategic rules
that reasoning is truly ampliative [Hintikka, 1999, p. 101]. And Peirce contends that
whenever the objective of the reasoning involves new hypotheses, a move is ampliative
only if it is abductive. Hintikka’s proposed ‘solution to the problem of abduction,’ is
that ‘[a]bductive ‘inferences’ must be construed as answers to the enquirer’s explicit or
(usually) tacit question put to some definite source of answer (information)’ [Hintikka,
1999, p. 102]. Note here the similarity of question-generating diagnostic models, for
example, KMS.HT. (See Peng and Reggia [1990, pp. 40–41].) Hintikka discerns this
method of reasoning in the Socratic elenchus, and in much later writers such as R.G.
Collingwood [1946] and Hans Gadamer [1975].
It is clear that the interrogative element of abduction is explicitly recognized by
Peirce.
The first starting of a hypothesis and the entertaining of it, whether as a
simple interrogation or with any degree of confidence, is an inferential
step which I propose to call abduction [Peirce, 1931–1958, p. 6.525].
12 And: Deduction is reasoning which proposes to pursue such a method that if the premisses are true the
conclusion will in every case be true [Eisele, 1985, p. 37].
4.7. HINTIKKA
85
Then, as we have seen,
This will include a preference for any one hypothesis over others which
would equally explain the facts, as long as this preference is not based
upon any previous knowledge bearing upon the truth of the hypothesis, nor
[sic] on any testing of any of the hypotheses, after having admitted them
on probation. I call such inference by the peculiar name, abduction, because its legitimacy depends on altogether different principles from those
of other kinds of inference.
In addition,
It is to be remarked that, in pure abduction, it can never be justifiable to
accept the hypothesis otherwise than as an interrogation. But as long as
that condition is observed no positive falsity is to be feared [Buchler, 1955,
p. 154].
Peirce did not explicitly identify an abductive inference with a question-answer step
in an interrogation. Consequently we lack Peircean answers to fundamental questions.
Can we say in a principled way what the best questions are in such an enquiry? How
is one to select the best questions from a set of possible questions?
In attempting to answer these questions, Hintikka proposes abandoning the idea
that abduction is inference of any kind. Rather, he says, ‘Abduction should be conceptualized as a question-answer step, not as an inference in any literal sense of the word’
[Hintikka, 1999, p. 105]. A virtue of the suggestion is that it entirely vindicates Peirce
in his claim that abduction is ‘altogether different’ from deduction and induction.
An accessible treatment of Hintikka’s model of question-answer enquiry is Hintikka and Bachman [1991], Hintikka and Halonen [1995], as well as the aforementioned Hintikka, Halonen and Mutanen [2001]. A similar approach is developed in the
erotetic logic of Kuipers and Wisniewski [1994].
86
CHAPTER 4. NON-EXPLANATIONIST ACCOUNTS
Chapter 5
A Sampler of Developments in
AI
5.1
Explanationist Diagnostics
In this chapter we shall briefly review some representative treatments of abduction by
AI researchers. Josephson and Josephson take an explanationist tack, whereas Peng
and Reggia’s1 approach integrates explanationist and probabalistic elements. In the final section, we shall make some effort to adjudicate the rivalry between explanationism
and probabilism.
Much of the recent — and most promising work — has been produced by computer
scientists and, like Gabbay, by logicians who also work in the AI research programme. 2
Most of the contemporary work to date tends to concentrate on hypothesis generation
and engagement. We begin by reviewing a representative system for dealing with a
class of problems for which hypothesis-generation and hypothesis-engagement are an
efficient way of finding the best solutions (or explanations). The system in question is
that of Josephson and Josephson [1994, ch. 7].3 We should be careful about what we
intend by the representativeness claim. Josephson and Josephson [1994] is representative of a the explanationist approach to formalizing diagnostics. It is an approach that
meets with severe complexity-problems. The work is more representative in the former
respect than the latter; but it is well to emphasize early in the proceedings the general
problem posed by complexity in formal reconstructions of actual human performance.
Josephson’s and Josephson’s is an approach with some notable antecedents; e.g.
Miller, Pople and Meyers [1982], [Pearl, 1987], de Kleer and Williams [1987], [Reiter, 1987], Dvorak and Kuipers [1989], Struss and Dressler [1989], Peng and Reggia
[1990], and [Cooper, 1990], among others.
1 Other important, indeed, seminal contributions to probabilistic AI are Pearl [1988] and [2000]. A useful
review of the whole sweep of developments in theories of abduction is Aliseda-LLera [1997].
2 See, for example, contributions to logic programming such as Eshgi and Kowalski [??].
3 The authors of Josephson and Josephson’s chapter 7 are Tom Bylander, Dean Allemang, Michael C.
Tanner and John R. Josephson.
87
CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
88
We here adopt the notational conventions of Josephson and Josephson [1994]. ­
denotes a fact or a datum, ® a class of data; ¯ denotes a particular hypothesis and °
a class of these. ° itself can be considered a composite hypothesis. An abduction
problem AP is an ordered quadruple, ± ®e² ³ ³ , °2² ³ ³ ´ µ ´ ¶· ¸ . ®2² ³ ³ is a finite set of the
totality of data to be explained; °e² ³ ³ is a finite set of individual hypotheses; µ is a
function from subsets of °e² ³ ³ to subsets of ®e² ³ ³ (intuitively ° explains µ ¹ °fº ); and ¶· is
a function from subsets of °2² ³ ³ to a partially ordered set (intuitively, ° has plausibility
¶· ¹ °fº ). In AP the requirement that unit values of ¶· be partially ordered leaves it
under-determined as to whether ¶· ¹ °fº is ‘a probability, a measure of belief, a fuzzy
value, a degree of fit, or a symbolic likelihood’ [Josephson and Josephson, 1994, p.
160].4 Derivatively, AP is a logic which aims at a solution of an abduction problem.
(In terms of our triple AL, AP could be seen as a sub-theory of the first component.)
° is complete if it explains all the data; i.e. µ ¹ °fºf»¼®e² ³ ³ . ° is thrifty if the
data explained by ° are not explained by any proper subset of it (cf. Aristotle on
syllogisms); i.e. ½‚¾ °K¿O°S¹ µ ¹ °fºeÀOµ ¹ °fÁ º º . If ° is both complete and thrifty then
it is an explanation. An explanation ° is a best explanation if there is no other more
plausible explanation; i.e. ½¾ °fÁ ¹ ¶· ¹ °fÁ ºSÂö· ¹ °fº º . Note that since ¶· gives only
partial orderings, there can be more than one best explanation of a ­ or a ® . The
system AP was designed to model Reiter’s account of diagnosis ([Reiter, 1987]) and
Pearl’s approach to belief-revision ([Pearl, 1987]). We briefly sketch the connection
with Reiter’s theory.
Reiter on Diagnosis: A diagnosis problem is an ordered triple ± SD, COMPONENTS, OBS¸ . SD is a finite set of first-order sentences which describe the diagnostic problematic. OBS is a set of first-order sentences which report observations.
COMPONENTS is a finite assortment of constants, of which ab is a one-place predicate
leastÈAset
ļÀ COMPONENTS such that
Å Æ ÇUmeaning
ÈAÉ%Å<ÇUÊ ‘abnormal’.
Ë Ì ¹ Í º Î Í<ÏSÄ«AÐ diagnosis
ÇUÊ ½ Ë Ì ¹ Í º isÎ Íea ÏS
Ñ ÒfÓ%ÈAÔÕÔÖÅ× Ä2Ð is consistent. A
diagnosis problem can be modelled in AP as follows:
°e² ³ ³š»
®e² ³ ³š»
µ ¹ °fº%»
COMPONENTS
OBS
a maximal set ®pÀL®
such that
Å ÆLÇ ® Ç>Ø Ù ¹ ¯ºÚ ¯>Ï>° Ç ½ Ø Ù ¹ ¯ºÚ ¯>Ï>°2² ³ ³ × °
is a consistent set.
(Diagnoses are unranked in Reiter’s treatment; hence ¶· is not needed in the AP reconstruction).
In a good may respects AP is a simplified model. For example, both µ and ¶· are
assumed to be tractable, notwithstanding indications to the contrary ([Reiter, 1987]
ÛÜeÝÞ ß ß
á â äã å æ‚ç:è é ê é ë ë ë ì
íÞ ß ß
î
4 We give a formal model of abduction in Chapters ?? and ??. In this model, given elements
,
an abductive algorithm utilizing the proof theory of the logic will yield a family
of
possible hypotheses which explain . From such families it is easy to define the set
and a function as
described in the Josephson model. In this context, the ordering
can be meaningfully defined from the
logic involved and using the abductive algorithm available.
The Josephson model
is no longer an abstract model but a derived entity from our abductive mechanisms. As a derived entity better complexity bounds may be obtained, or at least its complexity
can be reduced to that of the abductive algorithms.
Û
ó ÝÞ ß ß é íÞ ß ß é î é ï ð ô
à
ï ð ñ íAò
5.1. EXPLANATIONIST DIAGNOSTICS
89
and [Cooper, 1990]). Even so, intractability seems to be the inevitable outcome above
a certain level of interaction among the constituents of composite hypotheses.
We say that an abduction problem is independent when, should a composite hypothesis explain a datum, so too does a constituent hypothesis; i.e. õöø÷Dö2ù ú ú û ü û öfý2þ
ÿ ü û ý ý . In systems such as AP, the business of selecting a best explanation resolves into two component tasks. The first task (1) is to find an explanation. The
second task (2) is keep on finding better ones until a best is identified. A number of
theorems bear on these two matters, the first seven on (1) and the next three on (2). In
the interests of space, proofs are omitted.
Theorem 5.1.1 In the class of independent abduction problems, computing the number
of explanations is #P-Complete (i.e. as hard as calculating the number of solutions to
any NP-complete problem).
In the case of independent abduction problems, sub-task (1) is tractable. Hence we
have
Theorem 5.1.2 For independent abduction problems there exists an algorithm for specifying an explanation, if one exists. The algorithm’s order of complexity is eû ý , where is the true complexity of ü , and indicates calls to ü .
An abduction problem is monotonic if a complete explanation explains at least as much
as any of its constituent explorations; i.e. õö ö÷ ö2ù ú ú û ö¼÷ ö¼ü û öfý÷ ü û ö ý ý .
Any independent abduction problem is monotonic, but a monotonic abduction problem
need not be independent.
Theorem 5.1.3 Given a class of explanations, it is, NP-complete to determine whether
a further explanation exists in the class of monotonic abduction problems.
Moreover
Theorem 5.1.4 In the class on monotonic abduction problems there also exists an
2û ý algorithm for specifying an explanation, provided there is one.
Let be a composite
exists
hypothesis. Then an incompatibility abduction
problem
with regard to if contains a pair of constituent hypotheses and . More generally, an incompatibility abduction problem is an ordered quintuple 2ù ú ú öeù ú ú ü ,
where all elements but are as before and is a set of pairs of subsets of ö2ù ú ú which
are incompatible with one another. We put it that õöK÷Döeù ú ú û û !"û #e÷Döfý ý$
ü û öfý<þ&% ý . In other words, any composite hypothesis containing incompatible subhypotheses is explanatorily inert. Any such composite is at best trivially complete and
never a best explanation. However, independent incompatibility problems are independent problems apart from the factor of incompatibility. In other words, they fulfill the
following condition:
õöW÷Lö2ù ú ú û û û #%÷ öfý ý¼ü û öfý%þ(' ü û ý ý )
Incompatibility abduction problems are less tractable than monotonic or independent
abduction problems.
90
CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
Theorem 5.1.5 In the class of independent incompatibility abduction problems it is
NP-complete to find whether an explanation exists.
From this it follows that it is also NP-hard to determine a best explanation in the class
of independent incompatibility abduction problems. This class of problems can be
reduced to Reiter’s diagnostic theory. In fact,
Theorem 5.1.6 In the class of diagnosis problems, it is NP-complete to determine
whether a diagnosis exists, depending on the complexity of deciding whether a composite hypothesis is consistent with SD * OBS. (Here, a composite hypothesis is a
conjecture that certain components are abnormal and the remainder are normal.)
Independent and monotonic abduction problems are each resistant to cancellation.
One hypothesis cancels another if and to the extent that its acceptance requires the loss
of at least some of the explanatory force the other would have had otherwise. A cancellation abduction problem can be likened to an ordered sextuple + ,$- . . / 01- . . / 2 / 34 / 2 / 2 5 ,
of which the first four elements are as before, and 2 is a function from 0$- . . to subsets
of ,1- . . , indicating the data ‘required’ by each hypothesis. This gives an extremely simplified notion of cancellation. Nevertheless, it is enough to be a poison-pill for them
all. It suffices, that is to say, for interpretability:
Theorem 5.1.7 It is NP-complete to ascertain in the class of cancellation abduction
problems whether an explanation exists.
It follows from Theorem 7 that it is NP-hard to find a best explanation in this class of
problems.
Thrift is also an elusive property, notwithstanding its methodological (and psychological) desirability. Indeed, it is as hard to find whether a composite hypothesis is
thrifty in the class of cancellation abduction problems as it is to determine whether
an explanation exists. In other word, both tasks are co-NP-complete in this class of
problems.
Up to this point, our theorems address the problem of whether an explanation exists
in various problem classes we have been reviewing. The remaining theorems bear on
the task of finding best explanations. In systems such as AP, finding a best explanation
is a matter of comparing plausibilities. There is a best-small plausibility rule for this.
The rule gives a comparison criterion for classes of hypotheses 0 and 06 . The rule
provides that these be a function from 0 to 06 which matches elements of 0 to not less
plausible elements of 06 . If 0 and 06 have the same cardinality, at least one element
in 0 must be more plausible than its image in 06 if 0 is to be counted more plausible
than 06 . On the other hand, if 0 is larger than 06 , it cannot be more plausible than
06 . Whereupon
Theorem 5.1.8 In the class of independent abduction problems, it is NP-hard to determine a best explanation, using the best-small plausibility rule.
However, the best-small rule is tractable where the individual hypotheses have different
plausibility values 7 8 / 9 9 9 / 7 : and the 7 8 are totally ordered. In that case,
5.2. ANOTHER EXAMPLE OF EXPLANATIONIST DIAGNOSTICS
91
Theorem 5.1.9 In the class of totally ordered monotonic problems satisfying the bestsmall criterion, there exists an ;$< =>
?
@A>B C@A=D E algorithm for determining a best
explanation.
It follows that if there exists just one best explanation in the conditions described by
Theorem 9, it will be found by an algorithm of the stated type. On the other hand, it
is notoriously difficult to determine whether the ‘just one’ condition is indeed met. In
fact,
Theorem 5.1.10 In the class of totally ordered independent abduction problems deploying the best-small rule, if there exists a best explanation it is NP-complete to ascertain whether another best explanation also exists.
These theorems establish that if we take abduction to be the determination of the
most plausible composite hypothesis that is omni-explanatory, then the problem of
making such determinations is generally intractable. Tractability, such as it may be,
requires consistency, non-cancellation, monotonicity, orderedness and fidelity to the
best-small rule. But even under these conditions the quest for the most plausible explanation is intractable. What is more, these difficulties inhere in the nature of abduction
itself and must not be put down to representational distortion. One encouraging thing
is known. If an abduction problem’s correct answer-space is small, and if it is possible to increase the knowledge of the system with new information that eliminates
large chunks of old information, then this reduces the complexity of explanation in
a significant way. By and large however, since there are no tractable algorithms for
large assortments of abduction problems (the more psychologically real, the more intractable), most abduction-behaviour is heuristic (in the classical sense of the term; see
section 11.4).
5.2
Another Example of Explanationist Diagnostics
In this section we briefly review the parsimonious covering theory of Yun Peng and
James Reggia. As was the case with Josephson and Josephson, our discussion will
be limited to a description of characteristic features of this approach, as well as representative problems to which it gives rise. An attractive feature of this account is its
sensitivity to the complexities induced by giving the theory a probabalistic formulation.
This is something these authors attempt, but with the requisite reduction in computational costs. In our discussion we shall concentrate on the non-quantitative formulation
of the theory, and will leave adjudication of the conflict between qualitative and probabalisitic methodologies for later in this chapter
This is an appropriate place to make an important terminological adjustment. Explanativism and probabalism are not mutually exclusive methodologies. There is a
substantial body of opinion which holds that explanations are instrinsically probabalistic; and that fact alone puts paid to any idea of a strong disjunction. Better that we
characterize the contrast as follows.
An explanationist is one who holds that meeting explanatory targets is
intrinsic to his enterprize and that probabalistic methods either do not conduce to meeting that target or are needed for hitting such targets.
92
CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
Probabilists are the duals of explanationists.
A probabilist is one who holds that, whatever his targets are (including
explanatory targets), they are only meetable by way of a probabalistic
methodology which is intrinsic to the enterprise, or they are best met in
this way.
Peng and Reggia are inference-to-the-best-explanation abductionists, who happen
to think it possible to integrate, without inordinate cost, their informal explanationism
into probability theory [Peng and Reggia, 1990, ch. 4 and 5]. The main target of their
theory are procedures for the derivation of plausible explanations from the available
data [Peng and Reggia, 1990, p. 1].
Parsimonious covering theory is a theoretical foundation for a class of diagnostic
techniques described by association based abductive models. An associative (or semantic) network is a structure made up of nodes (which may be objects or concepts or
events) and links between nodes, which represent their interrelations or associations.
Associative models are evoked by two basic kinds of procedure: (1) the deployment of
symbolic cause-effect associations between nodes, and a recurring hypothesize-andtest process. Association-based models include computer-aided diagnostic systems
such as INTENIST-1 (for internal medicine; Pople [1975] and Miller, Pople and Meyers [1982]); NEUROLOGIST (for neurology; Catanzarite and Greenburg [1979]); PIP
(for edema: Pauker, Gorry, Kassirer and Schwarz [1976]); IDT (for fault diagnosis
of computer hardware; Shubin and Ulrich [1982]), and domain-free systems such as
KMS.HT (Reggia [1981]); MGR (Coombs and Hartley [1987]) and PEIRCE (Punch,
Tanner and Josephson [1986]).
Given one or more initial problem features, the inference mechanism generates a
set of potential plausible hypotheses of ‘causes’ which can explain the given problem
features’ (Peng and Reggia [1990], 20). Thus associative networks involve what we
call a logic of discovery. Once the hypotheses have been generated, they are tested
in two ways. They are tested for explanatory completeness; and they are tested for
their propensity to generate new questions whose answers contribute to the selection
of a given hypothesis from its alternatives. This hypothesize-and-test cycle is repeated,
taking into account new information produced by its predecessor. This may occasion
the need to update old hypotheses; and new ones may be generated
Association-based abduction differs from statistical pattern classification, as well
as from rule-based deduction. In statistical pattern classification, the inference mechanism operates on prior and conditional probabilities to generate posterior probabilities.
In rule-based deduction, the inference mechanism is deduction which operates on conditional rules. In the case of association-based abduction, the hypothesize-and-test
mechanism operates on semantic networks. Both statistical pattern classification and
rule-based deduction have strong theoretical foundations—the probability calculus in
the first instance, and first order predicate logic in the second. The association-based
abductive approach has not had an adequate theoretical foundation. Furnishing one is
a principal goal of the parsimonious cover theory (PCT)
PCT is structured as follows. In the category of nodes are two classes of events or
states of affairs called disorders D and manifestations M. In the category of links is a
5.2. ANOTHER EXAMPLE OF EXPLANATIONIST DIAGNOSTICS
93
causal relation on pairs of disorders and manifestations. Disorders are sometimes directly observable, and sometimes not. However in the context of a diagnostic abduction
problem disorders are not directly scrutinizable and must be inferred from the available
manifestations. Manifestations in turn fall into two classes: those that are present and
those that are not. ‘Present’ here means ‘directly available to the diagnostician in the
context of his present diagnostic problem’. The class of present manifestations is denoted by FG
T U V AU W diagnostic
U YZ problem P is an ordered quadruple H IJ KLJ MNJ KOGQP , where ISR
J J X X X J T [V is[Wa finite,[non-empty
set of objects or states of affairs called disorY Z
J JXXXJ
ders, KSR
is a finite, non-empty set of objects or states of affairs
called manifestations, and M]\^I`_aK is a relation whose domain is D and whose
range is K , and I`_K is the Cartesian product of I and K . M is called causality.
KOG is a distinguished subset of K which isU csaid to[bed present U c
For any diagnostic problem b , andU c any and [, d effects ( ) is the set of objects
or states of affairs directly caused by
[ed and causes ( ) is the set of objects[ord states
of affairs which can directly cause
. It is expressly provided that a given
might
have alternative and even incompatible,
possible
causes.
This
leaves
it
open
that
P
may
[d
produce a differential diagnosis of an
Uc
The effects of a set of disorders effects (I1f ) is g
effects ( ), the union of all
effects
Uc
[d
h i j kl
. Likewise, the causes of a set of manifestations causes Kam is
[d
(
g
no j p!q
causes
), the union of all causes ( )
The set I1f!\"I is a cover of KAm\"K if Kam\ effects (I1f ). Informally, a cover
of a set of manifestations is what causally accounts for it
A set r]\&I is an explanation of KOG with respect to a problem P if and only
if E covers KOG and E meets a parsimony requirement. Intuitively, parsimony can be
thought of as minimality, or non-redundancy or relevance
A cover I$f of Kam is minimum if its cardinality is the smallest of all covers of KAm .
A cover of Kam is non-redundant if none of its proper subsets is a cover of KAm , and is
redundant otherwise. A cover I1f of KOG is relevant if it is a subset of causes KOG ,
and otherwise is irrelevant
Finally, the solution Sol(P) of a diagnostic problem P is the set of all explanations
of KOG
Two especially important facts about diagnostic problems should be noted
Explanation Existence Theorem. There exists at least one explanation for
any diagnostic problem.
Competing Disorders Theorem. Let E be an explanation for KOG , and let
UV
K(Gt
U Vs
UW
U V
U W
u
I . Then
effects
and
U W ( ) \LK(Gts effects( ) forU Vsome
u
(1)
and
are not both in E; and (2)
if
E
then
there
is
another
UW
UV
explanation E* for KOG which contains
but not
and which is of the
same cardinality or less.
k
Y
Cover Taxonomy Lemma. Let v be the power set of D, and let w n
x , w x ,
wy z
and w
x
be, respectively, the set of all minimal covers, the set of all non-
94
CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
redundant
covers, the set of all relevant covers, and the set of all covers of
{O|
for a given diagnostic problem P. Then }$~"€

~"‚ 
~"
~"ƒ „
We now introduce the concept of a generator. Let … † ‡ … ˆ ‡ ‰ ‰ ‰ ‡ … ‚ be non-empty pairwise
disjoint subsets of D. Then ŠN‹"ŒŽ … † ‡ … ˆ ‡ ‰ ‰ ‰ ‡ … ‚  is a generator. The class  Š!‹ 
generated by ŠN‹ is   ‘ † ‡ ‘ ˆ ‡ ‰ ‰ ‰ ‡ ‘ ‚$ ‘ ’ “ … ’ ‡ ”N•A–•A—˜
Here is an informal illustration of how Parsimonious Cover Theory deploys these
structures. We consider the well-known example of a chemical spill diagnosed by
KMS.HT, a domain-independent software program for constructing and examining abductive diagnostic problem-solving
The problem is set as follows: A river has been chemically contaminated by an
adjacent manufacturing plant. There are fourteen different kinds of chemical spills
capable of producing this contamination. They include sulphuric acid, hydrochloric
acid, carbonic acid, benzene, petroleum, thioacetamide and cesmium. These constitute
the set of all possible disorders ‘ ’ . Determination of the type of spill is based on the
following factors: pH of water (acidic, normal or alkaline), colour of water (green or
brown normally, and red or black when discoloured), appearance of the water (clear
or oily) radioactivity, spectrometry results (does the water contain abnormal elements
such as carbon, sulphur or metal?), and the specific gravity of the water. M is the set of
all abnormal values of these measurements
The diagnostician is called the Chemical Spill System, CSS. When there is a spill
an alarm is triggered and CSS begins collecting manifestation data. CSS’s objective is
to identify the chemical or chemicals involved in the spill
The knowledge base for CSS includes each type of spill that might occur, together
with their associated manifestations. For example, if the ‘ ’ in question is sulphuric
acid, the knowledge base reflects that spills of sulphuric acid are possible at any time
during the year, but are more likely in May and June, which are peak periods in the
manufacturing cycle. It also tends to make the water acidic, and spectrometry always detects sulphur. If the spilled chemical were benzene, the water would have
an oily appearance detectable by photometry; and spectrometry might detect carbon.
If petroleum were the culprit then the knowledge system would indicate its constant
use, heaviest in July, August and September. It might blacken the water and give it an
oily appearance, and it might decrease the water’s specific gravity. It would also be
indicated that spectrometry usually detects carbon
KMS.HT encodes this information in the following way.
Sulphuric Acid
[Description:
Month of year = May ™ š› , June ™ š›
pH = Acidic ™ š›
Spectrometry results = Sulphur ™ œ › ]
Benzene
[Description:
Spectrometry results = Carbon
Appearance = Oily ™ › ]
™ ›
5.2. ANOTHER EXAMPLE OF EXPLANATIONIST DIAGNOSTICS
Petroleum
[Description:
Month of year = July ž Ÿ , August ž Ÿ , September
Water colour = Black ž ¡ Appearance = Oily ž ¡ Spectrometry results = Carbon ž Ÿ Specific Gravity of Water = Decreased ž ¡ ]
95
ž Ÿ Here a is always, b is high likelihood, m is medium likelihood, l is low likelihood and n
is never. It is easy to see that the knowledge base provides both causal and noncausal
information. For example, that sulphuric acid has a manifestation pH = acidic is causal
information, whereas the information that sulphuric acid use is especially high in May
and June is important, but it does not reflect a causal association between sulphuric acid
and those months. Information of this second kind reports facts about setting factors,
as they are called in KMS.HT.
The set effects (¢ £ ) is the set of all manifestations that may be caused by disorder
¢ £ , and the set causes (¡¤ ) is the set of disorders that may cause manifestation ¡¤ . In
particular, effects (Sulphuric Acid) = ¥ pH = Acidic, Spectrometry Results = Sulphur ¦ ,
and causes (pH = Acidic) = ¥ Benzenesulphuric Acid, Carbonic Acid, Sulphuric Acid ¦ .
Since there are fourteen disorders in this example, there are § ¨ © possible sets of disorders, hence § ¨ © possible hypotheses. However, for reasons of economy, CSS confines
its attention to sets of disorders that meet a parsimony constraint. If we chose minimality as our parsimony constraint, then a legitimate hypothesis must be a smallest cover
of all present manifestations ªO« (a cover is a set of disorders that can causally account
for all manifestations.) The objective of the CSS is to specify all minimum covers of
ªO« .
When a manifestation ¡e¤ presents itself it activates the causal network contained
in the system’s knowledge base. More particularly, it evolves all disorders causally associated with ¡e¤ . That is to say, it evokes causes (¡e¤ ). These disorders combine with
already available hypotheses (which are minimum covers of previous manifestations)
and new hypotheses are formed with a view to explaining all the previous manifestations together with the new one ¡e¤ .
The solution to P is the set of all minimal covers. For reasons of economy, it is
desirable to produce the solution by way of generators.
As we saw a generator is a set of non-empty, pairwise disjoint subsets of D. Let the
subsets of D be A, B, and C. Then form the set of all sets that can be formed by taking
one element from each of the subsets A, B and C. The set of those sets is a generator on
¥­ A, B, C¦ . If the ith set in a generator contains ¬£ disorders, then the generator reflects
£ ¬£ hypothesis, which in the general case is significantly fewer than the complete
list.
CSS now continues with its investigation by putting multiple-choice questions to
the knowledge base. For example, it asks whether the spill occurred in either April,
May, June, July, August, or September. The answer is June. This enables the system
to reject Carbon Isotope, which is never used in June. It then asks whether the pH was
acidic, normal or alkaline, and is told that it was acidic. This gives rise to hypotheses in
the form of a generator that identifies the alternative possibilities as Benzenesulphuric
96
CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
Acid, Carbonic Acid, Hydrochloric Acid, Sulphuric Acid. Each of these is a minimum
cover.
Now there is a question asking whether Metal, Carbon or Sulphur were detected
Spectrometrically. The answer is that Metal and Carbon were detected. This now
excludes Sulphuric Acid because it is always the case that Specgrometry will detect
the presence of Sulphur, and did not do so in this case.
The presence of metal evokes four disorders: Hydroxyaluminum, Cesmium, Rubidium and Radium. None of these occurs in the present generator. So to cover the
previous manifestation (acidic pH) and the new manifestation, CSS must produce new
hypotheses involving at least two disorders.
The presence of Carbon, evokes six disorders: Carbonic Acid, Benzene, Petroleum,
Benzenesulphuric Acid, Thioacetamide and Chromogen. Carbonic Acid and Benzenesulphuric Acid are already in the present generator, so they cover both pH = Acidic and
Spectrometry = Carbon. Integrating these into the set of four new hypotheses, gives
eight minimum covers for three existing manifestations. These in turn give a generator
of two sets of competing hypotheses. These are ® Carbonic Acid, Benzenesulphuric
Acid ¯ and ® Radium, Rubidium, Cesmium, Hydroxyaluminum ¯ .
The system now asks whether Radioactivity was present. The answer is Yes. This
evokes four disorders: Sulphur Isotope, Cesium, Rubidium and Radium. Each hypothesis to date involves one of these disorders, except for the Hydroxyaluminum hypothesis (which is not radioactive). Thus Hydroxyaluminum is rejected and a new generator
is formed. It contains six minimum covers for the current manifestation
The system asks whether Specific Gravity was normal, or increased or decreased.
The answer is that it was increased. This evokes four disorders: Hydroxyaluminum,
Cesmium, Rubidium and Radium, which are also evoked by the radioactivity answer.
Accordingly, the solution to the chemical spill problem is given by a generator containing two sets of incompatible alternatives: ® Carbonic Acid, Benzenesulphuric Acid ¯
and ® Radium, Rubidium, Cesmium ¯ .
CSS is a very simple system. (A typical search space involves ° ± ² candidates and
higher.) Even so, it is faced with 16,384 distinct sets of disorders. Of these, 91 are
two-disorders sets. But CSS identifies the six plausible hypotheses (where plausibility
is equated with minimality). This is a significant cutdown from a quite large space of
possible explanations. This indicates that PCT does well where the approach discussed
in the previous section tends to do badly, namely, on the score of computational effectiveness. In the engineering of this system, finding plausible inferences just is making
the system computationally effective.
Peng and Reggia point out that for certain ranges of cases non-redundancy is a
more realistic parsimony constraint than minimality, and that in other cases relevancy
would seem to have the edge. (Peng and Reggia [1990], 118-120). Nonredundancy
was a condition on Aristotle’s syllogism, and it is akin to Anderson and Belnap’s fulluse sense of relevance and of a related property of linear logic. In standard systems of
relevant logic a proof ³ of ´ from a set of hypotheses µ is relevant iff ³ employs
every member of µ . Relevance is not nonredundancy, however; a relevant proof might
have a valid subproof, since previous lines can be re-used. If the logic is linear, all
members of µ must be used in ³ , and none can be re-used. Peng and Reggia’s nonredundancy is identical to Aristotle’s: No satisfactory nonredundant explanation can have
5.3. COHERENTISM AND PROBABILISM
97
a satisfactory proper sub-explanation. Relevance for Peng and Reggia is restricted to
causal relevance, and so captures only part of the sense of that capacious concept. If
we allow nonredundancy as a further sense of relevance, then the following structural
pattern is discernible in the Peng-Reggia approach. One wants one’s explanations to
be plausible. Plausible explanations are those produced parsimoniously. Parsimony is
relevance, in the sense of nonredundancy. Thus relevance is a plausibility-enhancer.
In later chapters, we will suggest that relevance and plausibility combine in a different
way from that suggested here.
5.3
Coherentism and Probabilism
5.3.1 The Rivalry of Explanationism and Probabilism
We briefly characterized explanationism and probabilism in the previous section. As
we say, they are perhaps not wholly natural rivals. There are ways of being an explanationist which involves no resistance to probabilistic methods, and there are ways of
being a probabilist which are wholly compatible with some ways of being an explanationist. For example, one way of being an explanationist about abduction is to insist
that all abductive inference is inference to the best explanation. But there is nothing in
this that precludes giving a probabilistic account of inference to the best explanation.
Equally, someone could be a probabilist about inference in general and yet consistently
hold that explanation is intrinsic to abudctive inference.
Even so, there are explanationists who argue that probabilistic methods are doomed
to fail or, at best to do an inadequate job. If this is right, then knowing it endows the
enquirer into abduction with some important guidance with regard to the resourcemanagement aspects of his programme. It is negative guidance: Do not give to the
calculus of probability any central and load-bearing place in your account of abductive
reasoning
In this section we examine one way of trying to make good on this negative advice.
In doing so, we develop a line of reasoning which we find to have been independently
developed in Thagard [2000]. Since our case heavily involves Thagard himself, and
pits him against the same position that Thagard also examines, we shall not give a
fully-detailed version of our argument, since the case we advance is already set out in
Thagard [2000]
Coherentism. Like the approach of Peng and Reggia, Thagard sees abductive inference as a form of causal reasoning [Thagard, 1992]. It is causal reasoning triggered
by surprising events (again, a Peircean theme), which, being surprising call for an explanation. Explanations are produced, by hypothisizing causes of which the surprising
events are effects
Thagard understands explanationism with respect to causal reasoning to be qualitative, whereas probabilism proceeds quantitatively. In recent years, both approaches
have been implemented computationally. In Thagard [1992], a theory of explanatory coherence is advanced. It can be implemented by ECHO, a connectionist program which computes explanatory coherence in propositional networks. Pearl [1988]
98
CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
presents a computational realization of probabilistic thinking. These two styles of computational implementation afford an attractive way of comparing the cognitive theories
whose implementations they respectively are. The psychological realities posited by
these theories can be understood in part by way of the comparison relations that exist between ECHO and probabilistic realizations. As it happens, there is a probabilistic
form of ECHO, which shows not only that coherentist reasoning is not logically incompatible with probabilistic reasoning, but that coherentist reasoning can be considered
a special case of probabilistic reasoning. What makes it so is that ECHO’s inputs can
be manipulated to form a probabilistic network which runs the Pearl algorithms. Coherentism is not the only way of being an explanationist, and Pearl’s is not the only
way of being a probabilist. But since the computer implementations of these two particular theories are more advanced than alternative computer models, comparing the
Thagard and the Pearl computerizations is an efficient way to compare their underlying
explanationist and probabilist theories as to type
The chief difference between explanationist and probabilist accounts of human reasoning lies in how to characterize the notion of degrees of belief. Probabilists hold
that doxastic graduations can be described in a satisfactory way by real numbers under conditions that satisfy the principles of the calculus of probability. Probabilism is
thus one of those theories drawn to the idea that social and psychological nature has a
mathematically describably structure, much in the way that physics holds that physical
reality has a mathematical structure. Explanationism, on the other hand, is understood
by Thagard to include the claim that human causal reasoning, along with other forms
of human reasoning, can be adequately described quantitatively. It rejects the general
view that reasoning has a mathematical structure, and it rejects the specific view that
reasoning has a structure describable by the applied mathematics of games of chance.
Disagreements between explanationists and probabilists are not new; and they precede the particular differences that distinguish Thagard’s theory from Pearl’s. Probabilism has attracted two basic types of objection. One is that the probabilistic algorithms
are too complex for human reasoners to execute (Harman [1986]). The other is that experimental evidence suggests the descriptive inadequacy of probablistic accounts (Kahneman, Slovic and Tversky [1982]), a score on which explanationist accounts do better
(Read and Marcus-Newhall [1993], Schank and Ranney [1991] [1992], and Thagard
and Kunda [1998]; see also Thagard [2000], 94). The importance of the fact that coherentist reasoning can be seen as a special case of probabilistic reasoning is that it
suggests that there may be a sense in which the probabilistic approach is less damaged
by these basic criticisms than might otherwise have been supposed. There is ample
reason to think that the sheer dominance of human reason is evidence of the presence
of the Can Do Principle, which we discussed in section 3.3.1. A theorist comports
with the Can Do Principle if, in the course of working on a problem belonging to a
discipline D, he works up results in a different discipline D*, which he then represents
as applicable to the problem in D. The principle is adversely triggered when the theorist’s attraction for D* is more a matter of the ease or confidence with which results
in D* are achieved, rather than their well-understood and well attested to applicability
to the theorist’s targets in D. (A particularly dramatic example of Can Do at work, is
the decision by neoclassical economists to postulate the infinite divisibility of utilities,
because doing so would allow the theory to engage the fire-power of the calculus.)
5.3. COHERENTISM AND PROBABILISM
99
5.3.2 Explanatory Coherence
The conceputal core of the theory of explanatory coherence (TEC) is captured by the
following qualitative principles [Thagard, 1989; Thagard, 1992; Thagard, 2000].
1. Symmetry. Unlike conditional probability, the relation of explanatory coherence
is symmetric.
2. Explanation. If a hypothesis explains another hypothesis, or if it explains the evidence, then it also coheres with them and they with it. Hypotheses which jointly
explain something cohere with each other. Coherence varies inversely with the
number of hypothese used in an explanation. (cf. the minimality condition of
Peng and Reggia [1990].)
3. Analogy. Similar hypotheses that explain similar bodies of evidence cohere with
one another.
4. Observational Priority. Propositions stating observational facts have a degree of
intrinsic acceptability.
5. Contradiction. Contradictory propositions are incoherent with each other. (It
follows that the base logic for TEC must be paraconsistent lest an episodic inconsistency would render any explanatory network radically incoherent.)
6. Competition. If both P and Q explain a proposition, and if P and Q are not
themselves explanatorily linked, then P and Q are incoherent with each other.
(Note that explanatory incoherence is not here a term of abuse.)5
7. Acceptance. The acceptability of a proposition in a system of propositions depends on its coherence with them.
TEC defines the coherence and explanation relations on systems of propositions.
In ECHO propositions are represented by units (viz., artificial neurons). Coherence
relations are represented by excitatory and inhibitory links.
ECHO postulates a special evidence unit with an activation value of 1. All other
units have an initial activation value of 0. Activation proceeds from the special evidence unit to units that represent data, thence to units that represent propositions that
explain the data, and then to units representing propositions that explain propositions
that explain data; and so forth. ECHO implements principle (7), the Acceptability rule,
by means of a connectivist procedure for updating the activation of a unit depending
on the units to which it is linked. Excitatory links between units cause them to prompt
their respective activation. Inhibitory links between units cause them to inhibit their
respective activation.
Activation of a unit ¶ · is subject to the following constraint:
¶ · ¸ ¹ºt» ¼½t¶ · ¸ ¹ ¼ ¸ ¹˜¾a¿ ¼ À
5 Propositions
net· (max-a·
¸ ¹ ¼ if netÁ1Â"Ã
, otherwise net·
¸ ¶ · ¸ ¹ ¼ -min))
are explanatorily linked if one explains the other or jointly they explain something else.
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CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
in which d is a decay parameter (e.g., 0.5) that decrements every unit at each turn of
the cycle; j is the minimum activation (-1); max is maximum activation (1). Given the
weight wÄ Å the net input to a unit (netÅ ) is determined by
netÅQÆÈÇ
Ä É Ä Å Ê Ä (t).
In ECHO links are symmetric, but activation flow is not (because it must originate in the evidence unit). ECHO tolerates the presence of numerous loops, without
damaging the systems ability to make acceptability computations.
Updating is performed for each unit u Ä . For this to happen the weight of the link
between u Ä and uÅ (for any uÅ to which it is linked) must be available to it; and the
same holds for the activation of uÅ . Most units in ECHO are unlinked to most units.
But even if ECHO were a completely connected system, the maximum number of
links with n units would be n ËNÌ . Updating, then, requires no more than n*(n Ë!Ì )
calculations. However, uncontrolled excitation (the default weight on excitatory links)
gives rise to activation oscillitations which preclude the calculation of acceptability
values. Thagard reports experiments which show high levels of activation stability
when excitation, inhibition and decay are appropriately constrained. Accordingly, the
default value for excitation is 0.4; for inhibition (the default value of inhibitory links)
the value is -0.6; and 0.5 for decay [Thagard, 2000, p. 97]. Thagard also reports [2000,
p. 97] experimental evidence which indicates that ECHO’s efficiency is not much
influenced by the size or the degree of connectivity of networks. Larger networks with
more links do not require any systematic increase in work required by the system to
make its acceptability calculations.
5.3.3
Probabilistic Networks
A clear advantage of probabilistic approaches to reasoning have over explanationist
approaches is a precise theoretical vocabulary, a clear syntax, and a well-developed
semantics. Degrees of belief are associated with members of the [0-1] interval of the
real line, subject to a few special axioms. A central result of this approach in Bayes’
ÍÎ Ï ÐÑ Ò Ó ÍÎ Ï ÐÓÔÍ˜Î Ï Ò Ñ ÐÓ
theorem:
Æ
ÍÎ Ï Ò Ó
The theorem provides that the probability of a hypothesis on a body of evidence is
equal to the probability of the hypothesis alone multiplied by the probability of the
evidence given the hypothesis, divided by the probability of the evidence alone. The
theorem has a certain intuitive appeal for the abduction theorist, especially with regard to the probability of the evidence relative to the hypothesis, which resembles the
abduction schema of 1.1 in an obvious way. The Bayesian approach to human reasoning is, however, computationally very costly (Harman [1986] and Woods and Walton
[1972]). Probabilistic updating requires the calculation of conjunctive probabilities,
the number of which grow exponentially with the number of conjuncts involved. Three
propositions give rise to eight different probability alternatives, and thirty would involve more than a billion probabilities [Harman, 1986, p. 25]. It is also possible that
coherence maximization is computationally intractable, but provided the system has
5.3. COHERENTISM AND PROBABILISM
101
the semidefinite programming algorithm, TEC is guaranteed that the system’s optimality shortfall will never be more than 13 percent [Thagard and Verbeurgt, 1998]. In their
turn, Pearl’s network, in common with many other probabilistic approaches, greatly
reduces the number of probabilities and probability determinations by restricting these
determinations to specific classes of dependencies. If, for example, Y depends wholly
on X and Z depends wholly on Y, then in calculating the probability of Z, the probability
of X can be ignored, even though X,Y and Z form a causal network.
In Pearl’s network, nodes represent multi-valued variables, such as body temperature. In two-valued cases, the values of the variable can be taken as truth and falsity.
ECHO’s nodes, on the other hand represent propositions, one each for a proposition
and its negation. In Pearl networks edges represent dependencies, whereas in ECHO
they represent coherence. Pearl networks are directed acyclic graphs. Its links are antisymmetric, while those of ECHO are symmetric. In Pearl networks the calculation of
probabilities of a variable D is confined to those involving variables causally linked to
D. Causally independent variables are ignored. If, for example, D is a variable whose
values are causally dependent on variables A, B, and C, and causally supportive of the
further variables E and F, then the probabilities of the values of D can be taken as a
vector corresponding to the set of those values. If D is body temperature and its values
are high, medium and low, then the vector (.6 .2 .1) associated with D reflects that the
probability of high temperature is .6, of medium temperature is .2, and of low temperature is .1. If subsequent measurement reveals that the temperature is in fact high, then
the associated vector is (1 0 0). If we think of A, B and C as giving prior probabilities for the values of D, and of E and F as giving the relevant observations, then the
probability of D can be calculated by Bayes’ Theorem.
For each of Pearl’s variables X, ‘(x)’ denotes the degrees of belief calculated for
X with regard to each of its values. Accordingly, BEL(x) is a vector with the same
number of entries as X has values. BEL(x) is given by the following equation:
BEL(x) ÕtÖa×ØÙ
ÚÛ
×eÜÙ ÚÛ
in which Ö is a normalization constant which provides that the sum of the vector entries is always 1. Ø (x) is a vector representing the support afforded to values of X by
variables that depend on X. Ü (x) is a vector representing the support lent to values of X
by variables on which X depends.
It is known that the general problem of inference in probabilistic networks is NPhard [Cooper, 1990]. Pearl responds to this difficulty in the following way. He considers the case in which there is no more than one path between any two nodes, and
produces affordable algorithms for such systems [Pearl, 1988, ch. 4]. If there is more
than one path between nodes, the system loops in ways that destabilizes the calculation of BEL’s values. In response to this difficulty, procedures have been developed for
transforming multiply-connected networks into uni-connected networks. This is done
by clustering existing nodes into new multi-valued nodes
Pearl [1988, ch. 4], discusses the following case. Metastatic cancer causes both
serum calcium and brain tumor, either of which can cause a coma. Thus there are two
paths between metastatic cancer and coma. Clustering involves replacing the serum
calcium node and the brain tumor node into a new node representing variables whose
values are all possible combination of values of the prior two, viz.: increased calcium
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CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
and tumor; increased calcium and no tumor; no increased calcium and tumor; and no
increased calcium and no tumor. Clustering is not the only way of dealing with loops in
probabilistic networks. Thagard points out [2000, p. 101] that Pearl himself considers
a pair of approximating techniques, and that Lauritzen and Spiegelharter [1988], have
developed a general procedure by transforming any directed acyclic graph into a tree
of the graph’s cliques. Hrycej [1990] describes approximation by stochastic simulation
as a case of sampling from the Gibbs distribution in a random Markov field, Frey
[1998] also exploits graph-theoretic inference pattern in developing new algorithms
for Bayesian networks.
Recurring to the earlier example of D depending on A, B and C, and E and F depending on D, D’s BEL values must be computed using values for A, B, C,E and F only.
To keep the task simple, suppose that a, b, c, d, and e are, respectively, the only values
of these variables. Pearl’s algorithms require the calculation of d’s probability given
all possible combination of values of the variables to which D is causally linked. If we
take the case in which the possible values of D are truth (d) or falsity (not-d), Pearl’s
algorithm requires the calculation of eight conditional probabilities. In the general case
in which D is causally linked to n variables each with k variables, the system will have
to ascertain kÝ conditional probabilities. Where n is large, the problem of computational intractability reappears, and occasions the necessity to employ approximation
procedures. Even if n isn’t large, the question isn’t whether the requisite numbers of
calculation of conditional probabilities can be done by humans, that rather whether
they are done (in that number) by humans.
5.4
Pearl Networks for ECHO
There are important differences between ECHO and Pearl. There are also some significant similarities. If ECHO is faced with rival hypotheses, it will favour the ones
that have greatest explanatory power. The same is true of Pearl networks. ECHO likes
hypotheses that are explained better than those that aren’t. Pearl networks have this
bias, too. Thus there is reason to think that there may be less of a gap between ECHO
and Pearl networks than might initially have been supposed. Accordingly Thagard
discusses a new network, PECHO. PECHO is a program that accepts ECHO’s input
and constructs a Pearl network capable of running Pearl’s algorithms. (For details see
Thagard [2000, pp. 103–108 ].) The theory of explanatory coherence which ECHO
implements via connectionist networks, as well as by the further algorithms in Thagard and Verbeurgt [1998], can also be implemented probabilistically. That alone is a
rather striking result. Some of the older objections to probabilism held that the probability calculus simply misconceived what it is to reason. The integration achieved
by Thagard suggests that the inadequacies which critics see in probabilism need to
be treated in subtler and more nuanced ways. It remains true that PECHO is computationally a very costly network. Vastly more information is needed to run its update
algorithms than is required by ECHO. Special constraints are needed to suppress loops,
and it is especially important that the simulated reasoner not discard information about
co-hypotheses and rival hypotheses. But given that what coherentists think is the right
account of causal reasoning is subsumed by what they have tended to think is the wrong
5.4. PEARL NETWORKS FOR ECHO
103
account of it, computational cost problems occur in a dialectically interesting context.
How can a good theory be a special case of a bad theory? In general when a bad theory
subsumes a good, it is often more accurate to say that what the bad theory is bad at is
what the good theory is good at. This leaves it open that the bad theory is good in ways
that the good theory is not. This gives a context for consideration of computational cost
problems. They suggest not that the probabilistic algorithms are wrong, but rather that
they are beyond the reach of human reasoners.
If one accepts the view of Goldman [1986] that power and speed are epistemological goals as well as reliability then explanationist models can be
viewed as desirable ways of proceeding apace with causal inference which
probabilistic models are still lost in computation [Thagard, 2000, p. 111].
Thagard goes on to conjecture that ‘the psychological and technological applicability of
explanation and probabilistic techniques will vary from domain to domain’ [Thagard,
2000, p. 112]. He suggests that the appropriateness of explanationist techniques will
vary from high to low depending on the type of reasoning involved. He considers the
following list, in which the first member is explanationistly most appropriate, and the
last least:
Social reasoning
Scientific reasoning
Legal reasoning
Medical diagnosis
Fault diagnosis
Games of chance.
We find this an attractive conjecture. It blunts the explanationist complaint that
probabilism is just wrong about reasoning. It suggests that there is something for probabilism to be right about, just as there is something for explanationism also to be right
about. It suggests, moreover, that they are not the same things. The fact that coherentist networks are subcases of probabilistic networks, when conjoined with the fact
that what is indisputably and demonstrably troublesome about probabilistic networks
is their inordinate informational and computational complexity and, relatedly, their
psychological unreality, occasions a further conjecture. Systems that are well-served
by explanationism are systems having diminished information-processing and computational power, and have psychological make-ups that systems for which probabilism
does well don’t have. This, in turn, suggests that the variability which attaches to the
two approaches is not so much a matter of type of reasoning as it is a matter of type of
reasoner. In chapter two, we proposed that cognitive agency comes in types and that
type is governed by levels of access to, and quantities of, cognitively relevant resources:
information, time and computational capacity. If we take the hierarchy generated by
this relation, it is easy to see that it does not preserve the ordinal characteristics of Thagard’s list. So, for example, scientific reasoning will be represented in our hierarchy of
agencies by every type of agent involved in scientific thinking, whether Joe Blow, Isaac
Newton, the Department of Physics at the University of Birmingham, all the science
departments of every university in Europe, NASA, neuroscience in the 1990s, postwar economics, and so on. This makes us think that we need not take our hierarchy
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to be a rival of Thagard’s. Ours is an approach that also explains the dialectical tension represented by the subsumption of ECHO by probabilistic networks. It explains
why explanationism is more appropriate for reasoning by an individual agent, and why
probabilism could be appropriate for an institutional agent. And it explains why it is
unnecessary to convict either account of the charge that it simply doesn’t know what
reasoning is.
On the other hand, even for a given type of agency there can be variation in the
kind of reasoning it conducts. Individual reasoners undertake cognitive tasks for which
social reasoning is appropriate, but they also try their hands at games of chance. A
medical diagnostician need not always operate under conditions of battlefield triage;
sometimes he has plenty of time, an abundance of already-sorted information, as well
as the fire-power of his computer to handle the statistics. What this suggests is that the
fuller story of what model is appropriate to what reasoning contexts will be one that
takes into account the type of agent involved and the type of reasoning he is engaged
in. It is a story which suggests the advisability of criss-crossing a kinds-of-reasoning
grid on a kinds-of-resources grid — of laying a Thagard hierarchy across one such as
our own.
Things are less good, however, when we move from the solo reasoning of agents to
interactive Þ -agent reasoning. It is a transition that bodes less well for the probabilistic
approach than for various others. It is a feature of Thagard’s hierarchy that, whereas
at the bottom objective probabilities are in play, at the top we must do with subjective
probabilities. But subjective probabilities are notorious for their consensual difficulties
in contexts of Þ -agent reasoning. This is an invariant feature of them irrespective of
whether the interacting agents are Joe Blow and Sally Blee or the Governments of
Great Britain and France. Then, too, an individual reasoner may want to reason about
the state of play in Monte Carlo, which is reasoning of a type for which the probabilistic
approach is tailor-made. Even so, the individual gambler cannot run the probabilistic
algorithms willy-nilly.
It remains the case that at least part of the reason that explanationism gets the nod
over probabilism in the case of individual reasoning is that in general individual reasoners are constitutionally incapable in reasoning in the way that probabilism requires,
but is capable of reasoning as coherentism requires. Part, too, of the reason that probabilism is psychologically inappropriate for individuals is that agency-types for which
probabilism is appropriate don’t (except figuratively) have psychologies; hence have
no occasion to impose the constraint of fidelity to psychological reality. When PECHO
took ECHO into probabilistic form, it endowed PECHO-reasoners with probabilized
counterparts of reasonings that an ECHO-reasoner is capable of. But it also endowed
the PECHO-reasoner with two qualities which in the general sense are fatal for the individual reasoner, a capacity to run algorithms on a scale of complexity that removes it
from an individual’s reach, and a hypothetical psychology which no individual human
has any chance of instantiating. Thus the ECHO-reasoner can get most things right
which PECHO can get right, but it can’t get them right in the way that PECHO does.
The heart and soul of this difference is that the way in which the PECHO-reasoner
must operate to get things right involve its having properties which no individual can
have, in fact or in principle. Institutional reasoners are another matter, subject to the
quantifications we have already taken note of.
5.4. PEARL NETWORKS FOR ECHO
105
As it would now appear, an individual’s cognitive actions are performed in two
modalities, i.e., consciously and subconsciously. Bearing in mind the very substantial
difference in information-theoretic terms between consciousness and subconsciousness, this is a good place to remind the reader of the possibility that accounts that are
as computationally expressive as probabilistic models might more readily apply to the
subconscious cognitive processes of individual agents, which appear to be processes
that aren’t informationally straitened in anything like the way of conscious processes.
It is less clear that unconscious cognitive systems have structures that are not all realistically represented as structures in the real line. but it is harder to make a change
of psychological unreality stick against probabilism precisely because the true psychological character of the unconsciousness of individuals is itself hardly a matter of
consensus, to say nothing of theoretical clarity.
It is true that some theorists have no time for the idea of unconscious reasoning (cf.
Peirce’s insistence at Peirce [1931–1958, 5.109 and 2.144] or for unconscious cognition of any sort). Our present suggestion will be lost on those who share this view. On
the other hand, if one’s epistemological orientation is a generally reliabilist one, it is
difficult to maintain that the idea of unconscious reasoning is simply oxymoronic. We
ourselves take the reliabilist stance. If cognition and good reasoning are the products
of processes that are working as they should (i.e., normally, as in Millikan [1984] there
is nothing in this basic idea to which consciousness is either intrinsic or exclusive).
5.4.1 Mechanizing Abduction
It is often rational for a agent to invoke an hypothesis on the basis of its contribution,
if true, to some cognitive end that the agent desires to attain. This is abduction. Abduction is a fallible enterprise, something that even the modestly reflective abducer is
bound to have some awareness of. He will thus call down his hypotheses even in the
teeth of the possibility that they are false. In general, the falsity of the hypothesis does
not wreck the implication which the abducer seizes upon as warranting the inference to
its truth or its likelihood. In many accounts of implication, the falsity of the hypothesis
strengthens the implication to the object of the abducer’s desired outcome. The fallibility of abduction provides for the possibility that the implication that the abducer seeks
to exploit will have the form of a counterfactual conditional:
If H were the case, E would also be the case.
Counterfactuals also crop up in another way. Whether he does so consciously or
not, the abducer is faced with the task of hypothesis selection. Essential to that process
is the cutdown of sets of possibilities to sets of real possibilities; thence to sets of
relevant possibilities; thence to sets of plausible alternatives; and finally, if possible, to
unit sets of these. At each juncture there is an elimination strategy to consider: Does the
candidate H in question imply a falsehood F, never mind that it genuinely gives us E?
If so, then, provided the implication holds, the agent will have reasoned conditionally
in this form:
Even though H is false, it remains the case that were H true then E would
be true.
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We may take it, then, that real-life abducers routinely deploy counterfactual conditionals. A psychologically real account of what abducers do must take this fact into
account.
Computer simulations of what abductive agents do are attempts at producing mechanical models that mimic abductive behaviour. A model gives a good account of
itself to the extent that is mimicry approximates to what actually happens in real-life
abduction. In particular, therefore, such a model works to the extent that it succeeds
in mechanizing counterfactual reasoning. Can it do this? People who are disposed to
give a negative answer to this question are drawn to the following question: What is involved in expressly counterfactual thinking when it is done by real-life human agents?
It appears that the human agent is capable of producing some important concurrences.
For one, he is able to realize that P is true and yet to entertain the assumption that P is
not true, without lapsing into inconsistency. Moreover, the human agent seems capable
of keeping the recognition that P and the assumption that not-P in mind at the same
time. That is, he is able to be aware of both states concurrently. Thirdly, the human
agent is capable of deducing from the assumption of not-P that not-Q without in doing
so contradicting the (acknowledged) fact that Q might well be true.
When the AI theorist sets out to simulate cognitive behaviour of this sort, he undertakes to model these three concurrences by invoking the operations of a finite state
Turning machine. Turning machines manipulate syntax algorithmically; that is, their
operations are strictly recursive. The critic of AI’s claim to mechanize counterfactual
reasoning will argue that no single information processing program can capture all
three concurrences. It may succeed in mimicking the first, in which the agent consistently both assents to P and assumes its negation, by storing these bits of information
in such a way that no subroutine of the program engages them both at the same time.
But the cost of this is that the second concurrence is dishonoured. The human agent is
able consciously to access both bits of information at the same time, which is precisely
what the Turning machine cannot do in the present case.
It is possible to devise a program that will enable the simulation of the first and the
second concurrence. The program is capable of distinguishing syntactically between
the fact that P and the counterfactual assumption that not-P, say by flagging counterfactual conditionals with a distinguished marker, for example ß . Then the program
could have subroutines which has concurrent access to “P” and “ ß not-P ß ”, without
there being any danger of falling into inconsistency. Here, too, there is a cost. It is the
failure of the program to honour the third concurrence, in which it is possible correctly
to deduce “ ß not-Q ß ” from “P”, “ ß not-P ß ” and “if not-P then not-Q”.
Of course, the program could rewrite “If not-P then not-Q” as “If ß not-P ß ’
then ß not-Q ß ’”. From the counterfactual assumption “ ß not-P ß ”, the deduction
of “ ß not-Q ß ” now goes through, and does so without there being any question of an
inconsistency on the deducer’s part.
Still, there is a problem. It is that ß -contexts are intensional. There are interpretations of P and Q for which the deduction of “ ß Q ß ” from “not-P”, “counterfactually
if P then Q” and “ ß P ß ” fails. Thus it is possible to assume counterfactually that
Cicero was a Phoenician fisherman, and that if Cicero was a Phoenician fisherman,
then Tully was a Phoenician fisherman, without its following that I assume that Tully
was a Phoenician fisherman. The notation “ ß Q ß ” expresses that Q is assumed. As-
5.4. PEARL NETWORKS FOR ECHO
107
sumption is an opaque context [Quine, 1960], hence a context that does not sanction
the intersubstitution of co-referential terms or logically equivalent sentences. (See here
[Jacquette, 1986].) Thus à -inference-routines are invalid. Their implementability by
any information processing program that, as a finite state Turing machine must be, is
strictly extensional dooms the simulation of counterfactual reasoning to inconsistency.
We should hasten to say that there are highly regarded efforts to mechanize reasoning involving counterfactual or belief-convening assumptions. Truth-maintenance
systems (TMS) are a notable case in point. ([Rescher, 1964], [Doyle, 1979]; see also
[de Kleer, 1986], [Gabbay et al., 2001] and [Gabbay et al., 2002].) The main thrust
of TMSs is to restore (or acquire) consistency by deletion. These are not programs
designed to simulate the retention of information that embeds belief-contravening assumptions and their presentation to a uniformly embracing awareness. The belief that
P is not inconsistent with the concurrent assumption that not-P. There is in this no occasion for the consistency-restoration routines of TMS. Thus à -contexts resemble contexts of direct quotation. Such are contexts that admit of no formally sound extensional
logic [Quine, 1960; Quine, 1976]. No strictly extensional, recursive or algorithmic
operations on syntax can capture the logic of counterfactual reasoning. Whereupon
goodbye to a finite state Turning machine’s capacity to model this aspect of abductive
reasoning.
Named after the German word for assumption, ANNAHMEN is a computer program adapted from Shagrin, Rapaport and Dipert [1985]. It is designed to accommodate hypothetical and counterfactual reasoning without having to endure the costs of either inconsistency or the impossibility of the subject’s access to belief-contravening assumptions and the beliefs that they contravene. ANNAHMEN takes facts and counterfactual assumptions and conditionals as input. The latter two are syntactically marked
in ways that avoid syntactic inconsistency.
This input is then copied and transmitted to a second memory site at which it is
subject to deduction. The previous syntactic markers are renamed or otherwise treated
in ways that give a syntactically inconsistent set of sentences. The next step is to
apply TMS procedures in order to recover a consistent subset in accordance with an
epistemic preference-heuristic with which the program has been endowed. In the case
before us, the TMS is Rescher’s logic of hypothetical reasoning, or as we shall say,
the Rescher reduction. From this consistent subset the counterfactual conclusion is
deduced by a Lewis logic for counterfactuals and syntactic markers are re-applied.
Then all this is sent back to the original memory site. It mixes there with the initial
input of beliefs and belief-contravening assumptions. ANNAHMEN can now perform
competent diagnostic tasks and can perform well in a Turing test [Turing, 1950]. As
Jacquette [2001] observes,
the functions RESCHER REDUCTION and LEWIS LOGIC call procedures for the Rescher-style reduction of an inconsistent input set to a logically consistent subset according to any desired extensionally definable set
of recursive or partially recursive heuristic, and for any desired logically
valid deductive procedure for detaching counterfactual conditionals, such
as David Lewis’ formal system in Counterfactuals [Lewis, 1973].
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The problem posed by the mechanization of counterfactual reasoning is that there
appeared to be no set of intensional procedures for modelling such reasoning which
evades syntactic inconsistency and which allows for what Jacquette calls the “unity of
consciousness” of what is concurrently believed and contraveningly assumed.. ANNAHMEN is designed to show that this apparent problem is merely apparent. The
solution provided by this approach is one in which the inconsistency that occurs at
memory site number two exists for nanoseconds at most and occurs, as it were, subconsciously. Thus counterfactual reasoning does involve inconsistency. But it is a
quickly eliminable inconsistency; and it does not occur in the memory site at which
counterfactual deductions are drawn. Inconsistency is logically troublesome only when
harnessed to deduction. It is precisely this that the ANNAHMEN program precludes.
It may also be said that the program is phenomenologically real. When human beings infer counterfactually, they are aware of the concurrence of their beliefs and their
belief-contravening assumptions, but they are not aware of the presence of any inconsistency. (Rightly, since the counterfactual inference is performed “at a site” in which
there is no inconsistency.)
The ANNAHMEN solution posits for the reasoning subject the brief presence of
an inconsistency that is removed subconsciously. It is some interest to the present
authors that the program implements the operation Putter-of-Things-Right of [Gabbay
and Woods, 2001b]. This is a device postulated for the human information processor.
What makes the ANNAHMEN proposal especially interesting in this context is that, in
effect, it purports to show that Putter-of-Things-Right is mechanizable.
Whether it is or not, we find ourselves in agreement with Jacquette in the case
of an ANNAHMEN approach to counterfactual reasoning. Jacquette shows that while
ANNAHMEN handles certain types of counterfactual reasoning, it fails for other types.
Further even though certain refinements to the ANNAHMEN protocols, in the manner
of Lindenbaum’s Lemma for Henkin-style consistency and completeness proofs or in
the manner of the Lemma for consistent finite extensions of logically consistent sets,
resolve some of these difficulties; they cannot prevent others [Tarski, 1956; Henkin,
1950]. We side with Jacquette in thinking that there “is no satisfactory extensional
substitute for the mind’s intentional adoption of distinct propositional attitudes toward
beliefs and mere assumptions or hypothesis”. We shall not here reproduce details of
Jacquette’s criticisms; they are well-presented in [Jacquette, 2001].
Our more immediate interest is in the recurring question of whether the logic of
down below is plausibly considered logic. In section ?? we briefly noted treatments
of abductive insight by connectionist models of prototype activation at the neurological level. (See again [Churchland, 1989; Churchland, 1995; Burton, 1999].) We were
drawn to the connectionist model because it seems to capture important aspects of abductive behaviour at subconscious and prelinguistic levels. If this is right, there are
crucial aspects of abductive practice which on pain of distortion cannot be represented
as the conscious manipulation of symbols. As we now see, the manner in which ANNAHMEN is thought to fail bears on this issue in an interesting way. At present there is
no evidence that conscious mental functions such as memory and desire, and even consciousness itself, has a unified neurological substructure [Kolb and Whishaw, 2001]. If
one subscribes to an out and out connectionist materialism with respect to these matters, we would have it that the phenomenological experience of a unified consciousness
5.4. PEARL NETWORKS FOR ECHO
109
is an illusion. If that were so, then it would hardly matter that a mechanized logic of
counterfactual reasoning is incompatible with the unity of consciousness.
Even if we allowed that phenomenologically unified manifestations of consciousness were not always or in all respects illusory, we have already seen in section ?? that
there are strong information-theoretic indications that the conscious mind is neither
conscious enough nor efficient enough for the burdens of a human subject’s total cognitive agenda. In the face of mounting evidence that substantial and essential aspects
of cognition operate down below, it seems an unattractive dogmatism to refuse to logic
any purchase there. Add to that, that plausible mechanization of cognitive processes
such as hypothetical reasoning require that we postulate subconscious and prelinguistic
performance in a way that downgrades the role of a unified consciousness, not only is
the logic of down below given some encouragement, but so to is its algorithmic character. So we think that it must be said by everyone drawn, for reasons of the sort we have
been examining, to the logic of down below is in all consistency pledged to reconsider,
with more favour than proposed by researchers such as Jacquette, the plausibility of
mechanical models of abductive practice.
110
CHAPTER 5. A SAMPLER OF DEVELOPMENTS IN AI
Chapter 6
Further Cases
6.1
The Burglary
6.1.1 Setting the Problem: The Logic of Discovery
As we have in our discussion of the Cut Down Problem, in chapter 2, it is natural to
think of abduction problems as involving what we shall call candidate spaces and resolution procedures. Relatedly, it is also natural to suppose that an abductive logic for
the class of problems we are interested in here will involve a triple á â1ã ä
ã åNæ , where â
is an explanation logic, ä a plausibility logic and å a relevance logic. We have already
made the point that although both relevance and plausibility are constituent features
of what we have been calling filtration structures, there is no empirical evidence that
real-life abducers select their hypotheses by constructing or even examining filtration
structures. Even so, it does appear that at least for large classes of abduction problems,
both relevance and plausibility are intrinsic to adequate solutions; and that for explanationist abduction problems, the same is true of the concept of explanation. We shall
develop these ideas by examining the following test case which, for reasons that will
quickly become evident, we call The Burglary.
It is late afternoon on an overcast November Thursday. John arrives
home at his usual time, and parks his car in the garage at the end of the
back garden. Carol, his school-teacher wife, always comes home about an
hour later. John’s practice is to enter and leave the house through the back
door. But since the garage is only big enough for a single car, Carol parks
in the street in front of the house, and enters and leaves the house through
the front door.
Dora helps with the housework every Tuesday morning. Apart from
her, no-one but Carol and John have occasion to be in the house or the
means to gain entry; the couples’ children are adults and have left the
family home long since.
Having parked his car, John leaves the garage and makes his way along
the path to the back door. He hasn’t taken many steps before he sees that
111
CHAPTER 6. FURTHER CASES
112
the back door is wide open.
The Explanation Logic
The open door is the trigger of an abduction problem in process. We also stipulate
that the trigger was counterexpected for John. It is not just that ‘the door is [or will
be] open’ was not in John’s database before the trigger presented itself; rather the
database contained at that time the generic fact that the back door is never open under
these conditions (weekday, end of the day, before Carol arrives home, when Dora’s not
there, etc.).1 The case, therefore, incorporates the Peircean element of surprise.
The immediate significance of the counterexpectedness of the trigger is that it gets
John’s attention and constitutes a problem for him. It is far from universally the case
that the presence of counterexpected situations is a problem for those who discover
them. So we must provide that John’s database at the moment of discovery contains
not only, ‘The door is never open’, but also, ‘The door is not supposed to be open’. We
now have the means to motivate the open door as problematic for John.
If this is not immediately clear, the reader may wish to revisit section 2.1.4 above,
where thermodynamic aspects of consciousness are discussed. Human beings process
vastly more information per second than enters consciousness at that time. In conscious
speech, this amounts to something like 16 bits per second, as against the 40 million bits
per second processed in the sensorium. A further, but related, aspect of consciousness
is that very often though not invariably, it is something that is jogged by attention, and
attention is jogged by a circumstance that stands out in some way. Consciousness thus
is a statistically abnormal condition for a human being to be in, odd though saying so
strikes the ear. (After all, we can have no awareness of the statistical norm.)
An abductive trigger is not just an occurrence of which an agent is conscious. It is
an occurrence of a type that rises to a second grade of statistical abnormality. It is an
event whose occurrence is not only noticed and attended to by the agent; its occurrence
is, as we said above, uncharacteristic in some sense.
We note in passing that not all abductions are solutions of abductive problems.
Occurrences can be entirely in-character and yet still enter into stable and sound explanations. And not all abudctive problems are of the kind we are about to examine,
as we shall see in due course. In particular then, we reject the element of surprise as a
necessary condition on abductive triggers.
John wants to figure out ‘what’s going on here!’. To this end, he determines a
space of candidates (though his determination, let us note, might well be virtual). The
candidates are candidates for the role of what explains or might explain the triggerevent. Since the trigger presents John with a problem, we can say equivalently that
the space of candidates contains alternative resolutions of John’s abduction problem.
It is an empirically pressing question as to how big, and how varied, candidate spaces
actually are or could be.
let ç be the non-monotonic
database available to John at the moment of observation. Weé êA
èNé ê 1 Formally,
ê door closed.
ë haveì
ç
door
closed.
John
observed
A
revision
is
needed.
Compare
this
case
where
é ê˜ë êì and ì is observed. Then an explanation is needed. The ‘story’ will not admit as ‘acceptable’
and ç ì
taking itself as explanation.
6.1. THE BURGLARY
113
We assume that John, like everyone else, has acquired, perhaps tacitly, an explanation logic. A principal function of an explanation logic is to specify candidate spaces
for abduction problems. In having called John’s situation a test case we have chosen
our words carefully. John’s situation is a test for our emerging model of abduction in
the sense that it presents details of what actually happens in a certain class of—but
by no means all — close-to-the-ground situations. The model we develop here is not
canonical for every type of abduction problem originating in the same trigger that got
our John-case up and running (see below).
One fact about John’s situation is that his candidate space is very small. Yet the
number of states of affairs which, if they obtained, would explain John’s trigger-datum
is very large. John’s actual candidate space is a meagre proper subset. This the model
must take into account. To this end, we put it that John has intermingled his explanation logic with a relevance logic. A relevance logic in the sense required here has
little in common with what so-called relevant logicians occupy themselves with — viz.
the consequence and consistency relations (e.g., Anderson and Belnap [1975], [Read,
1988; Dunn, 1986; Woods, 1989a]), but rather is a logic in the sense of a set of algorithms which enable the human agent to discount, for the most part automatically
and with considerable alacrity, information unhelpful to a task at hand, or a set of such
tasks (see, e.g., [Sperber and Wilson, 1995] and [Gabbay and Woods, 2002a]). Setting
a candidate space for an abduction problem involves performing a reduction from sets
of potential explainers to sets of candidates for the status of actual explainer. It is an abductively sensitive instantiation of the more comprehensive operation of cutting down
from irrelevant although theoretically applicable information to information relevant to
whatever the task at hand happens to be. Again, we emphasize that there is little known
empirically about how these processes work.
Relevance in our sense is not a dyadic relation on sentences. It is a set of ordered
triples í îï ðï ñQò , informally interpreted to mean:
Information î is relevant for agent ð to the extent that it advances or closes
ð ’s agenda ñ [Gabbay and Woods, 2002a].
Because these processes occur with great speed and mainly automatically, and because they produce such selectively scant outputs, we think of the human reasoner in
such circumstances as activating instrumental (and perhaps causal) algorithms. Accordingly, the model posits what we are calling an explanation logic, and a logic of
relevant logic in our present sense, a primary function of which is to fix candidate
spaces in appropriately economical ways.
We return to our case.
John is perplexed. The door is open when it shouldn’t be. What’s
happened? Perhaps Carol has come home early. Maybe Dora couldn’t
do her chores on Tuesday, and has decided to make things up today. Or
perhaps John, who is the regular user of the back door, forgot to close it
when he left the house this morning. On the other hand, it could be a
burglar!
We identify these four alternatives by the obvious names. John’s candidate space is the
set ó Carol, Dora, John, Burglar ô .
CHAPTER 6. FURTHER CASES
114
The Plausibility Logic
The description continues:
John entertains the candidate Carol. No, he thinks, Carol is at school.
She never comes home before 5.30 and it’s only about 4.40 now. John
also rejects Dora. Dora hasn’t missed a Tuesday in years and years, and,
in any case, she would have mentioned it if she were planning to come
today. As for John, John recognizes that he is sometimes forgetful, but
he’s never been forgetful about shutting he door when he leaves for his
office. This leave Burglar. ‘Oh, oh’, says John ‘I think we just might
have been burgled!’.
John has riffled through the candidate space and has rejected Carol, Dora and John.
The rejections are judgements of implausibility, each rooted in what appears to be a
generic belief. It is not impossible that Carol is in the house, but it is implausible because she never is in the house at this time on a weekday. We note that generic utterance
is not restricted to classes of individuals, but can apply with equal effect to individuals.
Of course, like all generic claims, these are a generalization that license defaults for
John; each is made in at least implicit recognition that they needn’t be disturbed by
true negative instances. The inference that it isn’t Carol since Carol never is at home at
such times, is made in recognition of its requisite defeasibility . This suffices to make
the rejection of Carol a rejection founded on implausibility. Accordingly, we posit for
John a plausibility logic. It provides that John possesses the inference schema
Implaus
If õ is in an agent’s candidate space with regard to an abduction problem ö , and the agent holds a generic claim ÷ concerning the subject
matter of õ , and ÷ is incompatible with õ , then ø infers that the implausibility of õ ’s occurrence defeasibly disqualifies it at a solution of
ö .
Having reasoned in the same way with regard to Dora and John and provided that
John’s candidate space has taken in no additional members, John opts for Burglar (see
just below). Speaking more realistically, he finds himself in the grip of Burglar. He
not only “thinks” that it may be so; he is seized with apprehension that it may be so.
We may now say that John’s plausibility logic contains an iterated Disjunctive Syllogism rule, which he applies at his problem’s resolution point:
ItDS
Either Carol or Dora or John or Burglar. Not Carol and not Dora
and not John. Therefore Burglar.
At a certain level of abstraction there is no harm in assuming an application of ItDS.
Something like it, though different, is actually in play, since ‘not õ ’ here means ‘õ is
not a plausible solution of . . . ’, and so does not strictly negate õ .
Candidate spaces have the potential for looping. If Burglar is the only candidate
not yet eliminated from John’s space, then if John sufficiently dislikes Burglar, he
may find that the candidate space has taken in a new member, e.g., ‘The Sears man
has come’. We put it that the explanation logic will tolerate only low finite looping at
6.1. THE BURGLARY
115
most. Another possibility is retraction. John might dislike Burglar and find that he
has revisited, e.g., Carol. Here, too, the explanation logic will permit only low finite
oscillation. In general it is realistic to allow for options in addition to the one-option or
no-option approach. By and large, the failure to find an option which the abducer likes
is a trigger of (cautious) investigative actions. We take up this point in our discussion
below of the Yellow Car.
The Structure of a Resolution Point
John’s solution of his abduction problem involved the elimination of all candidates but
one. Carol was eliminated by ‘Carol is never ever home early’, (ù!ú let’s call it), Dora
by its corresponding ù!û , and John by ùQü . This leaves Burglar, or ý for short. John
plumped for ý . But why? ý too is attended by its own ùQþ , ‘There are never any
burglaries in this neighbourhood’. Why didn’t ùQþ cancel ý , just as ù!ú cancelled ÿ ,
ù!û cancelled , and ùQü cancelled ? It is evident that John’s eliminations have left
him with the following termination options:
ù
þ cancels ý , and there is neither retraction nor looping. The abduction
problem crashes.
ù
þ cancels ý and either retraction or looping occurs. The abduction problem is renewed short of resolution.
Although ý is a negative instance of ùQþ , it does not cancel it (recall that ùQþ
is a generic claim), and there is neither retraction nor looping. ý solves the
abduction problem.
How to get Determinacy out of Indeterminacy
It is necessary thatthe
give abducers the means of selecting from
plausibility
logic
multiples such as . How does an abducer know when to bet his confidence in the exhaustiveness of his candidate space against the fact that the solving
candidate is a negative instance of generic claim in which the abducer also has confidence? Or how does he know when to bet in reverse; that is, when to bet his confidence
that the last surviving member of the original candidate space is indeed cancelled by
the generic claim of which it is a negative instance? Similarly, how does the abducer
know when and when not toretract
a prior decision of candidate cancellation in or that involves
der to evade selection of a
a candidate which is not only a negative
instance, but which he now thinks is cancelled by it?
It is useful to repeat that our present purpose is to model John’s actual case. It
is not casuistry that we are intent on practising, but rather specifying a paradigm of a
nontrivial class of abduction problems. It is also worth noting that in real-life abductive
situations such as that of John, it is rarely helpful for John to ask himself why he went
-route rather then the -route. There is ample evidence that the routines
the
of actual abductive practice are not much accessible to human introspection. So the
question before us constitutes an abduction problem all of its own for the theorist who
is trying to figure out John’s abductive situation.
CHAPTER 6. FURTHER CASES
116
We put it that the plausibility logic favours the following inference schema which,
we emphasise, can only have defeasible legitimacy.
To the extent possible, favour the option that has an element of autoepistemic
backing. For example, in the case we are investigating, the generic claims
about Carol’s never coming home early, is likely to be underwritten by two
factors of autoepistemic significance. One is that if it were indeed true that
Carol never comes home early, this is something that John would know. And if
today were to be an exception to that rule, this too is something that John may
well have knowledge of.
bids John to favour Burglar since, Carol, Dora and John are subject to the
requisite autoepistemic factors. Burglar is different. Although John, and everyone
else, knows that burglaries rarely happen in this neighbourhood, it is not true that its
being so is something that John would know (as opposed to happens to know). Nor is
it true that if today is an exception, this too is something that John would know. John’s
is an urban, professional, two-income neighbourhood. There is not much interaction
with neighbours. (People are seldom at home, it would seem.) Neither John nor they
know their neighbourhood well enough to be able to say with conviction, ‘Burglaries
would not happen here’, i.e., “It would be out of character for a burglary to occur in
this part of town.” (Thus, the out-of-character is different from the improbable).
Human reasoners are natural conservatives. They lean towards explanations that
deviate from the normal as little as is consistent with getting an explanation. It is the
sort of favouritism that Quine had in mind in proposing his maxim of minimal mutilation as a principle guiding scientific theories in handling observationally recalcitrant
data. It is a maxim which favour adjustments to a theory as minimal as get the the
problem solved.
We assume such a principle to be in play in John’s plausibility logic. We assume
it is in the form of least possible deviation from the norm. Having it in play seems
to count against John’s selection of Burglar. For if there were indeed a burglary,
this would be quite a deviation from the normal, whereas if Carol had come home
unexpectedly, or Dora had switched her day without telling anyone, or John for the
first time in his life had forgotten to shut the door, these would be lesser deviations
from the norm.
It falls to John’s plausibility logic to hazard an answer to the following question.
Given that one of and does obtain, is there a way of selecting
one as the most plausible?
Whether the logic is capable of furnishing an answer in every case, it would appear
that if there is a least plausible of the present lot it is . If so, isn’t it the ‘wrong’ answer
for John?
We note that in finding thus for , the logic equates least plausiblity with greatest deviation from a contextually indicated norm. But this is not an answer to John’s
abduction problem. John’s abduction problem is to determine what best explains the
open door, which was John’s trigger. The abductive task was not to determine who
most plausibly was in the house. The door is wide open in late afternoon of an overcast
6.1. THE BURGLARY
117
day (in a community in the Northern Hemisphere, let us now say if it weren’t — abductively — clear before). Given that Carol were home early or that Dora had switched
her day, how plausible is it that the door is left wide open on either of these accounts,
when a burglar would have had no particular motivation to close the back door of the
empty house that he had burgled without incident earlier in the day?
This leaves John with the task of deciding between Burglar and John. In opting for
Burglar he was certainly not making a canonical choice. A different person in exactly
the same situation except for this difference might well have opted for the explanation
in which he did indeed forget for the first time in his life to close the door. But in John’s
actual situation, the autoepistemic factor is clinching:
If I had left the door open, I would now have remembered. But I don’t, so
I didn’t.
Alternatives
Consider a second, hypothetical, case, somewhat like the John-case (which was a real
situation). Recall, we wanted our model to capture John’s actual case: but it will be
instructive to examine the effects of even fairly small differences. Everything is as
before, except for factors we shall now consider. Carol is in John’s candidate space.
As John begins the winnowing process, he recalls that Carol is in Vancouver assisting
their daughter with a new baby. There is nothing generic or defeasible about this,
nothing that would justify the tentativeness of a plausibility judgement. Carol’s being
in Vancouver is a fact. It is a fact incompatible with Carol. End of story.
Bearing in mind that an abductive logic includes the triple an explanation logic,
a plausibility logic, a relevance logic , it is appropriate to ask at which stages a circumstance of a test case should be modelled. The present example presents us with
two modelling options. One is to have the plain fact that Carol is in Vancouver bear
on the model at the candidate space-construction stage. If it is a fact that Carol is in
Vancouver and John knows it, then Carol doesn’t belong in the candidate space to begin with. On the other hand, since John has appeared briefly to have forgotten Carol’s
whereabouts only to have recalled them after the candidate space was fixed, there is a
supplementary mechanism at work at the stage of winnowing.
SupplMech: If is the candidate space of a subject in relation to an abduction
problem , if is a fact incompatible with , if did not know (recall) during the construction of the candidate space, but called at a subsequent stage in the abduction process, then categorically and
summarily removes from the candidate space.
How reasonable is it that a plausibility logic would embed a rule such as SupplMech?
We think it inadvisable to press the matter over-much. Either SupplMech is in the
plausibiltiy logic, and permits candidate exclusion on factual grounds as limiting cases
of implausibility determinations; or SupplMech is an external supplementation of the
plausibility logic.
Our second hypothetical case raises further questions about how John actually operated in the test case we are attempting to model. For example, why have we represented
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CHAPTER 6. FURTHER CASES
John’s selection of Burglar as the result of the exhaustive elimination of the candidate
alternatives on grounds of their implausibility? Why did we not instead represent his
choice as having turned on his finding Burlar to have greater plausibility than its rivals? Why can’t the processor of an abduction problem produce a solution by opting
for the most plausible alterative in his candidate space, without the nuisance and the
cost of running thorough a number of implausibility-determinations? There is no reason in principle why this couldn’t happen. (In fact, there is rather a lot of evidence from
empirical investigations of learning that suggests that this is often what does happen.)
When it does, his abduction exhibits the following structure:
Corollary 6.1.1 If !#" $ % % % $ !'& ( is the candidate space of a subject ) in relation to
an abduction problem * , and ) judges !'+ to be more plausible than any other, then
!'+ gives the resolution point of * for ) . If !+ is chosen in the manner above, then
) is committed to finding the other !-, less plausible than !'+ but not necessarily
implausible.
We see that the winnowing of candidate spaces can proceed by way of judgements
of negative plausibility and judgements of positive plausibility. Seen the first way
we are reminded that membership in an abducer’s candiate space is not a matter of
plausibility. It is a matter of relevance. However, seen in the second way, we are
reminded that membership in a candidate space is not forbidden to relevant potential
explainers whose occurrence is also plausible.
Then, too, there is the following kind of situation. John comes home to find thick
black smoke pouring from the house. ‘Christ’, says John, ‘the house is on fire!’. The
occasion is backed by a generic claim of which it is a positive instance, and it instantiates no more plausible a generic claim.
Abductions of the negative plausibility sort resemble belief-revision by way of
backwards propagation [Gabbay, 2000]. Such structures have had a lot of attention
from AI researchers, and are comparatively well-understood. For all their apparent
economic advantages, it is not as easy to model abductions via judgements of positive
plausibility; nor is it at all easy to show this kind of approach to economic advantage.
We close this section with two related matters. Why, we again ask, did we represent John as finding Burglar plausible just on the grounds that he found its rivals
implausible? It is a question invited by a tempting confusion which we should try to
discourage. So, to repeat: we do not argue that John’s actual selection of Burglar
hangs on a favourable judgement of plausibility. John’s decision is captured by the
following
Choice Rule:
If a subject ) has a candidate space !#" $ % % % $ !& ( in relation to an
abduction problem * , and if ) has rejected all the !'. but !-, for
their implausibility, it suffices for the selection of !. that nothing
else of explanatory potential has sufficient plausibility for solution.
(Assuming solvability, no retraction and no looping).
6.2. THE PICCADILLY LINE
6.2
119
The Piccadilly Line
This, too is a real case. Names might have been changed in the interest of privacy, but
we see no need to do so. Dov is an academic at King’s College. Bas is an academic at
Princeton. Dov and Bas have known one another’s work for years, and they see each
other from time to time at conferences. Only two weeks previously they were keynote
lecturers at the Society for Exact Philosophy Conference in Lethbridge. It was pleasant
for them to have renewed their acquaintance.
Dov lives in North London. Greater London is made up of five zones, with central
London the inner dot and outer London a large belt called zone five. Dov lives at the
juncture of zones three and four. Early each morning he boards a Piccadilly Line train,
which gets him into central London by about 8.00.
Today Dov boards the train at the usual hour. As he takes his seat, he sees that
directly opposite, immersed in a book, is Bas, or someone who looks just like Bas.
Now Bas is a well-travelled person. Like nearly everyone in his situation, a trip to
London would be a trip to central London — to the University of London, to the West
End, and so on. Bearing in mind that the man who looks just like Bas was already
seated when Dov boarded, Bas had to have originated his journey in zone four or five.
But again, zones four and five hold no professorial, cultural or touristic interest for
people like Bas. So Dov concludes that this isn’t Bas after all. (Actually he came to
see that it was not Bas.)
It bears on our discussion that Bas has a physical appearance that distinguishes
him almost as much as his professional work has done. He is suavely slender, with an
elegant van Dyck beard and kind of benign charisma. Bas’s look-a-likes do not come
thick on the ground. But look-a-like is what Dov thinks.
Dov’s thinking so is the result of how he handled an abduction problem. The trigger
was Bas’s appearance in the outer reaches of greater London. It is an appearance that
contradicts the pertinent generic facts. People like Bas come to London to give lectures
to the British Academy. People like Bas come to London for the Royal Ballet in Covent
Garden. People like Bas enjoy shopping in Jermyn Street. People like Bas prefer to
stop at good hotels or clubs where they may have reciprocal privileges. What’s more,
although it is true that Bas has lots of European relatives, they all live on the continent.
Still the man sitting opposite Dov is the spit-and-image of Bas. Part of Dov’s
abduction problem is to entertain what might bring Bas to North London at the break
of day.
Dov could not have had this problem had he not been prepared to consider alternatives. Two things are noteworthy about Dov’s situation. Although he is on friendly
terms with Bas, Dov does not greet him. Dov is not shy; so this is odd. Even so, Dov’s
problem is not to ascertain whether his fellow passenger is Bas, but rather to figure out
why Bas would be in this wholly counter-expected situation. Dov’s case contrasts with
John’s. John could bump into Carol in the most shockingly improbable place on earth
and not be in the least doubt that it is Carol that he sees.
Dov solved his abduction problem by inferring that his fellow traveller was only a
look-alike. In so doing, he overrode the best explanation of why this man looks exactly
like Bas, namely that he is Bas. Bearing on this was the failure of Dov’s predecessor
abduction problem, which was to sort out why Bas would be in this part of London. In
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CHAPTER 6. FURTHER CASES
the absence of further information — of information we could have got just by asking,
Dov had no answer to give. He had lots of possible answers, of course, including
some relevant possibilities. (Bas may have been lost; perhaps he has friends in North
London; perhaps he is an insomniac and is riding the trains to fill unwanted time.)
But Dov had no basis for winnowing these possibilities via considerations of plausibility and implausibility.
As the train sped on, Dov found himself adjusting to his newly arrived-at belief
in interesting ways. Bas is quite slender, and this man is quite slender; but not, as
Dov now sees, as slender as Bas. Bas has a beautifully sculptured van Dyck beard,
as does the man opposite; but it is not as elegantly shaped as Bas’s beard. The facial
resemblance is also striking, but not really all that close, as Dov now sees.
Bas’ look-a-like has been reading all this time. He now marks his page and looks
up. ‘Hi, Dov!’
Bas and Dov then fall into an amiable conversation, during which it came out that
Bas had attended a wedding in North London and stayed there with friends overnight.
He was on his way to Heathrow to catch the first flight to Leeds where he would give a
lecture that evening.
After cheerful good-byes, Dov got off at Holborn, and Bas resumed his reading.
Of course, Dov got it wrong. Abduction is a defeasible process; getting it wrong is
not anything to be ashamed of as such. Dov was put wrong by the tug of what he took to
be characteristic of (people like) Bas. It seems that Dov was running something like a
Bas-script. It is also possible that he was also running a first-thing-in-the-morning-onthe-southbound-Piccadilly-line script. At no point at which they might intersect would
we find Bas or anyone like Bas. (It would seem that not even Dov has a characteristic
place in Dov’s own Piccadilly line script). The pull of generically rooted expectation is
very strong. In Dov’s case it was strong enough to override the evidence of his senses.
This happened because Dov didn’t have a plausible account to offer which would have
overridden, in the reverse direction. He could not override, ‘This is not the place for
the likes of Bas’, because he couldn’t figure out why it would in fact have been Bas.
And, of course, he didn’t ask. Such is the tug of the characteristic.
We end this section on a cautionary note. We have made the point that although
abducers seem not to solve abduction problems by constructing or even inspecting
filtration structures, it does appear that the factors of relevance and plausibility (and,
certainly, explanatoriness) sometimes and somehow guide their efforts.
This may well be so. But why would we suppose that because this is the case
abductive agents are running relevance, plausibility and explanation logics? Logic, we
said, is a principled description of aspects of what a cognitive agent does. If it is true
that, in solving abduction problems, practical agents are drawn to relevant plausibilities
with explanatory force, relevance, plausibility, etc., enter in a nontrivial way into what
logical agents do.
By these lights, it is appropriate to postulate logics of these concepts. What is
problematic is the supposition that an agent’s subscription to them might be virtual. The
problem is mitigated by the RWR (representation without rules) approach to cognitive
modelling. On this approach cognitive systems employ representational structures that
admit of semantic interpretation, and yet there are no representation-level rules that
govern the processing of these semantically interpretable representations [Horgan and
6.3. THE YELLOW CAR
121
Tienson, 1988; Horgan and Tienson, 1989; Horgan and Tienson, 1990; Horgan and
Tienson, 1992; Horgan and Tienson, 1996; Horgan and Tienson, 1999b; Horgan and
Tienson, 1999a]. Critics of RWR argue that it can’t hold of connectionist systems
[Aizawa, 1994; Aizawa, 2000]. Since we want to leave it open that some at least of the
cognitive processing of practical agents, it matters whether this criterion is justified.
We think not, although we lack the space to lay out our reservations completely. The
nub of our answer to critics of the RWR approach is as follows.
1. Critics such as Aizawa point out that connectionist nets are describable by programmable representation level rules. They conclude
from this that connectionist nets execute these rules [Aizawa, 1994,
p. 468].
2. We accept that connectionist nets are descibable by programmable
representation-level rules. But we don’t accept that it follows from
this that connectionist nets should be seen as executing such rules.
We borrow an apt analogy from Marcello Guarini:
The orbits of the planets are rule describable, but the planets do not make
use of or consult rules in determining how they will move. In other words,
planetary motion may conform to rules even if no rules are executed by the
planets [Guarini, 2001, p. 291].
A full development of this defence can be found in this same work.
What, we were wondering, could a virtual logic be? We propose that a reasonable
candidate is the requisite description of a cognitive system seen as a connectionist net
that satisfies the condition of the RWR approach. It is a logic of semantic processing
without rules.
6.3
The Yellow Car
[TBA]
6.4
A Proof from Euclid
This subsection gives another illustration of the role of plausibility.
We draw from a paper by S. Marc Cohen and David Keyt [1992]
The example is from Euclid Elements. In Book I, Euclid gives five postulates for
geometry and his goal is to prove from these postulates how to construct an equilateral
triangle on a given finite straight line. The proof has a gap in it; we need to abduce extra
assumptions. The point we make is that the considerations and reasoning involved in
what assumptions to add (abduce) to make the proof correct, have to do with the general
body of knowledge of ancient Greek mathematics and culture rather than geometric
proof theory.
CHAPTER 6. FURTHER CASES
122
Here is what happens. We quote directly from the paper [Cohen and Keyt, 1992,
pp. 190–192 ], including the footnotes:2
The proof, divided into discrete steps, runs as follows:3
2
354
/10
1. Let
687
be the given finite straight line.
2. With centre
3)
3. With centre
3)
6
7
and distance
687
, let the circle
79:
be described. (Postulate
and distance
76
, let the circle
689;
be described. (Postulate
4. The circles cut one another at point
9
5. From the point to the points
joined. (Postulate 1)
6. Since the point
(Definition 154 )
7. Since the point
(Definition 15)
8. Therefore,
689
6
7
6
9
and
7
.
let the straight lines
96
and
97
be
is the centre of the circle
9:<7 , 86 9
is equal to
687
,
is the centre of the circle
968;= 79
is equal to
76
,
is equal to
79
. (From 6 and 7 by axiom 15 )
9. Hence the three straight lines
(From 6, 7 and 8)
10. Therefore, the triangle
6879
689 , 86 7
, and
79
are equal to each other.
is equilateral. (Definition 20).6
An obvious flaw in this proof and one that has often been pointed out7 is that no
justification is given for step (4). What guarantees that the two circles intersect at a
point and do not slip through each other without touching? Step (4) tacitly assumes
that Euclidean circles are continuous. Can this be proved? Is there anything in
Euclid’s postulates that ensures that Euclidean lines have no gaps?
Here are Euclid’s five postulates:
1. Let it be postulated to draw a straight line from any point to any point,
2. and to produce a limited straight line in a straight line,
2 The
footnote numbering is continuous with ours. The cross references in the text are those of Cohen and
Keyt and therefore may refer to text not quoted in this book.
3 Thomas L. Heath, The Thirteen Books of Euclid’s Elements, 2nd edn, Cambridge 1926. i. 241–2.
4 A circle is a plane figure contained by one line such that all straight lines falling upon it from one
particular point among those lying within the figure are equal.
5 Things which are equal to the same thing are also equal to one another.
6 Of the three trilateral figures, an equilateral triangle is the one which has its three sides equal . . . .
7 Heath, Euclid’s Elements, i. 25, 242; Felix Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, trans E. R. Hedrick and C. A. Noble (New York, 1939), 197; Howard Eves, A Survey
of Geometry, revised edn. (Boston, 1972), 321; Ian Mueller, Philosophy of Mathematics and Deductive
Structure in Euclid’s Elements (Cambridge, Mass., 1981), 28.
6.4. A PROOF FROM EUCLID
123
3. and to describe a circle with any center and distance,
4. and that all right angles are equal to one another,
5. and that, if one straight line falling on two straight lines makes the interior
angle s in the same direction less than two right angles, the two straight
lines, if produced ad infinitum, meet one another in that direction in which
the angles less than two right angles are.8
One way to settle the question of continuity is to see if there is an interpretation
of the concepts that enter into these five postulates and into the above proof under
which the postulates are true but under which step (4) is false. Such an interpretation can be found by exploiting the techniques of analytic geometry and set theory.
The basic idea is to give the relevant terms an arithmetic interpretation in the domain of rational numbers. The domain of rational numbers is chosen since a line
whose points correspond to rational numbers, though everywhere dense (between
any two points there is a third), is not continuous. (There is a gap, for example,
between all the points of such a line greater than the square root of 2 and all the
points less than the square root of 2.) Following this strategy each of the relevant
terms is assigned an arithmetic meaning that corresponds by way of a Cartesian (or
rectangular) co-ordinate system to the intended geometric meaning of the term.
Under this arithmetic interpretation the word ‘point’ means ‘ordered pair of
rational numbers’; ‘straight line’ means ‘set of points that satisfy an equation of
the form
’, and ‘circle’ means ‘set of points that satisfy an equation
of the form
’. (In these equations and in those that follow
‘ ’ and ‘ ’ are variables and all other letters are constants.) These two equations
are chosen since the graph of the first, using a Cartesian coordinate system, is a
geometric straight line and that of the second is a geometric circle. Finally, ‘line
intersects line
’ means ‘the intersection of the set of points identified with
and the set of points identified with line
is not the null set’.
line
Consider now Euclid’s five postulates. Postulate Four is provable,9 so it can be
set aside. The other four are all true under the foregoing arithmetic interpretation
when it is elaborated in an obvious way.
If the points mentioned in Postulate One are the two ordered pairs
and
, the following set is the straight line through these points (where Ra is the
set of rational numbers):
> ?@A BC@DFEHG
? I @B I @> ?@A BJ@DKELG
?
B
M8N
OP
M8N
OP
QR S TU S V
QR I TU I V
W Q ?CT B VKX ?CT B<Y-Z8[ \] U S^#U ? @L] R ^#R S B@R S U I_
I
_
I ^ R I U SKEHG ` a
Under the arithmetic interpretation Postulate One makes the true assertion that this
set has members. Postulate Two makes the true assertion that if
, then for
any given rational number, , the set contains an ordered pair whose first element
is larger than and a second ordered pair whose first element is smaller than , and
it makes a similar true assertion for the second element if
. (If
,
the line is parallel to the -axis; and if
, it is parallel to the -axis.)
In order to interpret Postulate Three it is necessary to specify that by ‘any
centre’ is meant ‘any point as centre’ and by ‘distance’ ‘distance between two
points’. If this is laid down, then under the arithmetic interpretation Postulate
three makes the true assertion that the following set, which is determined by the
c
B
8 Mueller’s
9 The
R SEH
b RI
c
U SKEHU I
U S-Ed
b UI
?
translation, Philosophy of Mathematics, 318–19.
proof, which goes back at least to Proclus, is given in Mueller, ibid. 22.
c
R SEdR I
CHAPTER 6. FURTHER CASES
124
e f g h i and the radius j is not null:
k e lCg m iFn lCg m<o-pq rs lt#f u vwLs mt#h u vFxLj v y z
(In the equation determining this set j v must always be a rational number though
equation for a circle with the centre
j
can be irrational.)
The fifth, or parallel, postulate is the most complex. Consider the two following sets:
{
k
o-pq r8| } lw#~ } m8w }KxH€ y
 x x k ee lClCgg m m iKKi n n lClCgg mm<o
p8q r8| v l w#~ v mw# v xL€ y
Postulate Five asserts that if A and B each have at least two members and if
| } ‚ | v'x„
ƒ ~ } ‚ ~ v , then …‡†'ˆ x„
ƒ € . This amounts to the claim that the equations
that determine A and B have a simultaneous solution in the domain of rational
numbers. It is established by showing that a linear equation that is satisfied by two
distinct ordered pairs of rational numbers has rational coefficients10 and that the
simultaneous solution of two linear equations with rational coefficients must be a
pair of rational numbers.11
All of Euclid’s postulates are true under the proposed arithmetic interpretation,
but under this interpretation the key step in Euclid’s proof of his first theorem and
and
are the end-points
the theorem itself are false. For suppose that
of the finite straight line upon which the equilateral triangle is to be constructed.
Then under the arithmetic interpretation the two circles are:
e t‰ g € i
e‰ g€ i
k e lCg m Ki n lCg m<o-pq rs lt‰ u v mv H
x Švy
k e lCg m Ki n lCg m<o-pq rs lw‡‰ u v w
w mv H
x Švy
But these sets do not intersect. (In the domain of real numbers the points of intersection are
and
.) Thus Euclid’s first proof cannot be repaired,
and the reason is that his postulates do not guarantee that the lines of his geometry
are continuous.
e€ g ‹ Œ i
€ g t‹ Œ i
Obviously we here have several options for abduction. The first is to assume that all
lines
continuous.
means we are working in a model of the plane of all points
 ŽK  ‘ arewhere
Ž  are This
real numbers. It is plausible for us to assume that Euclid has
these intuitions at the time. The second option is to assume, based on our knowledge
of the extensive use and attention the Greeks gave to ruler and compass constructions,
that Euclid had in mind that all ruler and compass construction
 Ž points
‘ sucharethatavailable.
ŽK d’H“ In,
modern“ terms this means that we are looking at all points
where is the extension field
rationals
operation of
Ž’#of“ the
Ž”–•<—™closed
˜<Ž’'under
“ (i.e.,thetheadditional
getting square roots, i.e., if
and
set of all numbers we
can get from rationals by the˜ operations of addition š , multiplication › , subtraction
œ , division  and squareroot .)
10 The
ž Ÿ } ¡ } ¢ and ž Ÿ v ¡ v ¢
£ ¡ }J¤-¡
£Ÿ v ¥ ¦8§ v ¤ Ÿ v ¥ ¨F§ Ÿ } ¡ v ¤-Ÿ v ¡ }©#ª «
equation for a straight line through the two points
Ÿ } Ÿ v ¡}
¡v
is:
Since
and
are all rational, the coefficients in the equation are rational.
11 A system of two linear equations can be solved by successive uses of addition and multiplication. Hence,
if all of the coefficients of each equation are rational, the solution must also be rational.
6.5. NEUROPHARMACOLOGICAL INTERVENTION
125
Which abduction hypothesis do we adopt? Do we add the axiom that the plane is
continuously full of points or do we add the axiom that the plane includes all rational
as well as rule and compass points?
No geometrical abductive process can decide for us. Only our second logic, our
reasoning based on ancient Greek culture can decide.
This example shows our second logic at play.
By the way, Plato’s work is full of arguments in need of abductive ‘help’ and the
Cohen and Keyt paper discusses the methodology of helping Plato. In our terminology, the paper discusses the options for a second logic for supporting abduction for
enthymemes in Plato. (We return to the problem of enthymemes in chapter 10.)
6.5
Neuropharmacological Intervention
Neural disorders are often the result of imbalances of neurochemicals, owing for example, to cell damage or degeneration. When the requisite neurochemical is irrevocably
depleted a serious neurological malady is the result. Parkinson’s disease is such a disorder.
Pharmacological remedies of disorders such as Parkinson’s disease are the object
of what is called rational drug design (RDD) (see, e.g., [van den Bosch, 2001], from
which the present section is adapted). In a typical RDD, information is collated concerning the requisite organic structures and of the success or failure of past pharmacological interventions. In wide ranges of cases, computer models are able to suggest
further chemical combinations and/or dosages. RDDs are sometimes supplemented or
put aside in favour of hit-and-miss strategies in which large quantities of chemicals
are produced and tested in vitro for their capacity to influence receptors in the targeted
neural structures. A third procedure is the generation of data from laboratory studies
of animals.
Following Timmerman et al. [1998], van den Bosch develops the Parkinson’s disease example with a description of an important class of subcortical nuclei called basal
ganglia, which are necessary for control in voluntary behaviour. When Parkinson’s disease strikes, a component of the basal ganglia, substantia nigra pars compacta (SNC),
is significantly degraded, a result for which there is no known cause. The SNC furnishes the neurotransmitter dopamine, whose function is to help modulate signals from
the cortex. Figure 6.1 schematises the function of dopamine.
In Parkinson’s disease the object is to stimulate chemically increases in dopamine
levels. L-dopa has had a mixed success in this regard. In the first five years, it is
highly effective, but with nausea as a significant side effect; and after five years of
use, its therapeutic value plummets dramatically. Parkinson’s research is, therefore,
dominated by the quest for alternative modes of dopamine receptor agonists, especially
those that interact with only particular dopamine receptors. Figure 6.1 charts the effect
of versions of these dopamine receptor agonists. Two receptors on the passage from
the stimulation to the SNR/GPi are indicated as D1 and D2. D1 occurs on the direct
route, whereas D2 occurs on the indirect route via the GPe. Both D1 and D2 are
receptive to dopamine, but differently. When dopamine stimulates D1, this excites the
relevant cell, whereas the stimulation of D2 inhibits it. A natural question is whether
CHAPTER 6. FURTHER CASES
126
Cortex
Glu + striatum
DA
D1(+)
D2(-)
thalmus
GABA(-)
GABA(-)
GPe
GABA(-)
STN
SNC
GABA(-)
GABA(-)
Glu(+)
SNR/GPi
brainstem
GABA(-)
Figure 6.1:
a combined excitation/inhibition convergence is necessary for a favourable outcome.
Van den Bosch reports studies that show that compounds that stimulate D1 but not D2
receptors are ineffective [van den Bosch, 2001, p. 32].
The object of a drug design is to discover a successful agonist for the disorder in
question. In Parkinson’s research, as in many other sectors of neuropharmacology, this
is done by modelling what of the patient’s neurological endowments are known to be
of relevance to dopamine stimulation, together with what our various pharmacological
regimes are known to stimulate such receptors. When these two knowledge bases are
coordinated in the appropriate way, it is often possible to model the resultant dynamic
systems by a qualitative differential equation (QDE). An example of how a QDE might
model the basal ganglia as schematized in Figure 6.1 is sketched in Figure 6.2.
M+ and M- are, respectively, monotonically increasing and decreasing functions.
Variables have initial values of high, low or normal and dynamic values of increasing,
steady or decreasing. Variables are interconnected in such a way that their differentials
determine a value for the combination. These determinacies are subject to a qualitative
calculus. If a variable ¬ is a differential function over time of variables ¬ ­ and ¬ ® and if
¬ ¯ ° ±³² , then an increase in the value of ¬ ­ and ¬ ® will drive up the value of ¬ ¯ , but ¬ ¯ ,
will not be assigned a value when ¬ ­ increases and ¬ ® decreases. Such indeterminacies
flow from the qualitative features of QDEs.
A qualitative state of a system described by a QDE is an attribution of vari-
6.5. NEUROPHARMACOLOGICAL INTERVENTION
d/dtf (SNC)
M+
127
M+
d/dt a(L-DOMA,striatum)
d/dta(dopamine,striatum)
M+
M+
f(striatum)
M-
M+
d/dtf(striatum -D2-to-Gpe)
d/dt f(striatum-D1-to-SNR/GPi)
M+
M+
d/dta(GABA,SNR/GPi)
M+
d/dtf(Gpe)
M-
M-
d/dtf(SNR/GPi)
M+
d/dta(glutamate,SNR/GPi)
Figure 6.2:
able values to all variables of the system, consistent with the constraints in
the QDE. Given a QDE and a set of known initial values, a set of all consistent system states can be deduced, together with their possible transitions.
When a calculated value is unknown, all possible states are included in the
set. This set is complete, but is proved not always correct since spurious
states may be included as well [van den Bosch, 2001, p. 34]
Figure 6.2 graphs a part of a QDE (which includes a part of the model of basal ganglia
schematized in Figure 6.1) together with the operation of dopamine. It charts the firing
rates (f) of nuclei and neural pathways and amounts (a) of neurotransmitters contained
in nuclei. Thus if the firing rate of the SNC increases this will increase the quantity of
dopamine in the striatum, which in turn depresses activation of the neural pathway that
signals to the GPe.
A drug lead is a specification of the properties a drug should possess if it is to hit
128
CHAPTER 6. FURTHER CASES
a given desired therapeutic target or range of targets. For every disease for which there
is some minimal degree, or more, of medical understanding, a profile exists, which is
a qualitative specification of the disease. For every profile, the researcher’s goal is to
supply a drug lead. The search goal
is to find those variables by which one can intervene in the profile in such a
way that the pathological values of the variables associated with a disease
are reversed. The goal set is defined to consist of the variables of the
disease profile with an inverted direction of change, i.e., if a variable is
lower in the pathological profile, it is included in the goal to increase that
variable value [van den Bosch, 2001, p. 35], emphasis added.
The (ideal) goal of such a search is to ‘find a minimal set of variables such that a
manipulation of the variable values propagates a change in direction of the values of
the variables of the goal set’ [van den Bosch, 2001, p. 35]. Because of their qualitative
nature, QDEs are not typically complete, so that the specification of all possible desired
value changes of the goal will not be possible. Accordingly QDE search models are
taken as approximations to this ideal [van der Bosch, 1999; van den Bosch, 1997;
van den Bosch, 1998].
The search tasks can now be described. The starting point in a QDE model and
whatever is known of the initial values of variables. Next a goal is selected. A goal is
a set of desired values. The the searcher backward chains from the values set up by the
goal to ‘possible manipulations of the variables’ [van den Bosch, 2001, p. 36]. Because
QDE methods provide only approximately good (or bad) outcomes, approximation
criteria are deployed to measure the closeness of fit between values sought by the goal
and values actually produced by the manipulation of variables. The search is successful
when it finds a set of manipulations that best approximates to the production of the
greatest number of goal values, with least collateral damage. A successful search is
thus a piece of backward chaining successfully completed. It is a case of what van den
Bosch calls inference to the best intervention [van den Bosch, 2001, p. 34].
In the case of Parkinson’s disease, one of the goal values is a reduced activation
frequency of the SNR/GPi than is present in pathological circumstances. A search of
possible manipulations discloses two salient facts. One is an increase (´ ) of L-dopa in
the striatum. The other is a reduced firing rate (µ ) of the indirect channel between the
striatum and the GPe induces a suppression of the firing rate of the SNR/GPi. It turns
out that a ¶-· agonist can produce this decrease, but with lighter consequences for other
such (i.e., ‘dopaminergics’) than dopamine.
QDE searches bear some resemblance to computational diagnostic procedures. In
each case, they tell us nothing new about the diseases in question. Their principal
advantages are two. One is that these models are means of explicitizing both what is
known of the relevant symptoms and how one reasons about them diagnostically and/or
interventionistically. Another is that when QDEs are modelled computationally, they,
like computerized diagnostic techniques, are especially good at teasing out all consequences compatible with integrity constraints and design limitations. In each case, the
backward chaining component is modest. It is little more than manipulation by way of
the appropriate monotonically increasing and decreasing functions of variables already
known to be interconnected. However, a third feature of the QDE search methodology
6.6. THE ECONOMY OF RESEARCH
129
considerably enhances its status as an instrument of abductive reasoning. For all their
incompleteness and, relatedly, their disposition towards misprediction, drug lead exercises can also be used as proposals for experiments, and thus satisfy at least Peirce’s
idea that decisions about where to invest research resources are an essential part of the
logic of abduction. Research-economic abduction is discussed in the next section.
6.6
The Economy of Research
We owe it to Peirce who originated, and took pride in, a subdiscipine he would call the
economy of research or, as it has also come to be known research economics [Peirce,
1931–1958, pp. 7.139–7.157 and 7.158–7.161 ]. If we believe Rescher, Peirce came
to rest on a settled use for ‘abduction’ and ‘retroduction’. ‘Abduction’ names the processes of hypothesis-generation and hypothesis-engagement. ‘Retroduction’ denotes
the process of hypothesis-discharge, although, as we have seen, there is some doubt
as to whether Peirce himself would venture beyond what we have called quasi- or
Popperian-retroduction.
Peirce was quick (and early) to appreciate that for any body of data, infinitely many
alternative, and at times rival hypotheses conform equally well to it. Data always underdetermine theory. The theorist is landed with the vexatious question of which hypothesis from the infinity of possible and equally conforming alternatives to take seriously.
The theorist, like all the rest of us, is met with a Cutdown Problem, the problem of
selecting from hypotheses which ‘inundate us in an overwhelming flood . . . ’ [Peirce,
1931–1958, 5.602]. Peirce writes,
if two hypotheses present themselves, one of which can be satisfactorily
tested in two or three days, while the testing of the other might occupy a
month, the former should be tried first, even if its apparent likelihood is
a good deal less . . . . In an extreme, where the likelihood is of an unmistakably objective character and is strongly supported by good inductions,
I would allow it to cause the postponement of the testing of a hypothesis.
For example, if a man came to me and pretended to be able to turn lead into
gold, I should say to him, ‘My dear sir I haven’t time to make gold’. But
even then the likelihood would not weight with me directly, as such, but
because it would be a factor in what really is in all cases the leading consideration in Abduction, which is the question of ‘Economy — Economy
of money, time, thought and energy [Peirce, 1931–1958, 5.598–5.600 ].
Peirce is right to distinguish two different aspects of thinking with hypotheses, on
which economic considerations have a bearing. The one factor belongs to what we are
calling the logic of hypothesis-engagement. The question it seeks to answer is which
of the up to an infinite number of possibilities to forward for testing, i.e., engage. The
other aspect concerns retroduction proper, or the logic of hypothesis-discharge. It deals
with the question of the conditions under which an hypothesis can be forwarded categorically. As Peirce sees it, the logic of engagement is subject to three criteria, each
economical.
CHAPTER 6. FURTHER CASES
130
Now economy . . . depends upon three kinds of factors: cost; the value of
the thing proposed, in itself; and its effect upon other projects [Peirce,
1931–1958, 7.220].
In a subsequent passage, Peirce suggests the following classification:
Economical Considerations
regarding the ‘experimental character’ of hypotheses
¸
cheapness
¸
intrinsic value
¸
– naturalness
– likelihood12
Relation of Hypotheses
– caution
– breadth
– incomplexity
[Peirce, 1931–1958, 7.223]
Bearing positively on the engagement of our hypothesis is a wider spread of factors
than are caught in the passages we’ve so far quoted: conformity to the data, explanatory
power, novelty, simplicity, detailed reliability [Peirce, 1931–1958, 1.85–1.86 ], precision, thrift [1931–1958, 4.35] coherence with other received vision, and — to a degree
— intuitive appeal (But see again [1931–1958, 1.20]). For although an hypothesis, intuitiveness involves the ‘trifling matter of being likely’ [1931–1958, 1.85–1.86 ], Peirce
also allows that
if there be any attainable truth, . . . , it is plain that the only way in which it
is to be attained is by trying the hypothesis which seem reasonable [Peirce,
1931–1958, 2.776].
In pressing his case for the economics of research, Peirce is not using the name of economics in any loosely figurative way. His is a more literal cast of mind. By economics
he means economics; he means that ‘economical science is particularly profitable to
science . . . ’ [Peirce, 1931–1958, 7.161]. Peirce postulates for the economics of research a principle of diminishing returns [1931–1958, pp. 1.22, 1.84], according to
which ‘the expected marginal utility of further investigation decreases as experimentation continues’ [Rescher, 1978, p. 70]. (See also Peirce [1931–1958, pp. 7.139], his
paper of 1879, “Economy of Research”.)
12 However,
as Rescher rightly says, ‘Peirce does not rate the value of mere likelihood (antecedent personal
probability, as it were), for it must be conditioned by considerations of the design of experiments’ [Rescher,
1978, p. 111, nt. 108]. Rescher cites a ‘very Popperian passage’ [Rescher, 1978, p. 111, nt. 108]:
The best hypothesis, in the sense of the one most recommending itself to the inquirer, is one which
can be most readily refuted if it is false. This far outweighs the trifling merit of being likely.
For after all, what is a likely hypothesis? It is one which falls in with our preconceived ideas,
. . . [which] errors are just what the scientific man is out gunning for more particularly [Peirce,
1931–1958, 1.20].
6.6. THE ECONOMY OF RESEARCH
131
It is well to emphasize that Peirce’s contributions to the logic of abduction are
offered in the context of the philosophy of science. In terms of the hierarchical conception of agency which we developed in chapter 2, Peirce’s proposals are directed to
theoretical agents, not practical. Still, although theoretical agency is defined by the
extent to which agents command epistemic and economic resources that others lack
or lack to the requisite extent, Peirce judges it to be essential to scientific competence
that its more abundant resources not be squandered. Even so, differences persistent
for theoretical and practical abducers. One of the most central of these differences
is the role of plausibility (or what Peirce calls ‘likelihood’). On the whole, Peirce is
not much impressed by the criterion of plausibility in science. In this he anticipates
Quine’s rejection of intuitions in logic and the philosophy of mathematics (and philosophy generally, come to that.
We should also again emphasize that the focus of Peirce’s economics of research is
on the matter of hypothesis-engagement rather than hypothesis-discharge.
Another point on which we see theoretical and practical agency diverging in a rather
natural way is over the issue of generality. Like Popper after him (and with scant acknowledgement), Peirce insists that science has a large stake in general propositions,
the affordably wider whose generality, the better. This turns on a point noticed in
chapter 1. General propositions are fragile. They are felled by a single true negative instance. Their fragility conduces to the realization of another Peircean (and Popperian)
desideratum, namely, that the best hypothesis for science to venture with are those that
are most open to falsification if false. General propositions are universally quantified
conditionals, and so answer to this desideratum with extraordinary fidelity. We have
said why we think that the practical agent is less well served by the universally quantified conditional, and why he might be expected to show a preference (even if tacitly)
for the generic proposition. Unlike general propositions, generic statements are compatible with (kinds of ) true negative instances. This allows Harry to be simultaneously
wrong about Pussy the tiger and yet right to hold that tigers are four-legged. If Harry,
the practical agent, is wrong to think that Pussy is four-legged, then, on discovering
the error, he has to modify his beliefs appropriately. But he does not have to modify
his generic belief that tigers are four-legged, which entails a considerable economy for
Harry. Science faces a bigger clean up bill whenever a universally quantified proposition is overturned by a counter-example. Not only must prior singular beliefs (one or
more) be corrected, but so too the discredited generalization — something that is not
always easy to do. (Holism is one way — not acceptable to Popper and presumably not
to Peirce either — of minimizing this further cost. But the costs of holistic adjustment
itself are usually also very high.)
For Peircean science, the more generality the better. For one thing, the more general
a proposition the more informative it is. And the more a proposition is informative, the
closer it lies to optimal testability. These are virtues that any Popperian would be quick
to proclaim. Peirce goes them one better. Not only are generalities more informative
and more testable, they are also cheaper. That is, generalities over comparatively large
domains are cheaper to test than generalities over smaller subdomains. To see that this
is so, consider the following four claims, each a subset of a certain domain of ¹ s. For
correctness, let ¹ be a human being over 90 years of age. (We adapt this example from
[Rescher, 1978, pp. 80–83].)
CHAPTER 6. FURTHER CASES
132
1. All (10) º s in the center of the city of Groningen have property »
2. All (1000) º s in the Province of Groningen have »
3. All (100,000) º s in North Western Europe have »
4. All (100,000,000) º s who have lived have »
Which of these generalizations is the cheapest to test? We assume sample adequacy:
i.e., that given an appropriate degree of randomness that an adequate sample of a ¼ membered population ½ is a ¼ -membered subset of it. We also assume that in selecting
¾ individuals from a population ½ , the mean cost goes down with the cardinality of
¿ by a mass-production effect. Rescher proposes, and we agree, that the average
unit cost is À–Á ÂÃ Ä Å ¾ as regulated by the production principle that the cost of the
¼JÆ Ç unit proportions to È Â ¼ . Accordingly, when case 4 is compared with case 1, we
obtain 10,000,000 more in the range of the quantifier for a 500 fold increase of testingexpenditure.
Rescher sketches a second example. Suppose that for a given abduction trigger
É there
exist 110 alternative hypotheses falling into two subgroups of 10 and 100
respectively. In group A the probability of each relative to the evidence in hand is .05;
in group B it is .005. In Group A each hypothesis has a ‘generality value’ of 20; but in
B it is 80. The cost of testing Group A hypotheses is, say 10 units each; group B tests
cost 1 unit each. Finally, we assume that our testing budget runs to a total of 120 units.
Our task is to determine values for º and Ê in the test instruction: ‘Examine º
hypotheses from A and Ê hypothesis from B.’ The expected value as regards generality
is fixed by the equation
Ë Ì Í ºKÎ Ï Ì ÐÑ–Ë Ì Ì Í ÊJÎ Ò Ì ÐKÓ º Ñ–Ë Ô Ê
Given that
Ê
º
ÕÖÈ Ì
ÕÖÈ Ì Ì
È ÌÑ ÊØ×ÖÈ Ï Ì
we obtain the following linear-programming problem:
Accordingly, º Ñ‡Ë Ô Ê ‘maxed out’ with a value of 42 at (2,100). If we work through
in a similar way all of the B hypotheses and two random selections from A, hypotheses
of greatest range have the clear economic advantage.
This is significant. It allows us undogmatically to replace Popper’s purely logical
concern for universality-for-its-own-sake with . . . the . . . economic precept ‘Maximize
generality subject to the constraints of affordability” [Rescher, 1978, p. 83]. To put the
point anachronistically, it is here that Peirce breaks decisively with Popper. Popperian
affection for generality can be an unaffordable research luxury; better then to seek for
whatever generality the research programme is able to ‘pay for’.
6.6. THE ECONOMY OF RESEARCH
133
Ù
(2,1000)
Ù
= 100
à
Ú
= 10
(10,20)
Û Ü Ú-Ý
ÛßÜ
ÙÞ
Ú
134
CHAPTER 6. FURTHER CASES
Chapter 7
Plausibility
7.1
Historical Note
We have postulated that the factor of plausibility enters into an abducer’s reasoning
in ways that may be largely implicit. We have conjectured that a virtual logic is a
description of a cognitive agent that represents (certain of) his or its cognitive processes
as behaviour of a connexivist net that meets the RWR conditions. We also imagined
that one the ways in which a cognitive agent implements plausibility and relevance
considerations is might be captured by such a logic. (We also, by the way, now have
a serviceable specification of a virtual rule. A virtual rule is a law, soft law or tight
correlation that a cognitive system can conform to without executing.)
Whether this approach to virtual logics of plausibility and relevance is correct, it
remains the case that, either way, we owe the reader an account of what we take plausibility and relevance to be. To this end, plausibility will occupy us in the present chapter,
and relevance in the next.
In common usage plausibility is equated with reasonableness. The equivalence
originates in the concept of the reasonable (to eulogon). It comprehends Aristotle’s
notion of endoxa, or opinions held by all or by most or by the wise. These are opinions
endorsed by, as we would say, common knowledge, or by a received view or by the experts. To eulogon is discussed by the Skeptic Carneades in the last century B.C., in the
context of the evidence of the senses and the testimony of experts (See here [Stough,
1969]). A related notion is the Greek eikos which means “to be expected with some
assurance”, and it may be translated as “plausible-or-probable”. Rescher claims (and
we agree) that the one meaning of eikos is captured in the idea of approximate truth or
verisimilitude, which “ultimately gave rise to the calculus of probability”, though this
was not to be a Greek development [Rescher, 1976, p. 38, n. 1]. Aristotle contrasts
eikos with apithanon, which means “far-fetched” (Poetics 1460 á 27); he also distinguishes it from what is true. In criticizing his rivals, Aristotle says, “While they speak
plausibly (eikatos) they do not speak what is true (alēthē)” (Metaphysics, 1010 á 4).
Rescher opines that the Greek identification of eikos with the probable anticipates
the Latin probabilis, which means “worthy of approbation”, and he approvingly quotes
135
CHAPTER 7. PLAUSIBILITY
136
Edmund Byrne
[Probability] refers to the authority of those who accept the given opinion;
and from this point of view ‘probability’ suggests approbation with regard
to the proposition accepted and probity with regard to the authorities who
accept it [Byrne, 1968, p. 188]:
For a discussion of the emergence and development of the mathematical conception of
probability see [Hacking, 1975].
It is well to note a present-day use of “plausible” in the Bayesian analysis of prior
probabilities. This may or may not be a usage foreshadowed by Peirce’s understanding
of the plausible.
By plausibility, I mean the degree to which a theory ought to recommend
itself to our belief independently of any kind of evidence other than our
instinct urging us to regard it favorably [Peirce, 1931–1958, 8.223].
Rescher suggests that Peirce’s notion “seems closer to the idea of an à priori probability” than to the idea of being worthy of approbation [Rescher, 1976, p. 39, n. 1]. In
this we agree. If Rescher and we are correct, then we learn something important about
how Peirce understands abduction. Contrary to what we have been proposing, it is no
of what Peirce takes abduction to be to cut down a set such as â to a set such as
ãpart
. In other words, solutions of abduction problems are not shaped by what the abducer
takes to be plausible.
Here is a further remark of Peirce:
As we advance further and further into science, the aid we can derive from
the natural light of reason becomes, no doubt less, and less; but still science
will cease to progress if ever we shall reach the point where there is no
longer an infinite saving of expense in experimentation to be effected by
care that our hypotheses are such as naturally recommend themselves to
the mind . . . [Peirce, 1931–1958, 7.220].
This passage derives from about 1901. Ten years earlier Peirce was making a similar
point. He took it that evolution has rendered the axioms of Euclidean geometry “expressions of our inborn conception of space”. Even so, “that affords not the slightest
reason for supposing them to be exact”; indeed space may actually be non-Euclidean
[Peirce, 1931–1958, 6.29].
7.2
Characteristicness
In our discussion of previous cases, we entertained a loose connection between of
plausibility and characteristicness. The analysis of The Burglary (section 6.1) found
it natural to take judgements of characteristicness as a species of generic reasoning.
This in turn suggests a tie between plausibility and genericity. It would be well to
examine these connections more closely.
7.2. CHARACTERISTICNESS
137
Consider a candidate space ä for an abduction target å . When discussing The
Burglary, we proposed that propositions are excluded from ä where they are contraindicated by what is characteristic. Thus although “Carol has come home early” was
a member of John’s ä , it was excluded by the fact that it is characteristic of Carol not
to come home early on weekdays. This was the basis on which we then said that the
Carol conjecture was implausible.
Generic claims are also claims about what is characteristic. So it is natural to think
of characteristicness as intrinsically generic. A little reflection serves to call the claim
into doubt. Penguins don’t fly. This is a generic fact about penguins, and saying so
attributes what is characteristic of them. What, then, do we say about Harry’s pet
penguin, Hortense? Of course, Hortense doesn’t fly either, and that would seem to be
characteristic of her insofar as she is a penguin. Carol is unlike this. It is characteristic
of her not to come home early on weekdays; and Carol is a woman, mother, teacher and
lots of other things. But it is certainly not characteristic of women or mothers or even
teachers not to be home at 4:30 on weekdays. Carol does not imbibe this characteristic
from what is characteristic of any class or natural kind, of which she is a member.
If genericity is intrinsically of a certain kind of generality, then what is characteristic
of Carol in The Burglary case is not generic to Carol. On the other hand, even in
cases of particular characteristicness, the factor of generality is not lost entirely. If it
is characteristic of Carol not to be home early, then it is true to say that Carol doesn’t
generally come home early on weekdays. But the generality imputed by the use of the
adverb “generally” is not the generality of Carols (all Carols, so to speak) but rather is
all weekdays between September to July. All weekdays at 4:30 are such that Carol is
not home from school then.
What is characteristic of Carol is what Carol would or would not do. Something
that Carol would do is often what Carol has a standing desire to do or preference for
doing. What Carol is like is often just a matter of what she likes. Carol likes to do her
chores after school lets out at 3:30. Carol prefers to shop each day rather than weekly.
Carol likes to drop into Macabee’s Books for coffee and to browse. And she doesn’t
like to get home too early; it is less interesting to be home alone than drinking coffee
at Macabee’s.
Dora’s situation presents a different fix on characteristicness. Dora is never at
Carol’s house on Thursday, whether at 4:30 in the afternoon or any other time. It is
not just that Dora wouldn’t like to be there then (although in her actual circumstances
this is certainly true). The point rather is that Dora can’t be there on Thursdays. She has
other clients to whom she is committed on Thursdays. Dora is a good professional. She
knows that clients much prefer to have an assigned stable time at which Dora appears
in their households. Except where strictly necessary (and always with appropriate notice), Dora never changes these commitments (a further characteristic which turns on
what Dora is like, and in turn, in this instance, on what she likes). But the basic fact
is that Dora never comes to Carol’s house on Thursdays because she cannot. So externalities, as economists call them, can determine what is characteristic for an agent to
do.
What is characteristic of Dora is not characteristic of various of the class of which
she is a member. It may be characteristic of professional cleaners that they don’t deviate
from fixed commitments, week in and week out. But this flows not from what it is to
138
CHAPTER 7. PLAUSIBILITY
be a cleaner, but rather from what it is to have a good feel for customer relations. So
it would appear that what is characteristic of Dora is something generic to some of the
things she is, but not others.
It is widely supposed that “Birds fly” is a generic claim. For this to be true, it would
have to be characteristic of birds to fly. That is, flying would be tied up with what it is to
be a bird. Of course, this is not so. Penguins are birds, and penguins don’t fly. Similarly
for turkeys, and ostriches, and lots of other birds. What this suggests is that sentences
such as “Birds fly” do not express a generic proposition, but rather are under-expressed
instances of the quantified sentence “Most species of bird fly”. (We note, in passing
that “Most birds fly”, while true, is not what “Birds fly” asserts.) Generic sentences,
on the other hand, are not quantificational, although they imply quantified sentences.
“Penguins don’t fly” is a generic claim and true. It implies, but does not assert, that
most penguins don’t fly. The sense in which generic claims are generalizations is the
weak sense in which they imply generalizations, but are not equivalent to them.
One of the attractions of this approach to genericity is the light it throws on the
concept of default. Sometimes a default is understood as any claim which might in fact
be mistaken. This is an unwisely liberal use of the term. More soberly considered, a
default is the propositional content of an inference from a generic claim. If we infer
from “Tigers are four-legged” that Pussy the tiger is four-legged, that is a default.
Saying so is saying more than that it might be untrue. It is saying (or implying) that
it might be untrue in a certain way. If Pussy is not four-legged, then given that its
negation is a default, it is essential that although Pussy is not four-legged it remains
true that tigers are four-legged. Those that aren’t are so in ways that does no damage
this truth. They are non-four-legged adventitiously (a leg-trap casualty, for example)
or cogenitially (a birth-defect).
This is not to say that propositions such as “Tigers are four-legged” can’t be false,
that they can’t have genuine counterexamples. We might come upon a heretofore unknown species of five-legged tigers. This would falsify “Tigers are four-legged”, but
not “Most species of tiger are four-legged”. It would no longer be characteristic of
tigers that they are four-legged.
We now have the resources to define defaults.
Default: S is a default iff there is a generic claim G of which S is an
instance; and S is such that if G is true, S’s falsity does not falsify G.
The proposition that Carol is not now home at 4:30 is a default. It is an instance of
the generic claim that Carol doesn’t come home early on weekdays. Of course, upon
entering the house, John might see that today Carol has come home early. This might
be explained in either of two ways. Carol might be sick. Or she might tell John that
she’s grown weary of getting home so late, and has decided to come home straight from
school from now on. Each of these gives a different way for the default to be false. The
first way, it remains true that it is characteristic of Carol not to come home early. The
limits of the generic claim is undamaged. The other way is different. Carol’s early
home-coming instantiates a new policy that retires the old habit.
There is a third possibility. Carol’s early homecoming is inexplicable. “I don’t
know”, she says, “I just felt like it”.
7.2. CHARACTERISTICNESS
139
Let us briefly regroup. Our point of departure is The Burglar. Our present task is to
say something useful about the property of characteristicness.
We see that characteristicness is bound up with the concept of genericity and default. So, instead of attributing (as we did in 6.1) defaultedness to generic claims, we
proposed that its more fruitful application is to instantiations of generic claims. Thus,
the generic claim
(1) Tigers are four-legged
is fallible (though not a default), and its instance
(2) Henry is four-legged
is both fallible and a default.
Why? What’s the difference between the fallible non-defaultness of (1) and the
fallible defaultness of (2)? The answer is that (2) derives its plausibility from (1),
and is subject to downfall in the face of a single true counterinstance. But (1) is not
withdrawn for just any reason for which (2) is withdrawn. (1) remains true even though
(2) is false. (1) and (2) have different logics, as we might say. They are both fallible,
but in different ways. One could be mistaken about the four-leggedness of tigers, but
not every way of being mistaken about the four-leggedness of Henry is a way of being
mistaken about the four-leggedness of tigers. We require that there be a way of marking
this distinction. There is such a way: (1) is generic, and (2) is a default.
How does this play upon the abductively important notion of characteristicness? It
appears that the generic-default distincition partitions the charactertisticness property
in the following way. If (1) is true, then it is characteristic of tigers to be four-legged;
but it isn’t characteristic of Henry to be four-legged. Another way of saying this is that
the truth of (1) confers characteristic four-leggedness on tigers and yet that characteristicness is not closed under instantiation.
This makes The Burglary problematic. We said that (2) is not a statement of charcteristicness. And (2) is a singular statement. This might lead us to suppose that singular
statements can’t be statements of characteristicness. This is belied by
(3) Carol is characteristically not at home early on weekdays.
(3) is a singular statement, and a statement not obviously an instance of from any
generic statement. So, it would appear that it can’t be true that even where a singular
statement is a statement of characteristicness, it is not so because it is a default, as we
are presently defining it.
But, on the contrary, (3) is a statement of characteristicness and is a default, and yet
does not qualify as a default by intantiation from a generic statement that attributes the
same characteristicnes that (3) itself plainly attribute.
Of course, we tried earlier to find a generic statement of which (3) can be an instantiation. This required a certain regimentation (which, as Quine has taught us, is
tendentiousness in a good cause). We proposed that (3) be re-issued as
(5) It is characteristic of this day that Carol not be home early
which could then be taken to as derived from
CHAPTER 7. PLAUSIBILITY
140
(4) Weekdays are such that Carol is not home early on them.
It won’t work. Compare (1) and (2) with (4) and (5). In the case of the first pair,
(1) does all the work on behalf of characteristicness. Even if we held our noses and
allowed that (2) also attributes characteristicness, it would be clear that its functioning
thus is wholly parasitic on the characteristicness attribution embedded in (1).
With (4) and (5) it is the other way around. With (1) and (2) it is consistent to say
(6) Even though (2) is not a statement of characteristicness, (1) nevertheless is a statement of characteristicness
But we can’t say
(7) Even though (5) is not a statement of characteristicness, (4) nevertheless is a statement of characteristicness.
From this we may infer that:
(8) That (4) is a statement of characteristicness does not derive from any
generic claim of which it is an instantiation.
(9) Whether a statement is a default depends on its relation to attributions
of characteristicness. In particular, (2) is a default because it instantiates
a statement of characteristicness; and (4) is a default not because it instantiates a statement of characteristicness, but rather because it is itself a
statement of characteristicness.
Thus, on this view, defaultness and characteristicness occur in contexts linked by equivalence.
What now of plausibility? The Burglar example suggests that
(10) Defeasibly, possibilities that contradict (statements of) characteristicness are not plausible candidates.
7.3
A Cautionary Note on Plausibility
Let proposition æ be a candidate for consideration as a hypothesis ç . Bearing on this
issue is whether (or the extent to which) æ passes the plausibility test.
Some abductive logicians (e.g., Perice and Popper) are of the view that (contrary
to (10)), the plausibility of æ has nothing to do with whether to adopt it as an ç . The
reason for this is that it is not in the abducer’s remit to determine whether to infer æ ,
but only to determine whether to send æ to trial. Actual experience amply attests to
trial-determination that proves successful even where æ is implausible. So plausibility
is not needed at the beginning and it is not needed it the end — for if, in the end, æ is
inferred this will be because it has fared well under conditions of trial. In such cases, it
is not plausibility that counts, but inductive strength.
The kind of plausibility that is here given such short shrift we may call propositional plausibility. In 1900 the proposition that quanta exist was an implausible proposition, it was, as we sometimes remark, an implausible thing to say or suggest.
7.4. RESCHER’S PLAUSIBILITY LOGIC
141
This contrasts with what we might call strategic plausibility. Judgements of strategic plausibility are judgements in the form
è
is a plausible candidate for trial.
The difference is that propositional plausibility involves what it is reasonable to believe,
whereas strategic plausibility is a matter of what it is reasonable to do.
Peirce’s subdiscipline which he calls the economics of research is an effort to specify conditions on strategic plausibility wholly independently of considerations, if any,
of propositional plausibility.
A realistic guide to strategic plausibility involves extrapolation from known connections. This is well-illustrated by van den Bosch’s discussion of drug leads and drug
designs. We might call this quantitative extrapolation. Another involves analogical
extension. E.g., given that
éëêíì
ì-î
ì
î
î
suppose
ì î that resembles and ï<ï-ð resembles ð . Therefore, test ð with respect
to .
But what is this similarity? We return to this question two chapters hence.
7.4
Rescher’s Plausibility Logic
Rescher sees plausibility as essentially tied to authoritative sources, of which expertise
is an important instance. One of the more dramatic things about experts is that they
disagree. How might these disagreements be resolved?
We put it that experts on a given subject form a poset under the relation “has greater
expertise than”. Plausibility in turn can be construed in ways to be detailed below.
Suppose that an authority or group of authorities has made a number of pronouncements, and that the set of propositions vouched for by these authorities is collectively
inconsistent. Suppose further that we can rate the plausibility or reliability of these
authorities in some comparative way. Is there a rational way to deal with such a situation? One such procedure, developed by [Rescher, 1976], is called plausibility screening. Rescher’s method is to scan the maximally consistent subsets of the inconsistent
totality and give preference to those that include the maximum number of highly plausible elements. This general process can also function in other ways. For example, for
some purposes we might want to give preference to those sets that include as few lowplausiblity elements as possible. It depends on whether our goal is to maximize overall
plausibility or minimize overall implausibility. The two policies are not identical.
Suppose that we are given a fragment of what appears to be a thirteenth-century
manuscript on logic. It has been examined by three experts on historical manuscripts
on the logic of this period, Professors, ñ , ò , and ó . Let us say that we can rate their
respective pronouncements on a scale of one to ten as follows: ñ has a comparative
reliability of 8, ò has a rating of 5 and ó a rating of 2. (Even though ó ’s rating is low, it
must be emphasized that ó is a bona fide expert, and that is low rating is a comparative
matter.) Professor ò ventures the opinion that the manuscript was authored by William
of Sherwood, the thirteenth-century Oxford logician, or by William of Ockam, his near
CHAPTER 7. PLAUSIBILITY
142
contemporary. Professor ô asserts that if the document were authored by William
of Sherwood then it would definitely make reference to Aristotle’s doctrines on logic.
But, he adds, no such reference to Aristotle is made. Professor õ points out that if the
document was authored by William of Ockam, then from what we know of William of
Ockam, it would include references to Aristotle’s doctrines.
We are supposing, then, that these authorities vouch for the following propositions:
Authority ô (who has a reliability value of 8): ö³÷–ø , ùKø
Authority ú (who has a reliability value of 5): öLû#ü
Authority õ (who has a reliability value of 2): üë÷–ø ,
where
ö„ý
øëý
ü³ý
The manuscript was authored by William of Sherwood.
The manuscript makes reference to Aristotle’s doctrines on logic.
the manuscript was authored by William of Ockam.
We now put it that the reliability rating of a proposition is the same as the reliability rating of the expert who vouches for it. The sets of given propositions, þ öÿ÷
ø ùFø ö ü ü ÷ ø , is inconsistent, as a truth table will show, but it has four
maximally consistent subsets as follows:
1.
2.
3.
4.
þö ü ö³÷–ø üë÷–ø , rejecting ùFø
þ ö ü ö³÷–ø ùKø , rejecting üë÷–ø
þö ü üë÷–
ø ùF
ø , rejecting ö³÷–ø
þ ö³÷–
ø üë÷–
ø ùK
ø , rejecting ö ü
.
Notice that (1) and (3) both reject one of the highly rated pronouncements of ô . Therefore, given that we want to maximize plausibility, we can eliminate both (1) and (3) as
candidate subsets. Looking at the remaining two subsets, we see that we have a choice
between rejecting ü ÷dø (which has reliability (2)) and ö
ü (which has reliability
(5)). Again, since our policy is to maximize plausibility, we will want to reject any
alternative that excludes propositions of relatively high reliability. So the choice here
ü , a proposition that
is straightforward as well. We reject (4) because it excludes ö
is more reliable than ü ÷dø , the proposition excluded by (2). All told, then, the most
plausible maximally consistent subset is (2). Thus, on this model, the rational way to
react to inconsistency in this instance is to accept the pronouncements of ô and ú and
reject the opinion of õ . Note that the plausibility of (2) suggests that the manuscript
was authored by William of Ockam, since (2) logically implies ü .
In our example, we selected the most plausible subset by pruning the original, inconsistent set of data. We identified the most plausible maximal consistent subset of
as that which excludes propositions of least reliability. The method of plausibility
screening tells us even more, however. If we look at (2) and (4), the preferred subsets
of our example, we see that the two propositions ö ÷„ø and ùFø each appear in both
(2) and (4). In other words, no matter which of (2) or (4) we decide to accept, we are
going to accept ö„÷–ø and ùKø . These two propositions therefore constitute something
akin to a “common denominator.” We also see that in this case the pronouncements of
7.4. RESCHER’S PLAUSIBILITY LOGIC
143
X have a certain preferred status: no matter whether we reject the opinions of by
rejecting (2), or reject the opinions of . Plausibility screening also tells us that (2)
is not consistent with (4), as a truth table will show, and thus that we must choose between (2) and (4). As we say, in this case it is preferable to select (2); but in choosing
either (2) or (4) we will still be accepting the common subset
.
The method of plausibility screening consists of several steps: First, we take the
original set of pronouncements of our authorities and test them for consistency by
constructing a truth table. If the set is inconsistent, we then determine all the maximally
consistent subsets of this set. We then look over the alternatives and reject those sets
that tend to exclude the most reliable propositions until only one set is left. In the event
of a tie, we look to see if there is a common component among the tied sets. As well,
we can try to discover whether there is a common component among any number of
the maximally consistent subsets that tend to be preferable.
We could bend our example to form an illustration of a tie: suppose that
is
assigned a reliability value of 5. The (2) and (4) would be tied. Whichever set we
rejected, we would be rejecting a proposition of value 5. Both (2) and (4) are preferable
to (1) and (3), but we cannot narrow the field down to one proposition. In this context,
the best we can say is that we will want to accept
because it is common
to both (2) and (4).
Here is a second example, which we develop in a little more detail. Suppose we
have the following pronouncements of three authorities, , , and :
Authority
Authority
Authority
(who has a reliability value of 9):
(who has a reliability value of 7):
(who has a reliability value of 2):
! .
First, we construct a truth table with a column representing the truth values for each
proposition in the set of propositions stated by the experts:
"#$%$%"# !
(1) T T T
T
F
T
F
T
F
(2) T T F
T
T
F
F
T
F
(3) T F T
F
F
T
F
T
T
(4) T F F
F
T
T
F
T
T
(5) F T T
T
F
T
T
T
T
(6) F T F
T
T
F
T
T
T
(7) F F T
T
F
T
T
F
T
(8) F F F
T
T
T
T
F
T
Second, we can scan the truth table and highlight all combinations of true propositions, omitting only those that are proper subsets of others. Some rows, like (1), (2),
(3), and (7), will have true propositions in them, but these patterns of true propositions
will already be included in one or more of the other rows. For example, the true propositions in (1) form a subset of those in (5); the true propositions in (2) form a subset
of those in (6); the true propositions in (3) form a subset of those in (4); and the true
propositions in (7) form a subset of those in (8):
CHAPTER 7. PLAUSIBILITY
144
&"'%($&)'%*($')(%*&"&+'%*, &+'(1) T T T
T
F
T
F
T
F
(2) T T F
T
T
F
F
T
F
(3) T F T
F
F
T
F
T
T
(4) T F F
F
T
T
F
T
T
(5) F T T
T
F
T
T
T
T
(6) F T F
T
T
F
T
T
T
(7) F F T
T
F
T
T
F
T
T
T
T
T
F
T
(8) F F F
Third, we look at the truth table, one row at a time, and list those propositions
that are true in each highlighted row. For example, in (8) we see that the following
propositions are true:
,
,
,
, and
. Reading off the true
propositions for the remaining rows that contain highlighted propositions, we find that
we have a total of four maximally consistent subsets:
&).' *( '/).( *&
(4)
(5)
(6)
(8)
*0, &12'!-
)(4 &+'4 *, &1'- 5
3 *&()4 '
'
4
)(4 *&!4 &+'4 *0, &1'- 5
3 &)'4 '
*
(
3 &)'4 *(44 *'6&!)4 &(+4 *'&!4 4 **, , &&11''- - 5 5
3
Fourth, for each such maximally consistent subset, we list at the right those propositions of the original set that are not included.
(4)
(5)
(6)
(8)
)(4 &+'4 *, &1'- 5 , rejecting &)'4 *&
3 *&()4 '
'
4
)(4 *&!4 &+'4 *0, &1'- 5 , rejecting *(
3 &)'4 '
*
(
)(
3 &)'4 *(44 *'6&!)4 &(+4 *'&!4 4 **, , &&11''- - 5 5 , rejecting '
&
+'
3
, rejecting
.
Fifth, we scan the maximally consistent subsets in order to construct a preference ordering. The general rule here is that any set that rejects a highly reliable proposition
should be eliminated. Clearly we can eliminate (4) because it rejects
, which
has a value of 9. Likewise (5) must be eliminated because it rejects
, which also has
value 9. That leaves (6) and (8). Here the choice is also clear. Row (6) rejects
(which has a value of 7), whereas (8) rejects only
(which has a value of 2). On
the policy that plausibility is to be maximized, we therefore eliminate (6) and accept
(8).
Sixth, in the event of a tie, we look to see if there is a “common denominator” subset
that should be accepted even if it is necessary to reject all of the maximally consistent
subsets. Looking over the maximally consistent subsets, we see that the subset
is common to both (6) and (8). Furthermore, we see that
is common to all four maximally consistent subsets. These common
components could serve as tiebreakers, although in this case this is not necessary —
(8) stands out as the clear winner. Nonetheless, it is interesting to note that despite its
low individual plausibility,
is highly acceptable because it is “carried along”
in (8) and in every other maximally consistent subset.
Finally, it is worth emphasizing that there may exist undecidable cases. If authority
says and authority says , and if both authorities are equally reliable and this
is all the information we are given, then plausibility screening will not tell us which
*(
&+'
*, &91'-
;
'6).(
3 &)
'4 *(4 *&!4 *0, &81'- 5
3 *0, &1'!- 5
:
&7)'
<
*;
7.5. THE RESCHER AXIOMS
145
proposition to accept. It is simply a stalemate, and we have to wait for more information
before making our decision about whether to accept or reject .
Critics of Bayesianism will no doubt be quick to see a similar weakness in the
theory of plausibility screening. Just as there appears to be no general way to assign
prior probabilities, the same would appear to be the case for prior plausibilities, or what
Rescher calls plausibility-indices (see the section to follow). It is true, of course, that in
real-life we make these assignments when we are confident that they give approximate
expression to our intuitive judgements or our commonsense estimates of which is the
greater expert. But, it remains the case that there is no general method for plausibility
indexing even for classes of human claimants to expertise. Beyond that, it matters that
what we take as plausible ranges far beyond the reach of any human expert — a point
to which we shall briefly return.
Perhaps the greatest difficulty with setting prior plausibilities lies in the claim that,
in the absence of a general method, we do this intuitively or under the guidance of commonsense. What can “intuitively” mean here, if not “in ways that we find plausible”?
And what can the guidance of commonsense be here, if not the counsel afforded by our
judgements of characteristicness? Either way, we import the notion of plausibility into
our account of how to set prior plausibilities. The ensuing circularity requires that we
cannot embed these plain truths in our account of plausibility. This is consequential.
It leaves plausibility theory impotent to lay down principled conditions on plausibility
indexing.
On the positive side, the theory of plausibility adjudicates conflicting expert testimony eucumenically and realistically. Instead of an all-or-nothing favouritism for the
total testimony of the most qualified expert, the theory seeks to accept the biggest part
of the combined testimony of all the experts that it consistently can and the lowest cost
(i.e., rejecting claims of least plausibility). Even so, while the idea of saving as much
of the joint testimony of all the experts that consistency and plausibility-cost will allow
is a good one, Rescher’s specification of how this is achieved is problematic.
Consider two cases. Experts
and have been sworn in a criminal trial. Each
gives psychiatric testimony concerning the question of diminished responsibility.
has an expert raking of 8, and has a ranking of 7. In the first case, and agree on
everything each other says save for one proposition (e.g., that the defendant had at most
diminished responsibility for his action). In that case, the nod goes to ’s testimony.
In a second case, everything is as before except that and disagree on everything
each other testifies to. Here the nod also goes to . More generally, in all such cases,
the nod goes to the most highly ranked expert except when one of them introduces
testimony which is neither repeated nor denied by the other. Doubtless there are lots
of real-life cases in which this is so. But as our examples show there are other cases
in which the method of amalgamation is just the all-or-nothing strategy in favour the
most highly ranked expert’s total evidence.
=
>
?
?
>
>
7.5
>
?
?
>
>
The Rescher Axioms
Having sketched the method of plausibility screening, we shall now expound Rescher’s
view of plausibility in somewhat greater detail. Readers interested in the complete story
CHAPTER 7. PLAUSIBILITY
146
are advised to consult [Rescher, 1976]. We begin with the crucial notion of plausibility
indexing. The integer 1 denotes total reliability and gives effective certainty. Rescher
rates disciplines such as logic and mathematics as having reliability-value 1. In all other
cases, 1-n/n, n-1/n, n-2/n, . . . , 1/n denote diminishing degrees of positive reliability.
Thus even the least reliable of sources is reliable to some extent. 1/n denotes minimal
positive reliability.
Corresponding to a reliability index for sources is that of a plausibility index for sets
of propositions. Such sets Rescher calls p-sets; they are sets of propositions endorsed
by sources of positive reliability. Indexing of p-sets S is subject to four axioms. By
axiom 1 (metrization) every proposition in S has a logical value (
). Axiom
2 (L-truth maximization) every truth has plausibility values 1. By axiom 3 (compatibility), all propositions of plausibility 1 are mutually co-tenable. Axiom 5 (consequence)
provides that for any consistent sets of proposition in S and any proposition in S
and any proposition in S entailed by it, the entailed proposition cannot have a lower
plausibility value than any proposition in . Axiom 5 permits that a proposition and
its negation can have plausibility values as high as they come, short of 1. By axiom
6, differentially plausible rival propositions are to be adjusted in favour of the higher
(highest) plausibility value.
@ [email protected]
D
D
7.6
Plausibility and Presumption
In section 7.2 we considered the link between the plausible and the characteristic. In
the present section we shall examine briefly a proposed link between plausibility and
presumption. As it is developed in Walton [1992b] plausibility is subject to two restrictions, both of which we regard as inessential. One is that plausibility is a property
instantiated only (or paradigmatically) in dialogues. The other is that such dialogues
involve only (or paradigmatically) everyday matters. If plausible reasoning were intrinsically dialogical and instrinsically quotidian then Walton’s account would be of
little value in the explication of abduction, which is neither. But as we say, we see
nothing in Walton [1992] that shows these constraints to be essential; consequently we
shall decline to impose them. We agree however that, just as abduction is a certain
kind of enquiry, presumption can also usefully be relativized to enquiry. Thus whether
something is reasonably presumed or not will in general be influenced by the kind of
inquiry in process.
Walton’s central idea is that plausible reasoning “needs to be understood as being
based on and intimately connected to presumptive reasoning” [Walton, 1992, p. 4]
Presumptive reasoning, in turn, is held to contrast with both deductive and inductive
reasoning. “Presumptive reasoning”, says Walton, “is inherently subject to exceptions,
and is based on a defeasible general premise of the form ‘Typically (subject to exceptions), we can expect that if something has property F, it also has property G’” [Walton,
1992, p. 4]. If it appears unclear as to why presumptive reasoning can be neither deductive nor inductive, Walton says that “deductive reasoning is based on a universal
general premise of the form ‘All F are G’, and inductive reasoning is based on a probabilistic or statistical general premiss to the effect that most, or a certain percent [sic] of
things that have properly F also have property G” [Walton, 1992, p. 4]. We won’t take
7.6. PLAUSIBILITY AND PRESUMPTION
147
the time to subject these caricatures to the derision that they so richly deserve. Let it
suffice that properly understood, the contrast class for presumption is neither deduction
nor induction, and that, even as characterized by Walton, presumptive reasoning cuts
across the grain of these distinctions. What makes this so is the role of presumptive
premisses in arguments of all kinds. What Walton misses is that presumptive inference is two different things: (1) inference to a presumption, and (2) inference from a
presumption.
Walton distinguishes among three kinds of presumptions — required, reasonable
and permissible.
E
A proposition is a required presumption in an inquiry if and only if, (i)
whether or not is true is open to further argument or inquiry, and (ii)
may be inferred from what is presently known in the inquiry, and (iii)
must be inferred from what is presently known in every inquiry of this
type, in the absence of special circumstances [Walton, 1992, p. 53].
E
E
E
E
A proposition is a reasonable presumption in an inquiry if and only if the first two
conditions are met and “ may reasonably be inferred from what is usually expected
in relation to what normally can be expected to be found in this type of enquiry, in
the absence of exceptional circumstances” [Walton, 1992, p. 53].
is a permissible
presumption in an inquiry when “may be inferred from what is known, but does not
have to be” [Walton, 1992, p. 53]. We note in passing two unattractive features of
these definitions. One is their imprecision. There is, for example, no discussion of the
difference between “may be inferred” (condition (ii)) and “may reasonably be inferred
” (condition (iii) on reasonable presumption). Also unexplained is the meaning
of “must be inferred
” (condition (iii) on required presumption.) Apart from this
inexactitude there is also at least a hint of circularity. Thus something is a reasonable
presumption when it is reasonable to infer it.
Perhaps the greatest difficulty with this account is making sense of the tie between
presumptive reasoning and plausible reasoning. Though the connection is asserted,
it is not explicated.1 Even so, it may be possible to reconstruct the tie on the basis
of what Walton says about presumption. Two points are particularly important. One
is that the inquirer be undecided as to the truth or falsity of any proposition that he
holds presumptively. The other is that the proposition “may be inferred” from what
is known of situation in which the presumption arises. Jointly, presumptions are epistemically legitimate to hold even when they chance to be untrue. This legitimacy is
off-set by the tentativeness with which a presumption is held or forwarded. This goes
some way toward capturing Rescher’s idea that “plausibility is a weaker counterpart to
truth” [Rescher, 1995, p. 688]. Nowhere, however, does Walton speculate on the conditions under which our present state of knowledge makes it epistemically respectable
defeasibly to hold a proposition that might be false. This leaves his account of plausibility woefully underdescribed. A proposition is plausible when it is an epistemically
respectable possible falsehood. So is a probability statement in the closed interval [0,
1]; but Walton insists that statements of probability are not statements of plausibility.
E
E
FFF
E
FFF
1 The great oddity of [Walton, 1992b] is that, notwithstanding its title, Plausible Argument in Everyday
Conversation it is a book with almost nothing to offer in the way of a positive account of plausibility.
148
CHAPTER 7. PLAUSIBILITY
Plausibility, says Rescher turns on a claim’s credibility via the acceptance-justifying
backing that a duly weighted source (human, instrumental, or methodological) can provide. Thus if we think of informative sources as being graded by reliability, then the
plausibility of a contention is determined by the best authority that speaks for it. A
proposition’s plausibility accordingly depends on its probative status rather than on its
specific content in relation to alternatives [Rescher, 1995, p. 688].
Walton, too, acknowledges a role for expertise in presumptive reasoning (see, e.g.,
[Walton, 1992, pp. 259–262 ]). The difference is that Rescher makes the factor of authoritative reliability intrinsic to plausibility; whereas for Walton is a contingent matter,
only occasionally sufficient for plausibility.
Plausibility, we said, involves something characteristic embedded in the phenomena and possibilities under the abducer’s consideration. What is characteristic of Carol
is that she is never home early on weekdays. Of course on any given day, she might
in fact have come home early, but that is a possibility wholly computable with what is
characteristic of her. There is also a clear sense in which the singular statement “Carol
is not home now [at 4:30]” is a default from the generic claim that Carol doesn’t come
home early, and a sense, also, in which the default statement can be called presumptive. John presumes until he knows better that Carol has yet to come home. And what
he presumes on this occasion is plausible because the generic proposition that Carol
doesn’t come home early on weekdays is true and because it licenses the default in
question.
But these are interconnections that are not discussed by Walton.
Rescher’s notion of plausibility makes authoritative reliability the pivot. In his
conception, plausibility is an inherently economic matter. It is cheaper to make do with
the plausible than to hold out for the true. It is a conception tailor-made for our notion
of the practical agent, who must prosecute his cognitive and conative agendas under
press of scarce resources. The practical agent, we said, is someone who discharges
his cognitive tasks on the cheap. If part and parcel of those cognitive economies is
satisficing with regard to the plausible, it is reasonable to suppose that recognizing the
plausible is something that beings like us are reasonably adept at. Certainly one of the
economies practised by practical agents is reliance on the say so of others. Rescher
recognizes three classes of such reliance: human (“Is this the way to Central Station?”,
“Modern science has discredited the idea that intelligence is accurately measured by
IQ-tests), instrumental (“The thermometer registers 40 degrees C, so I’ve got a very
bad fever”), and methodological (“Strings are the simpler hypothesis”).
Rescher also sees a link between plausibility and presumption. He cites
The Fundamental Rule of Presumption A positive presumption always favors the most plausible contentions among the available alternatives. It
must stand until set aside by something yet more plausible or by direct
counter-evidence ([Rescher, 1976, p. 55]; cf [Rescher, 1977, p. 38]).
Although the leanness of his exposition makes it difficult to determine with confidence,
Walton appears to lodge his notion of plausibility in the conceptually prior idea of
presumption. Thus a proposition is plausible to the extent that it has the appropriate
presumptive bona fides. For Rescher, it is the other way round.
7.6. PLAUSIBILITY AND PRESUMPTION
149
For plausibility can serve as the crucial determinant of where presumption
resides [Rescher, 1977, p. 37].
A number of theorems are provable from the Rescher axioms. Here are some of the
more important ones. Proofs are easily reconstructed and are here omitted.
Theorem 1 If Q follows from P, then P’s plausibility value cannot be greater
than Q’s.
Theorem 2 Interdeducible propositions have the same plausibility values.
G H
I GJKHL M S, the plausibility value of I GJKHL
H . In symbols N GJHN OPQ RS N GN T N HN U
Theorem 4 For any G , H , and I GVHL M S, max S N G8N T N HN UXWN GVH6N
Theorem 5 For any G , H , I GJ9HL , I GV9HL M S, if NGYN ONXHZN , then
N GJH6N ON GN ON H6N WN GVH6N
Theorem 6 For all [ , N \^]X_ `]^aKN W!N `![N ; and N `[N WN b ]X_ `]^aKN .
Theorem 3 For any , and
= the lesser of those of and
G
P-sets are not subject to deductive closure. However, when a p-set is consistent it is
possible to extend the plausibility index of S to cover S’s deductive closure. Let P be a
consequence of S (which may or may not occur in S). Then the sequences
G!c dc T Gd dc T e e e T G!f dc g
G c TG d TeeeTG f h
G c i T G d i T e e e T G!f j
i
are the propositions in S that imply P in the various rows of a plausibility matrix. Then
to determine the plausibility index of the deductive closure of , select the maximum
of the minima
G
PQ Rlk G k i
for each such sequence. Thus
N GN OP2[ ] j P Q Rlk G k i
The inconsistency of S is another matter, as we have seen in our earlier discussion
of plausibility screening. In the case of an inconsistent p-set S, “the automatic addition
of logical consequences must be avoided.” Plausibility screening is Rescher’s most
general method for handling inconsistent p-sets, although other possibilities are also
discussed [Rescher, 1976, ch. 5 and 6]. Since our main interest in this chapter is the
explication of plausibility, we shall not expound these alternative possibilities.
Central to Rescher’s conception, as with some of the ancients, is that the reasonable
is not to be equated with the probable in its Bayesian sense. The difference between
plausibility and probability is attested to in a number of ways, some of the more significant of which are their respective treatments of the logical operators. The calculus
of plausibility (as Rescher calls it) is not closed under negation. If
is known,
this is not generally sufficient for the determination of
. But the probability
of
the probability of . The negation operator always raises or lowers
I mGL2O/o p
G
NmGnN
NGZN
CHAPTER 7. PLAUSIBILITY
150
the probability of the proposition negated; and probability always degrades conjunctive
probability. Thus, where and are independent,
. But
negation does not in general raise or depress plausibility nor does conjunction degrade
plausibility. Nor is
definable when is inconsistent, whereas inconsistent
sets are freely open to plausibility indexing.
There is much to approve in Rescher’s account of plausibility, especially in the
contrast he draws between plausibilities and probabilities. There is less to be said for
how prior plausibilities are handled. Rescher is quite right to have accorded epistemic
significance to authoritative sources. You turn on the radio and hear that Kabul has
been bombed. So, you know that Kabul has been bombed, albeit defeasibly. You ask
whether this is the way to Central Station, and you are told that it is. So now you know,
albeit defeasibly. You consult the bruised horizontal slash in a prairie sky, and you
know, albeit defeasibly, that both and wind and temperature will soon rise. You see the
spots on your child, and report chicken pox to your pediatrician. John sees the open
door, and assures himself that it wouldn’t have been Carol’s doing; John’s knowledge
of Carol’s habits is authoritative. You hear the drumming on the roof and, looking out,
you see that it is raining hard. But now your source is impersonal; it is the evidence
of your senses. You want to know what your bank balance is. So you balance your
chequebook. Here, too, your source is impersonal. It is the evidence produced by your
calculations.
This is problematic. Impersonal epistemic sources are patches of evidence or arithmetic rules, or some such thing. So far as is presently known, evidence is good as
opposed to bad, or better rather than worse, when it affects conditional probabilities
in requisite ways. But Rescher wants his plausibilities to be different from probabilities. He owes us a nonprobabilitistic account of how evidence gets to be authoritative.
This has yet to materialize. In its absence, we can only conclude that the concept of
authoritative source has too short a leash to tie down the more far-ranging concept of
plausibility.
q
r
Kq s t qy r!v
7.7
qKs t qur!vwqs t q!v x0qs t r!v
r
Relishing the Odd
It is widely appreciated that a key difference between a philosopher of science such as
Carnap and a philosopher of science such as Popper is that, all else being equal, the
former espouses that scientific enquiry accord a preference to the more probable hypotheses, whereas the latter, Popperian, approach is to favour less probable hypotheses.
(See here [Popper, 1962, p. 219].) This latter is also Peirce’s predilection, as we have
seen.
Experiment is very expensive business, in money, in time and in thought;
so that it will be a saving of expense, to begin with that positive prediction
from the hypothesis [under investigation] which seems least likely to be
verified. For a single experiment may absolutely refute the most valuable
of hypotheses, while a hypothesis must be a trifling one indeed if a single
experiment could establish it [Peirce, 1931–1958, 7.206, cf. 1.120].
7.7. RELISHING THE ODD
151
z{ | z}
z~
Consider, then, three hypotheses,
and , having the probability values respectively of .1, .2 and .7. (Here we draw from [Rescher, 1978, pp. 84–87].) For present
purposes, these prior probabilities can be conceptualized in any intuitively attractive
way that permits the probabilities of test comparisons to be fixed by the relative weights
of the priors. The problem that faces the researcher is what to test first — probables
against probables, or improbables against improbables? Figure 7.1 gives the outcome
possibilities.
€  z{ ‚  zƒ vs z„ … …
z{†z}‡z~
Winner
Test
vs
vs
vs
z{
z{
z}
z}
z}
z~
.33
.13
–
.66
–
.22
–
.87
.78
Figure 7.1:
Of course, there is always the question of testing-costs. Rescher proposes that, all
else being equal, the cost of pairwise tests goes down monotonically as the distance
enlarges between the likelihoods of the compound hypotheses. This is an idea captured
by the following proportionality principle:
z„ …ˆ ‰0Š zƒ | z!„ … … and hence ‹
Œl Ž ‰9 zŠ{  vs
z
ƒ
Š zƒ | z!„ …‹
 pr  zƒ ‚ zƒ vs| z!z!„ „ … … where
‹ pr  z!„ ‚ zƒ vs z„ … 
cost
This gives the following test costs:
Œ
Test
Cost
vs
.67
vs
.26
vs
.44
If we accept the Popperian direction to test first the least likely, then we test in accordance with the flow chart
z{
z{
z{
z}
z~
z~
Œ
Œ
z{
vs
z}
‘ ’ ’ z{
vs
z~
‘ “ “ z}
vs
z~
with these expected costs:
‘ ’ ’  ‘ “ ” Œ• ‘ – “ Œ … • ‘ “ “  ‘ “ ” Œ!• ‘ — — Œ …‹ Ž ‘ ˜ — Œ
On the other hand, if we follow the Carnapian preference for giving earlier attention to
the more probable then we have a test history
z}
vs
z~
‘ – – z{
vs
z{
‘ ” ™ z~
vs
z{
CHAPTER 7. PLAUSIBILITY
152
and an expected test cost of
š › › œ š   ž!Ÿš ¡ ž ¢XŸš ¡ £ œ š   ž!Ÿš › ž ¢¤š ¡ £ žlš
Here, as we see, the nod goes to the Carnapians. However, it is easily confirmed that it
would go to the Popperians if the initial likelihoods were somewhat different, with
having (as before) likelihood .1, and
and
having the new values, respectively,
of .4 and .5. In this case the Popperian strategy carries a test cost of 1.17 , whereas the
Carnapian or orthodox strategy costs out at 1.24 .
What this shows is that whether to go with hypotheses that strike one as reasonable is not something that admits of definitive prior determination once the economic
considerations of test costs are admitted to the equation. It also bears emphasis that
the issue of whether to favour more rather than less likely hypotheses is, as discussed
here, a question for the logic of hypothesis-engagement, not the logic of hypothesisdischarge.
It is clear that the initial probabilities whose research economies we have been
sketching are not plausibility values in the sense of Rescher [1976], which we met
with earlier in the present chapter. Plausibility values do not compound in the way that
these priors do. So we may not suppose that the economic lessons to be learned about
favouring or not-favouring the initially more reasonable hypothesis are lessons that
convert willy-nilly to the same question for hypotheses of greater (or less) plausibility.
Indeed, not only may we not suppose this, we must suppose the contrary. As Rescher
develops it, presumption should always favour the more plausible. Although he doesn’t
say so in [Rescher, 1976], what he goes on to emphasize in [Rescher, 1978] suggests
an economic caveat: in the matter of what to presume, favour the more plausible over
the less as much as costs allow. However, even with the caveat, there is little doubt
that Rescher does think that the plausible is to be preferred over the implausible, and
that whatever else might be said about the economic constraints under which such
preferences are exercised, whether to make an exception to this rule is never something
to be decided (as with prior probabilities) by the comparative plausibility values alone
of pairwise comparisons.
The concept of presumption is nearly enough our concept of hypothesis-engagement.
In both Rescher’s accounts and our own, engagement favours the plausible. In both
approaches, a plausibility is an idea fraught in ways that we have yet to examine. Ambiguities lurk, some of which are not honoured in Rescher’s’ plausibility logic. We
have been considering that for at lest very large classes of abduction problems, a solution is got by the systematic cutdown of large sets of possibilities to sets of relevant
possibilities; to sets of plausible candidates; thence, if possible, to unit sets thereof. Let
us say that full engagement is achieved at this point. If the process produces a unit set,
the next step is to submit it to whatever test is available for it. If the process terminates
in a larger subset, the next step is to test in a similar way its individual members. It
is well to bear in mind the wide variety of test conditions, ranging all the way from
single-proposition observational corroboration to nothing more than the fact that the
hypothesis in question better generates or conduces to generating some desired consequence relation for the problem’s target. In so saying we meet with two sobering
thoughts, one old and the other new. The old sobriety is that in actual practice, full
¥§
¥¦
¥¨
ž
ž
7.8. THE FILTRATION STRUCTURE AGAIN
153
engagement is not the process by which hypotheses are selected. The newer sobriety is
that propositional plausibility is not positively correlated with abductive progress.
In Rescher’s plausibility logic, plausibility is always a property of the propositional
content of an hypothesis. But as Peirce has made us aware it is perfectly possible to
want to engage an hypothesis not for the plausibility of what it says but rather because
of the plausibility of its fit with the the type of abduction problem at hand. In the
oft-quoted example from Sherlock Holmes, after you have discarded all the rival alternatives, it is advisable to deploy the one remaining no matter how implausible. The
distinction embedded in this advice is that between the plausibility of what a hypothesis
says and the plausibility of entertaining and even engaging it.
7.8
The Filtration Structure Again
© ªl« ¬K« ­« ®¯
We have repeatedly asserted the existence of the filtration-structure
. We
have said that is produced from by a theory of relevance serving as a filter on it,
and that is got from by a theory of plausibility, also acting as a filter. It is time to
reveal something of these purported filters.
We know that every abductive problem can be reduced to the following essentials.
­
¬
ª
¬
°
i. The abductive agent has a target which he desires to be the consequent of an appropriate -statement (i.e., which he desires to be
yielded in some appropriate way.)
ii. No such statement is constructible from , i.e., what the abducer
presently knows (or takes himself to know, or is at least prepared to
assert).
which either in
iii. At this juncture, the abducer’s task is to find an
concert with or with some proper subset of gives a
of which
is desired -consequence. (Where delivers on its own, then
provided is consistent with , we say that
likewise
yields .)
±
²
°
±
°
³
²
²
³
³
²
²´
°
² ´^µ.²¶· ³9¸
°
In the general case, the sets of wffs that yield significantly outrun the
are a proper subset of this larger set . Here we meet with a problem.
ª
²
’s. The
²
’s
a. The proper subset relation is here governed by the presence of successful s. When the s are antecedently known, it is a straightforward matter to reduce to the desired, and less yieldy, proper subset
(the set of ’s that are -successful for ). One of our problems is to
explain how the abducer was able to think of these s. (Concerning
which see Peirce’s “guessing instinct”.)
b. On the other hand, sometimes the abducing agent won’t have any
hypotheses antecedently in mind. What is he to do?
³
²
±
­
³
°
³
7.8.1 Some Suggestions
A.
­
, of course, exists independently of whether the abducer is acquainted with it (or
could be) or any of various subsets of it.
CHAPTER 7. PLAUSIBILITY
154
If
¹
exists independently,
so too does its minimalization
¹Xº
. We define
¹Xº
as
»
the set of distinct consistent non-redundant subsets
.
»¼Y½
Intuitively, every
¾¿ with respect to ½
of
¹
»
such that
will be a member of .
B. Not conversely, however. How does the abducer home-in on the
¾¿ ’s in »
?
We have suggested that, at least for large classes of cases, cutdowns are achieved by
application of relevance — and plausibility — filters, often in that very order.
This seems not to work of for the present problem, certainly for our conception of
relevance:
À
Â
Á
Information is relevant for a cognitive agent with respect to an agenda
and background information * to the extent that in processing , is
affected in ways that advance or close A.
À
À Á
Ã
The abducer’s overall agenda is to determine whether to infer . (He will settle for
sending to trial as a second-best outcome.) At this juncture, the abducer has an
interest in zeroing in on relevant members of
sets. Let be a member of such a
set. Then is relevant in the agenda-relevance sense if on being informed that ,
is able to close his agenda, which is to determine whether to conclude that .
But the same is true of every member of a
set, . If informed that , now
set is relevant.
knows what to infer with respect to . Thus every member of a
(We return to this difficulty in chapter 8.)
Upon reflection it is apparent that minimalization on delivers much, if not all, that
is required of relevance. Premiss-irredundancy was imposed by the originator of logic;
and it is arguable that, though Aristotle didn’t use the word “relevant”, this is the first
invocation of a relevance constraint in the history of logic.
What minimalization gives is relevance in that sense. Thus the relevance-filter has
already been applied. The relevant subsets of are the members of
. These in turn
are the smallest sets that deliver the goods for
Each of these sets is what we have been calling a
with respect to a . Since
each
is a variation on an original , it is clear that for each
it is possible to find
its associated .
This suggests a method of hypothesis-specification:
Ã
¹Xº
Ã
ȼ
Ä
Á
Ã
Ã
Ä
ȼ
à Á
Ä Á
»
»
¾¿
¾
¾
½
ȼ
¾¿
¾¿
½
¾¿ , the à in question is fixed by the difference between ¾¿
¾
If ¾¿Å¾ÆÇ È0É , for some wff È , then ÃnÅ6È . If ¾¿Å6¾ÊÇ È0É , then ÃZÅ7Ë È
should not be regarded as true Ì . If ¾¿XÅ8¾8ÊÇ È0ÉÆ9Ç Í!É , for some Í incompatible
with È , then Ã/ÅË ÎÈ0Ì ; and so on.
HS: For any
and .
It won’t work. The proposal is liberal to the point of promiscuity. Consider a
concrete case, adapted from Paavola [2001].
7.9. A LAST WORD
155
Someone might want to know why Atocha, a resident of Mexico City, is
presently in Groningen. One possibility is that she has been invited to
join Theo’s team, to work on abduction. Another is that the University of
Groningen has given her an honorarium of 50 million guilders. We lean
towards the first suggestion and reject the second on grounds of plausibility
and implausibility.
This suggests a method of repair of the over-liberal device for finding hypotheses.
Apply the method as indicated above. Then submit the resultant wff to a
plausibility test. If it passes the test, it qualifies as an . If more than one
wff passes this test, pick the more (or most) plausible if these is one. If
not, send both (or all) to trial.
Ï
Our present problem is to say how beings like us are able to think up hypothesis. This is
the problem of hypothesis-generation. On the current proposal, hypothesis generation
is not in the general case independent of and prior to hypothesis-engagement (or even
hypothesis-discharge).
Peirce and others postulated an innate flair for guessing right or approximately
right. But on the present suggestion, hypothesis-generation requires no more than the
ability to
i. construct irredundant
Ð
-derivations; and
ii. make plausibility determinations
This might be said to give us the edge; i.e., it is a better account than the guessing
account. Why? We have some degree of theoretical and practical understanding of
both irredundant -derivations and plausibility. There is at present nothing like it for
guessing. A further virtue of this approach is that it cuts relevance down to size, and
makes it just what it needs to be and no more for the job at hand.
Where does this leave us? It leaves us with something to say of a notion of relevance
that suffices for the existence of filtration structures. It also leaves us with something
to say about conditions under which the purported plausibility filter might work. At
least, we have something to say about it if we accept Rescher’s background notion
of epistemic authoritativeness. But, as we have seen, authoritativeness is not a broad
enough notion for abduction. It remains true, even so, that we know a good deal more
about filtration structures than we know about guessing. Yet it appears that what we
know of the real-life hitting of relevant and plausible targets is not a jot greater than
what we know of guessing.
Ð
7.9
A Last Word
We have been suggesting a robust connection between the plausible and the characteristic, and, in turn, between the characteristic and the generic. The ties between the
latter two is hardly one of straight equivalence. Even so, there appear to be ways in
which singular statements of what is characteristic can be paraphrased so as to be derivable from properly generic claims. Preserving a role for genericity in the analysis of
156
CHAPTER 7. PLAUSIBILITY
characteristicness turns out to be a welcome bonus. For, as we indicated in chapter
2, a ready capacity for generic judgement seems to be one of the dominant skills of
practical agents in tight cognitive economies.
Plausibility then falls out as what conforms to what is characteristic; with the implausible contradicting the characteristic. In some respects this approach to plausibility
fares no better than Rescher’s. But perhaps two advantages might be claimed. One is
that judgements of characteristicness may guide us without circularity in setting prior
plausibilities. The other is, that since an account of characteristicness does not depend
on a source-based notion of epistemic authority, we evade the problem of extending
that notion to impersonal contexts.
Chapter 8
Relevance and Analogy
Ñ
Anything of the nature M would have the character , taken haphazard;
S has the character
Provisionally, we may suppose S to be of the nature of M.
Ò
Ñ
[Peirce, 1992, p. 140]
8.1
Agenda Relevance
Ó7ÔÖÕ ×lØ ÙØ ÚØ ÛÜ
Û
ß
ß
Ý
Ó
be a filtration structure for an abductive target . If is a
Let
full-engagement filter then
is a unit set
. Where a filtration structure exists,
the desired
or set of s has a precise location there; but as the empirical record
attests, there is no reason to believe that practical reasoners specify that
(or those
s) by constructing the structure in which it (or they) are uniquely lodged. This is so,
irrespective of whether the plausibility filter is taken propositionally or strategically.
Given the ways in which actual reasoners proceed in real-life situations, there is no
strong evidence to suggest that they invariably, or even frequently hinge their plausibility determinations on prior judgements of relevance. This might well be so even if, for
every proposition judged plausible enough to engage hypothetically there exists some
or other conditions of relevance which that proposition satisfies. What is far from clear,
however, is whether the relevance of those propositions is something that the abducer
satisfies himself about prior to a judgement of plausibility. Perhaps it is also true of the
actual practice of practical agents that they would not judge a possibility to be plausible enough for hypothetical engagement unless that possibility stood in the requisite
relevance-relation to the abductive task at hand. If this were so, it would fall to the abductive theorist to give an account of relevance even if it were the case that judgements
of relevance are not intrinsic to the abducer’s task.
In the previous chapter, relevance lightly entered our discussion in two different
senses. In one sense, relevance is just premiss-irredundancy. In the other, it is what
we call agenda relevance, to which we now turn. Agenda relevance is explained at
length in our book of the same title [Gabbay and Woods, 2002a]. A good deal of
Þ ß9à
ß
ß
157
158
CHAPTER 8. RELEVANCE AND ANALOGY
the case that we there make for agenda relevance turns on advantages that we claim
over rival accounts, and the attendant capacity that our account has for incorporating
such attractive features as these alternative views may possess. Our present task is
not to convince readers of the superiority of our account; readers can consult Gabbay
and Woods [2002a] if they are interested. It suffices for our purposes here to have an
account of relevance that appears to do real work in the theory of abduction.
We pause here to give a thumb-nail sketch of the theory of agenda relevance (AR).
We begin with a set of adequacy conditions.
AC1.
The theory should not make excessive provision for relevance. That is, it
should guarantee the falsehood of “Nothing is relevant to anything” and of
“Everything is relevant to everything”.
AC2.
It should acknowledge that relevance is context-sensitive. Things relevant in
some circumstances aren’t relevant in others.
AC3.
It should honour the comparative nature of relevance. That is, it should provide that some things are more (or less) relevant than others.
AC4.
It should provide for a relation of negative relevance, distinct from the idea of
mere irrelevance.
AC5.
It should describe a connection, if there is one, between relevance and truth.
AC6.
The theory should help elucidate the fallacies of relevance.
AC7.
The theory should make a contribution to territorial disputes between classical
and relevant logic.
AC8.
It should make a contribution to a satisfactory account of belief-revision.
AC9.
The account should invesitgate the suggestion that relevance is instrinsically
a dialogical notion.
AC10. It should provide a common analysis of relevance; i.e., it should stay as close
as possible to the commonsense meanings of “relevance”.1
The notion of adequacy conditions is rather loosely intended. Some will weigh
more heavily than others. Some, such as AC1, would seem to have the full force of
necessary conditions. Others, for example AC6 weigh less heavily. AC6 says that a
theory of relevance should try to elucidate the fallacies of relevance, so called. A theory
of relevance that failed it could be regretted on the score of incompleteness, or some
such thing, but it would be over doing it to give it up as a thoroughly bad job. So our
adequacy conditions include both necessary conditions and what we might think of as
1 It should be remarked that AC10 makes it unnecessary to qualify AC1 in some such fashion as, “to the
extent to which a theory of relevance offers a common analysis of it”. AC10 guarantees that this should be
the case. (It is easy, by the way, to see that AC8 and AC9 are special cases of AC10, but we will let the
redundancy stand.)
8.1. AGENDA RELEVANCE
159
desiderata. Perhaps our Enrichment principle falls in between AC1 and AC6 on the
question of necessity, but there is no denying its attractions.2
The basic idea of AR is that information is relevant for a cognitive agent when and
to the extent that it acts on the agent in ways that advance or close one or more of the
agent’s agendas. Agendas are here understood as the sorts of thing that information
can bring to completion. So we may think of agendas as cognitive tasks. Information
is understood in the tradition of the Carnap and Bar-Hillel [1952] notion of semantic information, in a version in which changes of belief are manifested in changes of
subjective probabilities [Jamison, 1970, p. 30]. This can be illustrated as follows.
Let be an agent’s belief states prior to the reception of some information I and
* are his beliefs afterwards. Then it is possible to specify the quantity of information furnished by I, or contained in I, as a strictly increasing function of the distance
between and *; i.e.,
á
á
á
á
â ãXäå æ çè8éê í å é å 7
é ë$
ìç ç î á ë áï!î
é7ëì
é
where is the number of virtually exclusive and jointly exhaustive possible state of
affairs.
Likening an agenda to a cognitive task can be misleading. It may suggest that
intrinsic to agendas is the conscious intention to perform them. We want to discourage
the suggestion. We require an account of agendas that enables us to preserve what
is perhaps the basic pair of pre-theoretical data about relevance. One is the striking
success with which beings like us evade irrelevancies and stay on point as we make
our way through life’s rich complexities. The other is the extent to which irrelevance
evasion is achieved automatically and subconsciously. We should also note the sheer
primitiveness of our relevance-managing skills, which are evident in infancy and are
up and running rather robustly in early childhood. 3
Agendas, then, are the sorts of things that can be possessed unawares and whose
advancement can proceed without being noticed. This requires that we leave room in
AR for a conception of agenda-management describable by rules that aren’t actually
executed, in the spirit of neural nets meeting the RWR conditions.
Seen our way, relevance is a causal relation and a pragmatic one. It is causal because the relevance of information is always a matter of affecting agents in ways that
advance agendas. It is a pragmatic relation because relevance is always relevance for a
user of information. Thus, as a first pass,
I is relevant for S with respect to A to the extent that I is processed by S
in ways that advance or close A.4
ð0ñ
ð
2 Recall that Enrichment is a principle that bids the constructor of a theory
by drawing upon the
resources of another theory
to try to contrive the new theory in ways that abet new results in the borrowed
theory.
3 Here, in passin g, is a small abduction of our own. The more primitive a cognitive skill is the more
resistant it is to theoretical articulation. This might explain why there has been so little theoretical work done
on relevance, comparatively speaking. Abduction itself is similarly situated.
4 We do not here invoke the distinction often drawn by psychologists between being affected by information and processing it. Ours is the broader of this pair.
CHAPTER 8. RELEVANCE AND ANALOGY
160
ò
Let us think of agendas as involving sets of endpoints preceded by sequences of
effectors
of . Effectors effectuate s and thus may be seen as causal or constitutive matrices. Accordingly,
ó0ô ò
ò
Definition 8.1.1 (Execution-Agendas) An execution-agenda is a structure
õ2ö ÷ùø úû ü óý ö þ þ þ ö ó0ÿ ö ò in which X is an agent, D is a disposition to perform or an interest in performing in
such a way as eventuates in a state of affairs denoted by ‘N’, and M is a performance
manual for producing the causal or constitutive matrix
.
ü óKý ö þ þ þ ö óKÿ ö ò
Because they possess the appropriate deontic character, execution-agendas can readily
be described as things that can be offered to agents, or even urged upon them; and they
can readily be described as things that an agent might seek out and want to have.
Trailing along is a simplified conception in which an agenda is simply an appropriately ordered set of actions of a suitably situated agent that eventuates in the realization
of a targeted state of affairs. These we might call action-agendas, and describe their
basic structure as follows:
Definition 8.1.2 (Action-agendas) An action-agenda is a structure
õ2ö ÷/ø ü ý ö þ þ þ ö ÿ ö where X and D are as before, the ý are actions or states of affairs which if performed
by X, would eventuate in state of affairs .
Seen this way, an action-agenda is an execution-agenda except that the performance
manual is replaced by the actions or states that it proposes. The choice between the execution and action conceptions of agendas is a matter of how psychologically complex
a conception is best for the theory of agenda relevance. We ourselves lean toward the
simpler position represented by the action conception of agenda. But we acknowledge
that in particular contexts it may prove beneficial to appropriate into service the other
conception. Our default position, however, is that agendas are action-agendas.
We also note that the dispositional component of action-agendas
allows us to say that Harry can have a disposition to perform the action
even though
he might not have consciously brought them to mind. We may also speak in an informal way of Harry’s being disposed to advance or close the agendas of which this
disposition is a component. But in both these latter cases, the dispositions are oblique
rather than transparent.
It is well to note that action agendas are subject to constraints. For example, we
must decree that the action/state of affairs components of an action agenda are nonmonotonic. For it is not the case that any set of actions or states eventuating in a
given state eventuates in that same state when supplemented arbitrarily many times by
arbitrarily selected new actions or states. And, of course, if we wanted our agendas
to be linear, this would be automatic provision for them to be relevant in the sense
of Anderson and Belnap. The agendas would be relevant, even though the relevance
would not be agenda relevance in our sense.
õ2ö ÷/ø ü ý ö þ þ þ ö ÿ ö ô
8.1. AGENDA RELEVANCE
161
It is also necessary to emphasize that action-agendas are not intentional objects
in the sense of being the objects or conscious choice or conscious intent. They are
however intentional objects in Husserl’s sense. They possess actions or states that are
objects of an agent’s disposition to realize or interest in realizing, where the disposition and the interest need not be consciously held or consciously present. Even so
those actions or states are what the relevant dispositions are dispositions toward and
the relevant interests are interests in. So they are Husserlian intentionalities.
Readers interested in a somewhat more formal sketch of agendas and agendarelevance are invited to consult the Appendix to the present chapter.
We note that it is easy to extend action-agendas by addition of the appropriate
temporal indicators.
We turn now to a smattering of claims endorsed by AR.
Proposition 8.1.3 (Agenda-Override) Information can sometimes override an agenda.
When this happens it can also set up a new agenda, toward the closure of which it might
itself contribute significantly. Information is relevant for an agent whose agenda it
overrides. It is also relevant if it creates for him a new agenda, irrespective of whether,
having done so, it advances the new agenda. (Advancing, here, is the partial closure
of an agenda, not the setting of it up.)
Proposition 8.1.4 (Degrees of relevance) I is relevant for S with regard to A to the
extent that it closes A.
Proposition 8.1.5 (Degrees of relevance) I is relevant for S with regard to A ,. . . , A
to the extent that I closes agendas, = 0 , . . . , n.
Let us say here that a linked argument is one in which there are no redundant premisses [Thomas, 1977, pp. 36–38]. Linked arguments resemble deductions in a linear
logic. By extension, information can be said to be linked for an agent with respect
to an agenda to the extent that advances or closes but no proper subset of does
either.
Proposition 8.1.6 (Linkage) Information for S is usually linked-like for S.
Proposition 8.1.6 may strike some readers as excessive. Certainly it is not strictly
the case that agents confine themselves to irredundant information. But there is ample
empirical evidence to suggest that in the general case practical agents process information in ways that tend to subdue redundancy.
Proposition 8.1.7 (Integration) Linked information is integrated information.
Definition 8.1.8 (Relevance) I is relevant for S with regard to A to the extent that
information I is information for S, I I is linked for S and I I advances or closes A.
This suggests that the prior definition of relevance was too coarse-grained. Relevance
was a set of triples. As we now see, it is better to take it as a set of quadruples: I, X,
I , A . Our quadruple answers well, in I , to the idea of background information. If
we think that by and large information is relevant only in the company of background
information, we now have the means of representing the fact abstractly.
We now have the means to characterize
CHAPTER 8. RELEVANCE AND ANALOGY
162
Definition 8.1.9 (Parasitic relevance) I is parasitically relevant for X with regard to
A iff for some I for X, I I advances or closes A and I I is not linked for X.
Definition 8.1.10 (Cumulative relevance) I is cumulatively relevant for X with regard to A to the extent that I is the consistent union of subsets I , . . . , I such that for
each I , I is relevant for X with regard to A to some or other degree , where in each
case is less than the degree to which I itself is relevant to X with regard to that same
agenda (but with k sufficiently high throughout).
Intuitively, cumulativeness falls midway between linkedness and convergence with respect to agenda closure. Cumulativeness is an approximation to convergence. Cumulative relevance is a good thing. It is defined for summations of intuitively relevant items
of information that jointly increase the degree of relevance.
Let us say that an argument is convergent just in case, for some relation (e.g.,
entailment) every non-empty subset of its premisses bears to its conclusion [Thomas,
1977, pp. 38–39]. by extension information is convergent for an agent with respect
to agenda iff every nonempty subset of advances or closes to degree (with
sufficiently high). Accordingly,
Definition 8.1.11 (Hyper-relevance) I is hyper-relevant for X with respect to A iff
there is a subset of I that is convergent for X with respect to A and no proper subset of
I is parasitically relevant for X with respect to A.
We may expect, in turn, that proper subsets of hyper-relevant information will
sometimes be cumulative. Hyper-relevance is cumulative relevance taken to extremes.
A young barrister is pleading his case before the local magistrate. It is an impressively
strong case, but, callowness being what it is, the rooky-lawyer goes on and on. The
judge intervenes. “Would counsel be good enough to pause so that I might find for
him?” It is a common enough mistake among the dialectically anxious. Cumulative
relevance gives way to convergence and thence to hyper-relevance.
The relevance that we have been describing is what we call de facto relevance.
Information is relevant if, as a matter of fact, it acts on an agent in ways that advance
agendas. It may be that there are cases in which information actually relevant in this
sense, might be thought not to have been properly relevant or really relevant. This
suggests a contrast between de facto relevance and normative relevance, concerning
the latter of which we are governed in what we say by our prior subscription to the
Actually Happens Rule introduced in chapter 1. The rule provides, as a first pass,
that how cognitive agents actually perform is reliable indication of how they should
perform. By these lights, information that is actually relevant is relevant properly so.
This, of course, is too strong. There are cases in which what closes an agent’s
agenda shows not that the information wasn’t relevant in fact, but rather that there is
something wrong with the agent. Can we get from this a serviceable notion of normativity?
Definition 8.1.12 is normatively relevant for with regard to an agenda
ground information iff is de facto relevant for agents with agenda
ground information .
and backand back-
8.2. THE ROLE OF RELEVANCE IN ABDUCTION
163
The “plural” ‘agents’ in the RHS of the biconditional is spurious. We do not here
mean “more than one agent”, but rather agents (where “agents” functions as “tigers” in
“Tigers are four-legged”).
What is normative about this proposed conception? We might answer by repeating
the point that a generic proposition asserts something that is characteristically the case.
With regard to the present issue, the definition proposes that normative relevance is
what is characteristically relevant in fact for cognitive agents having that agenda and
that background information.
It is not wild-eyed naturalism to suggest that what is characteristic of human information processors is normative for them. Of course, any such claim is defeasible. Its
particular instances are instances of defaults.
But aren’t there lots of well-known cases in which what is characteristic for human
agents is pretty bad logically? How about our disposition to commit all those fallacies
that informal logicians are so het up about?
We have tried in chapter 2, to undermine the idea that these ways of reasoning
which indeed are characteristic of practical agents are fallacies. In other words, we
ourselves incline to a panglossian view of such matters, as we have said.
8.2
The Role of Relevance in Abduction
Agenda relevance is always relative to an agenda. If an abducer’s agenda is to determine for some (or set of s) whether to conjecture it (or them) then the information
that is agenda-overriding. It “tells” the agent that is true; so there is no point any
longer in conjecturing it. This gives an interesting result.
If is a successful resolution of an agent’s engagement problem, then
information
is the case is agenda relevant for that agent only in
the limiting sense of being an overrider of his agenda.
"! #
On the other hand, if the abducer has a discharge-problem, then it is trivial that
$%! #
is the case is relevant with regard to that agenda, since it closes
it.
Of course, there are other tasks involved in the process of abduction. It is typical, if not
invariable, of abducers to want to determine whether there are reasons short of its truth
to conjecture that . Consider some typical values of .
$%! 2. $%! 3. $%! 1.
is part of the domain of the
is plausible
#
is in the minimalization of
&
-relation
#
'(# .
It is easy to see that these are cases in which is relevant to such an agenda. Even so,
it may be that abductive agents restrict their selections of
to propositions that are
either propositionally or strategically plausible. but as we have seen, it needn’t be the
case that in abducing
there is any judgement made by the abducing agent as to its
plausibility. The same is true of relevance.
CHAPTER 8. RELEVANCE AND ANALOGY
164
The most important point about the interplay of relevance and abduction has yet to
be made with adequate emphasis. Consider again the filtration structure
.
We said that is the set of relevant possibilities with regard to some abduction target .
This is certainly the case for the irredundancy conception of relevnace; but note well,
that on this conception, relevance is a property of itself rather than the members of
. In a loosely conversational way, we speak of the members of as the relevant
possibilities. In greater strictness, all that this can mean is that they are members of an
irredundant domain of the -relation.
Much the same is true of agenda relevance. Let be the set
.
is the case is agenda relevant to any abductive agenda,
Then no information
save the agenda of determining whether to discharge. It is well to be clear. What we
have shown is that short of overriding, there is no abductive task other than hypothesisdischarge to which the members of are relevant in our sense. But it is one thing to
say that the
aren’t relevant. It is another thing entirely (and false) to say that abductive agendas are routinely advanced and closed by information other than that which
conveys that the
are the case. So, unless we are careful to note its highly particular
conception of relevance, it is simply wrong to say that a relevance filter is needed for
the solution of the Cut Down Problem.
)+*-, .0/ 1$/ 23/ 465
7
1
1
1
8
C$*-D :FE
G
1
1
9 :<; / :>= / :>? / @ @ @ / :>A B
1
:FE
:>E
8.3
Analogical Argument
In section 4.7 we sketched an approach to the solution of hypothesis-generation and
engagement problems proposed by Lindley Darden. We reproduce that sketch here
[Darden, 1976, p. 142]
problems
posed by
fact
plausible
solution to
this problem
H H generalize
HHHHHHHI
K H particularize
H H H H H H H H HH
Darden’s Schema
general form
of the problem
H H analogize
H H H H H H H toH H I
general form
of solution
to problem
K H construct
H H H H H H HH
J
general forms of
similar problems
with solutions
general forms of
other KNOWN
solutions
Darden’s Schema reflects Hanson’s influence, as evidenced by the following summary, reproduced from section 3.4.
Hanson’s Schema
L0; / L(= / L(? / @ @ @ are encountered.
2. But L0; / L(= / L(? / @ @ @ would not be surprising were a hypothesis of : ’s type to obtain. They would follow as a matter of course from something like : and would
1. Some surprising, astonishing phenomena
be explained by it.
: ;
L0; / L(= / L(? / @ @ @
3. Therefore there is good reason for elaborating a hypothesis of the type of
for proposing it as a possible hypothesis from whose assumption
might be explained [Hanson, 1961, p. 33].
8.4. THE META APPROACH
165
Darden bids the abducer to analogize. Hanson bids him to reason from types. As it
happens, there is a ready-built theory of analogical argument that gives instruction on
both these matters. It is the Meta-Argument Theory of Analogical Argument originated
by [Woods and Hudak, 1989].
8.4
The Meta Approach
Consider what may have been one of the most commented-on analogical arguments of
the century just passed [Thomson, 1971, p. 49].
You wake up one morning and find yourself back in bed with an unconscious violinist. He has been found to have a fatal kidney ailment, and
the Society of Music Lovers has canvassed all the available records and
found that you alone have the right blood type to help. They have therefore kidnapped you, and last night the violinist’s circulatory system was
plugged into yours, so that your kidney can be used to extract poisons from
his blood as well as your own. The director of the hospital now tells you,
“Look, we’re sorry the Society of Music Lovers did this to you-we would
never have permitted it if we had known. But still, they did and the violinist now is plugged into you.” Is it morally incumbent on you to accept
this violation? No doubt it would be very nice if you did, a great kindness.
But do you have to accede to it? What if it were not nine months, but nine
years? Or longer still? What if the director of the hospital says, “Tough
luck, I agree, but you’ve now got to stay in bed with the violinist plugged
into you, for the rest of your life. Because remember this: All persons
have a right to life, and violinists are persons. Granted you have a right
to decide what happens in and to your body, but a person’s right to life
outweighs your right to decide what happens in and to your body. So you
cannot ever be unplugged from him ... .
Thomson develops her analogical argument as follows. If you judge The Violinist to
be a good argument, then there is another argument similar to The Violinist which,
by parity of reasoning, you must also judge good. This other argument (call it The
Pregnancy) is one that concludes that maintaining a pregnancy arising from rape is not
morally obligatory. Thomson is here invoking, without naming it, the Fundamental
Rule of Analogy, requires that similar cases be treated similarly.
What the Fundamental Rule does not make clear is what the relevant similarities
between The Violinist and The Pregnancy would consist in. This is answered by the
Meta-Argument Theory of Analogical Argument (MATAA). It would be well to bear in
mind at this juncture that Darden’s schema has the reasoner generalizing and Hanson’s
schema has the reasoner arguing from type. so consider the following schema which
we’ll call The Generalization.
The Generalization
Human beings
M<N
and
M>O
are so related that
CHAPTER 8. RELEVANCE AND ANALOGY
166
P>Q ’s consent, P<R
1. without
dency;
PFQ
has placed
in a state of vital depen-
2. the period of dependency is indeterminate (perhaps nine months, perhaps nine years, perhaps forever);
3. the dependency is a grievous impediment both to locomotion and to
(stationary) mobility;
4. the dependency constitutes a grievous invasion of privacy;
5. it is an invitation to social disaster, for
laughing stock;
PFQ
(and
PFQ ’s economic self-sufficiency;
6. it threatens
7. therefore, it would be morally permissible for
vital dependency.
P<R
P>Q
as well) is a
to terminate the
Suppose, for the moment, that Thomson is right in claiming parity between The Violinistand The Pregnancy, then MATAA furnishes the answer to what it is that these two
quite different arguments nevertheless have in common. The MATAA proposal takes
seriously the mention of cases in the Fundamental Rule to treat similar cases similarly.
Accordingly it is proposed that what The Violinist and The Pregnancy are case of is
a common deep structure represented here by The Generalization. thus an analogical
argument is a meta-argument; it is an argument about arguments. It is an argument in
the form
Meta-Argument
S
1. Argument possesses a deep structure that provides that the premisses of bear relation to its conclusion.
S
2. Argument
U
U
shares with
S
T
the same deep structure.
3. Therefore,
possesses a deep structure that provides that its premisses likewise bear to its conclusion.
T
U
S S
4. Hence, is an analogue of . and
by parity of reasoning, so-called.
S
U
are good or bad arguments,
T
U
In our example, argument is The Violinist. is a strong consequence relation. is
The Pregnancy. The deep structure is The Generalization.
The MATAA analysis requires that the analogizer generalize on an argument whose
assessment is already settled, and then instantiate to a different argument. To achieve
the desired parity, it is essential that the property that the original argument is assessed
as having (e.g., validity) is preserved by the generalization and preserved by the subsequent instantiation (Darden calls this “particularization”).
How does this bear on abduction? In a typical case the abducer has a target for
which he seeks an
such that for his belief-set ,
. His principal
task is to find the requisite . By Darden’s and Hanson’s lights, the abducer reasons
his way to
by analogy. Doing so involves generalization and instantiation, and
reasoning from type. If he proceeds in the manner prescribed by MATTA, the abducer
looks for a distinct consequence structure
PW
PW
P
X X"Y6Z PW [<\]V
V
8.5. APPENDIX
(1)
167
^<_a`b c+_ d$egfF_
that generalizes to
(2)
^<_a`b c+_ d$egfF_
in ways that preserve the truth of (1). A condition on this generalization is that the target
of the abducer’s present abduction problem instantiate the
of the generalization of
the original consequence structure. He then instantiates from
and
to the required
and . What this represents is a rather commonplace situation in which there is a
conclusion that you want to find some justification for. This is a premiss-selection task.
You look for a conclusion whose sole similarity to your intended conclusion is that it
is justified by premisses of a type (Hanson) that would likewise justify your intended
conclusion. So you look for the requisite similarities among putative premisses.
Except for tightly circumscribed situations, there are no algorithms for this that
abducers actually execute. On the other hand, the MATAA has some powerful advantages. One is that it reduces analogizing to target-property preserving generalization
and instantiation, which, given their utter commonness is a significantly simplifying
reduction. The other that it gives content to the schematic insights of Darden and Hanson.
f
^
f h
^Fh
c
8.5
ch
APPENDIX
A Formal model for Agenda Relevance
8.5.1 Introduction
This appendix sketches a generic formal model of agenda relevance the model is presented schematically in terms of components. The exact details can be fine-tuned for
any application area. The generic model will help explain conceptually how agenda
relevance is supposed to work.
We specify the following components.
Component 1: Base Logic
We give no details about the base logic except the following:
i
j
1. A language which the notions of a declarative unit (wff) and a database
are defined. We require that a single wff is a database ant that the empty
database is a database.
^
j
2. A notion of consistency of a database is available.
^lkmnj
3. A consequence relation of the form
is available satisfying certain conditions. We need not specify the exact axioms, but certainly
holds.
j"kmnj
CHAPTER 8. RELEVANCE AND ANALOGY
168
4. Note that we are not saying what it means for one database to extend another. We
do not need this concept for our generic model. Further note that a set of wffs
is not necessarily a database, nor have we defined what it means for a database
to contain . We can define, however,
by:
p
o
plqr o
p q r os t plq rnu vxw y z3{ | | l
u } $o ~
o
Component 2: Consistent input operator
p
o€ u$ ‚
p-ƒ o
u
, we have a function “+” which
Given a consistent database and a set of wffs
forms a new consistent database
.
We require at least the following. If is consistent, then
p-ƒ6u"qrnu
u
p-ƒ„un…np
(b) If is not consistent then
(a)
.
p"ƒ„u
p-†  $u ‚
Note that
is a combined revision + abduction + non-monotonic adjustment
operation. We can put more restrictions on “+” for example, if
is consistent
then
p%†  u$‚€‡ˆp-ƒ6u
p‰†  u$‚ may be meaningless (for example if the dataHowever, in a general logic,
structure are list). We leave the fine-tuning if “+” for later.
Two comments though.
p
Š , we should not expect any unnecessary connecplqrnu contains
Š ‹ and p-ƒ6u"q r ‹ .
Œ
1. If the language of
tion between
2. + is not necessarily a pure revision operator. It may be a combination of abduction and revision and non-monotonic default all done together. So, for example,
even when and are consistent, we may have
u
p
p%†  $u ‚Ž‡ˆ
 p%ƒ„u
but no equality, because + may involve additional abductive consequences of the
input.
Nor do we necessarily expect
 p-ƒ6u$aƒ ‹ …np-ƒ  u v ‘ ‚ ~
p-ƒ6u might be a set of databases.
3. +may not always be single-valued.
8.5. APPENDIX
169
Component 3: Basic actions
’0“ ”
We assume our language contains basic action notation of the form
etc. An action
has a precondition
and a postcondition we allow for an empty precondition (think
of it as
) and the empty postcondition (
, note that we may not know
what
is going to be) and for the empty action .
Given a situation or state adequately described by a database , if
then
the action can be taken and we move to a new state
.
Note that
, i.e. the result of the action is guaranteed to hold.
The above actions are deterministic. We may have non-deterministic actions whose
postconditions are sets of wffs
•0–
•0–+˜š™
›-œ„™
’
›F–<ž ˆ— –
—–
—(–˜š™

›
› –F˜n›-œ6— –
›Ÿž ‰•x–
—(–F˜%¡ ¢ £– “ ¤ ¤ ¤ “ ¢ ¥– ¦
and such that, once the action is taken, at least one of the postconditions will hold.
Given a , we may also allow for a probability distribution (dependent on ) on the
outcomes of the actions.
We leave all these details to the fine-tuning of our model.
›
›
8.5.2 The Simple Agenda Model
We assume we have available the components described in the previous section. We
now define some basic concepts.
Definition 8.5.1
1. Universe
Let be the set of all consistent databases. Let
iff for some
and
result of an action in performed on .
§
›F© ¨Fª
¨ be a set of actions. Define
’«¨$›¬ž %•0– ª ˜n›"œ6— – . This means that ª is the
¨
›
2. The model ­ ®>“ ¡ © ¨ ¦ ¯ is called the universe.
Definition 8.5.2 (Agendas)
1. A simple agenda is a sequence of sets of formulas
£
of the form ­ ° “ ¤ ¤ ¤ “ °3± ¯ . Recall that a set of wffs may not be necessarily a
database › , though we can write ›lž ˆ° as ›‰²>³ for all ³Œ«<° .
£
2. A sequence ­ › ´ “ ¤ ¤ ¤ “ ›>± ¯ of databases satisfies the agenda ­ ° “ ¤ ¤ ¤ “ °µ± ¯ if for
£
each ¶ “ ›F·µž ˆ°µ· . Think of ›F´ as now and ­ › “ ¤ ¤ ¤ “ ›>± ¯ as a future sequence.
3. A sequence of databases ­ › ´ “ ¤ ¤ ¤ “ ›F± ¯ is said to be a real history (from the
£
universe) iff for some actions ­ ’ “ ¤ ¤ ¤ “ ’ ± ¯ and for each ¸Ž¹„¶µ¹„º , we have:
›F· » £ ž n•0– ¼0½ ¾(¿ ›>·x˜n›>· » £ œ6— – ¼ ¤
£
4. We can also say, if all actions are deterministic, that ­ › “ ¤ ¤ ¤ “ ›>± ¯ is generated
£
by ­ ’ “ ¤ ¤ ¤ “ ’ ± ¯ starting at ›F´ .
£
£
£
5. We can also say that ›F´À “ ­ ’ “ ¤ ¤ ¤ “ ’ ± ¯ ¯ satisfy the agenda ­ ° “ ¤ ¤ ¤ “ °µ± ¯ if ­ › “ ¤ ¤ ¤ “ ›>± ¯
£
£
satisfies the agenda, where ­ › “ ¤ ¤ ¤ “ ›F± ¯ is generated by ­ ›F´ “ ­ ’ “ ¤ ¤ ¤ “ ’ ± ¯ ¯ .
CHAPTER 8. RELEVANCE AND ANALOGY
170
Definition 8.5.3 (Simple agents) A Simple agent
1. A set
Â
Á
has the following components
of actions which he can execute.
2. A family
Ã
of simple agendas of the form
ĵÅxÆ-Ç ÄµÅÈ É Ê Ê Ê É Ä3ËÅ Ì Í , Î3Æ-Ï É Ð É Ê Ê Ê .
ÑFÒ (a database).
4. Let Ç Ó È É Ê Ê Ê Ó Ë Í be a sequence of databases. We say that agent Á generate this
sequence as a future history of for some Ô È É Ê Ê Ê É Ô Ë available to Á (i.e. Ô
Å3Õ Â ),
Ç Ñ Ò É Ç Ô È É Ê Ê Ê É Ô Ë Í generates Ç Ó È É Ê Ê Ê É Ó Ë Í .
5. We say Á can potentially satisfy the agenda Ä"Æ¬Ç Ä È É Ê Ê Ê É Ä Ë Í if we the agent
can generate a future history Ç Ó È É Ê Ê Ê É Ó Ë Í which satisfies the agenda.
6. Á can partially satisfy ĄÆ-Ç Ä È É Ê Ê Ê É Ä Ë Í if he can generate Ó È É Ê Ê Ê É ÓŽÖ$Í(×ÙØ6Ú
which satisfy Ä È É Ê Ê Ê É ÄµÖ$Í .
3. A “now” point
Á
We are now ready to define a simple notion of relevance. Our agent assumes that
he is at a state (“now”)
. However, he may be mistaken. If he receives input
which is verified, he must assume his state is
. Now the agent may think
he can satisfy some agenda because a sequence of actions
is available
to him to create a suitable history.
This means that
. However,
. Thus the input is relevant to
his agendas.It can also work the other way round, more agendas may be satisfiable.
Ñ Ò
Ä
ÑFÒ<Ý Þ-ß0à á
Ñ Ò$Ü6Û%ÆnÑ È
Ñ È ÝÞ-ßxà á
ÇÔ È ÉÊ ÊÊÉÔ Ë Í
Û
ÁÆ"Ç Â É Ã<Í
Á
ÇÔ È É ÊÊ ÊÉÔ Ë Í
Û
Ñ Ò
Definition 8.5.4 (Simple agenda relevance) Let
be rooted at a state
.
Let
be a wff. Let
. Then
is relevant to at
iff for some
agenda
and some sequence of actions
from we have that
for
we have
Û
ÇÄ È É Ê Ê Ê ÉÄ Ë Í
â Õ ãä ÉÏ å
Ñ È ÆgÑFÒFÜ-Û
Û
Ñ Ò
Â
Ç Ñ>æ É Ç Ô È É Ê Ê Ê É Ô Ë Í Íaç è é ê ç ë ì ç3Ç Ä È É Ê Ê Ê É Ä Ë Í
while Ç Ý í î ï ð ñ È ò æ É Ç Ô È É Ê Ê Ê É Ô Ë Í Í does not satisfy Ä È É Ê Ê Ê É ÄµÖ$Í for some Ç Ä È É Ê Ê Ê É Ä Ë Í
Õ
à .
Note that the agent may still be able to satisfy all of his agendas but through a
different sequences of actions.
We conclude this section by explaining why we prefixed all our concepts by the
word “simple”.
First, the language was not assumed to be able to talk about the future i.e. we did
not require construct of the form
, with
ó
µô Û
ÑlÝ Þ-ôµÛnõÙö0÷>Ç Ñ-܄÷šÝÞnÛ$Í
.
We are also not allowing actions as part of the data, or more complex logics, as in
the formal chapters of Agenda Relevance.
8.5. APPENDIX
øFù
ú ø<û0ü ý ø>û0þ ý ø>ûxÿ ý øFù
¬ú €û0ü ý €û0þ ý µûxÿ ý 3ù
ý ý ý
ø>ÿ ˆøFù
171
Second, our present (“now”)
, has no history. We can define relevance relative
to a history say
and allow actions to have as preconditions the
and we can apply the action if
closing of agendas i.e. , and only then we more to .
This allows the closing of agendas to be preconditions for actions.
Third, agendas require only satisfaction of wffs. We do not require certain actions
to be performed. It may be part of the agenda that Harry be home but the agenda does
not specify by what action this is achieved.
ø<û
172
CHAPTER 8. RELEVANCE AND ANALOGY
Chapter 9
Refutation in
Hypothesis-Testing
9.1
Semantic Tableaux
Hypothesis-discharge involves either an active programme of attempted refutation or a
willingness to expose one’s hypothesis to such a test. On some accounts is refuted
when in a context such as there is a fact that contradicts !" . In other
cases is straightforwardly refuted by any observation directly incompatible with it.
There are some commentators for whom this is a blurred distinction at best. Bearing
on this is the extent to which observations are conceptually inert and on the extent to
which holism is true. We shan’t pursue these questions here, and shall concentrate
instead on what it is to refute s in contexts such as #$ , irrespective of how
dissimilar this is from the direct refutation of itself.
Semantic tableaux constitute a refutation method for designated formal languages
[Hintikka, 1955; Beth, 1969]. An attractive exposition is that of Smullyan [1968]. The
basic method has been adapted for abductive tasks by Atocha Aliseda [1997]. We quote
from this same work:
To test if a formula % follows from a set of premisses & , a tableau for
the sentence & '% is constructed. The tableau itself is a binary tree
built from its initial set of sentences by using rules for each of the logical
connectives that specify how the tree branches. If the tableau closes, the
initial set is unsatisfiable and the entailment &)(#% holds. Otherwise,
if the resulting tableau has open branches, the formula % is not a valid
consequence of & . A tableau closes if every branch contains an atomic
formula * and its negation [1997, p. 83].
According to the tableau rules, double negations are suppressed, conjunctions add both
conjuncts; negated conjunctions branch to two negated conjuncts; disjunctions branch
into two disjuncts; negated disjunctions add both negated disjuncts; and conditionals
reduce via negation and disjunction. Accordingly,
173
CHAPTER 9. REFUTATION IN HYPOTHESIS-TESTING
174
+,+-.)Negation
-"/"0.21
Conjunction
+4 -/"05.63 +-7 +0
-"8"0.)-7
Disjunction
0
+4 -8"05.29 1
9
3
-$.)0.6+-7 0
Conditional
+4 -.)0:5. 1
9
3
In languages in which all wffs have either disjunctive or conjunctive normal forms, the
above rules reduce to one or other of two types of rule, a conjunctive (; -type) rule, and
a disjunctive (< -type
.>rule).
= ?
= @
7
; .
< A <B
Rule 2. 4 < D:5 D
Suppose that C 4 D5 is a tableau for a theory
. then it is known that
D
D
a. If C
has open branches, is consistent. An open branch repreD
sents4 D
a 5 verifying model for .
b. If C
has only closed branches, is inconsistent.
Rule 1.
c. Semantic tableau constitute a sound a complete system for truth functional languages.
d. A branch of tableau is closed if it contains some formula and its negation.
e. A branch is open if it is not closed.
f. A branch E of a tableau is closed if (recall rules 1 and 2 just above)
for every ; occurring in E both ;,A and ;,B occur in E , and for every
< occurring in E , at least one of <A F <B occurs in E .
+-H closed or complete.
g. A tableau is completed
if every branch is either
IJh. A proof of D
a wff
i. A proof of
is a closed tableau forD
G K"L +. -M
is a closed tableau for
.
Aliseda construes abductions as a kind of extended tableau. An extended tableau
D - of adding new formulas to a tableau. Consider theDNcase
IL -
is the result
inM which for some
theory , is not a valid consequence. In the tableau for
there are open
D
branches.
Each such branch is a counterexample to the claim that is a consequence
- they
of . Of course, if further wffs were added to the open branches it is possible that
might now close. Consider a least class of D wffs that fulfill this function. Then would
be derivable from a minimal extension of . Finding such wffs is a kind of abduction.
Accordingly, Aliseda proposes [1997, p. 91] the following conditions.
9.1. SEMANTIC TABLEAUX
175
Plain Abduction OQP P RST"U VWXT"U YX S is closed. (Hence R[Z Y\JW .)
Consistent Abduction Plain abduction ]OQP RT^U YX S is open. (Hence R_
W .)
Explanatory Abduction Plain abduction ]O[P RTU VWX S is open (hence
R_W ) and OQP U Y,XT"U VWX S is open (hence Y`_W ).
Further restrictions are required. Added wffs must be in the vocabulary of R . Each such
wff must be either a literal, a non-repeating conjunction of literals or a non-repeating
disjunction of literals.
Given a R and a W , plain abductive explanations are those wffs that close the open
branches a of OQP RT$U VWX S . Consistent abductive explanations are subsets of wffs
which close some but not all open branches a of O[P R:S .
A total closure of a tableau is the set of all literals that close each open branch a .
A partial closure of a tableau is the set of those literals that close some but not all open
branches a . (Such closures are definable for both branches and tableau.)
The negation of a literal is its ordinary negation of atomic or the embedded atom if
negative. Thus V"b$WcVW or W .
We consider now an algorithm for computing plain abductions [Aliseda-LLera,
1997, pp. 102–103 ].
Input
A set of wffs representing theory R . A literal W representing the fact to be
explained.
Preconditions: R"_U WX Z R"_U VWX .
Output
Generates the set of abductive explanations Y,d Z e e e Z Y,f such that OQP P R$T
U VWX S TU Yg X S is closed and the Yg satisfy the previously mentioned lexical
and syntactic restrictions.
Procedures
Calculate R]VWhciU a,d
open branches
Atomic plain explanations
Z e e e Z afX
. Select those
ag
that are
1. Compute OO:jQP a,d Z e e e Z af S[chU kd Z e e e Z k l:XJc the total
tableau closure = the set of literals which close all branches
concurrently.
2. U k d Z e e e Z k l[X is the set of plain abductions.
Conjunctive plain explanations
1. For
m[n each open branch ag construct its partial closure
jQP ag S`c the set of literals that close that branch but
do not close any of the other open branches.
2. Determine whether all ag have a partial closure. Otherwise
there is no conjunctive solution (hence go to END).
176
CHAPTER 9. REFUTATION IN HYPOTHESIS-TESTING
3. Each oQp:q`r st u contains those literals that partially close
the tableau. To construct a conjunctive explanation take a
single literal of each oQp:q`r st u and form their conjunction.
4. For each conjunctive solution v delete repetitions. The
solutions in conjunctive form is vw x y y y x vz .
5. END.
Disjunctive plain explanations
1. combine atomic with atomic explanations, conjunctive with
conjunctive explanations, conjunctive with atomic, and each
atomic and conjunctive with { .
2. Example. Construct pairs from the set of atomic explanations, and form their disjunctions r | t }^| ~ u .
3. Example. For each atomic explanation, form a disjunction
with { (viz. r | t }"{u ).
4. The result of all such combinations is the set of explanations in disjunctive form.
5. END.
In the interest of space, we omit the specification of algorithms for constructing consistent abductive explanation. The interested reader should consult [Aliseda-LLera, 1997,
pp. 103–106 ].
Aliseda rightly asks “whether our algorithms produce intuitively good explanations
for observed phenomena” [Aliseda-LLera, 1997, p. 107]. In answering this question
it is well to emphasize the strength of the tableau approach to abductive inference.
Given that one accepts Aliseda’s Schema for Abduction, introduced in section 2.4, an
extremely attractive feature of the tableau method is that it makes transparent various
means for finding and engaging abductive hypotheses. This goes a long way towards
answering the objection (Reichenbach, Popper, etc.) that there can be no logic of
discovery. Yet, we observe in passing, we find it surprising that this issue is still in
doubt, given the striking success of machine learning protocols. (See here [Gillies,
1994].) Perhaps the greatest weakness of the tableau approach is that it is defined
for only a strong consequence relation in which €: . This means, in effect, that
the tableau method can’t work for our Expanded Schema for Abduction (section 2.5
above). Neither is there any obvious way to adapt the approach to this more general
end.
An open branch s in a tableau ‚[r ƒ„J… †{‡ u is a counterexample of the claim that
ƒ!{ . Alternatively, we could say that open branches are refutations of such claims.
Abduction can be seen in turn as a process of hypothesis-selection that cancels every
refutation. Abduction would, then, be construction of refutation-proof tableaux. This
is fine as far as it goes. On a careless reading, it might appear that the tableau method
is, for a certain range of cases in which €: , a complete system for Popper’s pair,
conjecture and refutation. This would be a mistake twice-over. The tableau method
leaves each of these targets unmet. In those cases in which there are alternative ways of
closing open branches in conformity with the requisite lexical, syntactic and minimality
constraints, the method gives no guidance as to which alternative should be engaged.
9.2. DIALECTICAL REFUTATION
177
So the conjecture problem is underdetermined by the tableau method. Worse still, the
tableau approach makes no contribution to the refutation problem in Popper’s sense.
Popper, like Peirce before him, sees refutation as the failure of a given hypothesis
under conditions of experimental trial. Refutation is thus successful falsification of
a hypothesis. But refutation in tableau models is an open branch, concerning which
hypotheses have no constitutive role. It is only in the process of cancelling a refutation
in the tableau sense that hypothesis enter the picture. When they succeed in this, i.e.,
when an hypothesis does cancel a refutation in the tableau sense, it remains wholly
open to the prospect of subsequent refutation in Popper’s sense. The short way of
saying this is that successful hypotheses are refutation-busters in the tableau sense. But
being so is no indication of their truth.
What, then, might a Popperian refutation look like?
In our discussion of Lakatos (section 4.6), we took note of an important criticism.
Popper is a justification abductivist, although is so with an important twist, as we shall
soon see. Popper holds that science is inherently abductive. It is a matter of making
and holding conjectures under attempts to refute them. Since a failure to be refuted is
always a failure to be refuted so far (i.e., until now) the conjectures of a scientific theory
are always defeasible. Yet-to-be-refuted is not decisive with regard to a conjecture’s
truth (or shelf-life). The closest that a conjecture can come to being confirmed is that
it has so far withstood serious efforts to refute it (which means that by the lights of
classical confirmation theory they aren’t confirmed at all). While being-not-yet-refuted
is not decisive vindication, being refuted is decisive repudiation.
A Popper sees it, a conjecture ˆ is refuted when it has a deductive consequence
which fails an appropriate test (in pure cases, when it is contradicted by an observation). Popper is drawn to what we could call the modus tollens theory of refutation.
Accordingly, if ˆN‰Š and Š is contradicted by a fact, then either ˆ is false or ‹ is,
where ‹ is the larger theory in which ˆ is embedded. Lakatos finds this too hard.
Sometimes, he says, ˆ is retained by finding something else for the discrediting fact
to discredit. Sometimes ‹ itself is retained even though ˆ is dropped or qualified,
or something else in ‹ is dropped or qualified. Either way, Lakatos is attributing to
theories some fairly flexible individuation conditions, which it is not to our purpose to
examine here in any detail. Instead we shall concentrate our attention on the question of
what is it specifically about a scientific conjecture C that a recalcitrant observation discredits? Is it ˆ itself? Is it the theory ‹ in which ˆ is embedded? Is it ‹ ’s background
theory Π?
9.2
Dialectical Refutation
There is nothing in what Popper writes about refutation in which this specificationproblem is addressed in a decisive way. It is interesting that in the theory of refutation
propounded by the father of logic, the specification-problem happens to have a principled resolution. It would be interesting to see whether Aristotle’s solution can be
generalized to contexts of scientific refutation.
Aristotle imbeds his account of refutation in contexts of dialectical argument. There
are two participants,  , a questioner, and Ž , an answerer. Ž proposes a thesis ‹ .  ’s
CHAPTER 9. REFUTATION IN HYPOTHESIS-TESTING
178
role is to question (erōtan)  , putting questions to him, answers to which are formatted
as simple (non-compound) declarative sentences, or propositions (protaseis) in Aristotle’s technical sense of the term.1  ’s answers are thus available to  as premisses of
a syllogism — let’s call it Ref — which it is  ’s role to construct. If Ref is constructed
successfully and if its conclusion is the contradictory ‘ ’“[” , of  ’s original thesis, then
 ’s argument is a refutation of “ .
It is necessary to say a brief word about how Aristotle marshalls the concept of syllogism. Syllogisms, here, are taken in a broad or generic sense, discussed by Aristotle
in the Topics and Sophistical Refutations. Syllogisms are always valid arguments; that
is, their premisses always necessitate their conclusions. Strictly speaking, there can
be no such thing as an invalid syllogism. To qualify as a syllogism, a valid argument
must meet three main conditions (and others that derive from them). First, a syllogism
must have multiple premisses and non-multiple conclusions. Second, a syllogism can
contain no superfluous premisses; that is, they are valid arguments whose validity is
lost under premiss-deletion. Third, if an argument is a syllogism then its conclusion
may not repeat a premiss; hence, in one well-recognized sense of the term, syllogisms
are required to be non-circular.
For a correct understanding of Aristotle’s logic it is essential that syllogisms not be
confused with (merely) valid arguments. Aristotle’s theoretical interest is concentrated
on syllogisms, not validities. Even so, it is interesting to speculate on two subsidiary
questions.
•
•
How did Aristotle understand the concept of validity?
Notwithstanding the preoccupation with syllogisity, does validity have any loadbearing role to play in Aristotle’s logic other than its being a condition on syllogisms?
We will assume without argument that Aristotle’s validity is classical (in the modern
sense). That is, an argument is valid if and only if the set of its premisses and the
negation of its conclusion is inconsistent.2 The second question we shall not here
pursue.3 For concreteness, we now imagine a simple case of refutation.  ’s thesis is
“ .  ’s refutation is
–—
’J“
–
—
in which and are  ’s concessions and ‘ ’“[” is syllogistically derived from them.
It is a noteworthy feature of the case — a feature which generalizes to all refutations
— that  ’s thesis “ cannot be a premiss of Ref. We cannot suppose that “ is either
a missing premiss of Ref or that the result of adding “ to Ref ’s premiss set is itself a
1 Aristotle’s propositions are statements in the form ‘All ˜ are ™ ’, ‘No ˜ are ™ ’, ‘Some ˜ are ™ ’ and
‘Some ˜ are not-™ ’.
2 The classical construal of Aristotle’s validity is considered in greater detail, together with some possible
counterexamples, in [Woods, 2001].
3 Here, too, see [Woods, 2001].
9.2. DIALECTICAL REFUTATION
179
syllogism. To see that this is so, it suffices to recall that Aristotle imposes on syllogisms the requirement that they contain no superfluous premisses. Syllogisity thus is a
nonmonotonic property for Aristotle.
If Ref is, as it stands, a syllogism, then its extension, š
›œ

ž ›
is not a syllogism (though it is a valid argument if validity is monotonic). 4 On the other
hand, if œ this new argument E were a syllogism then its subargument, i.e., Ref

ž ›
œ
œ
 ›:¡
would
would be a consistent set, whereas, by
œ be invalid. œ This
› being
 ›:so,
¡ Ÿ š
Ÿ
the
validity
of
,
is
an
inconsistent
set. But this is a contradiction:
›
 ›[¡¢
 ›[¡
Ÿ Ÿ . The contradiction is pointful: No argument is a syllogism if a premiss is the contradictory of its conclusion. That is, no argument of the
form £
¤
¥
¥
£¥
£
ž
¦
§
is a syllogism (or a refutation either).
Let us
œ revisit our original refutation Ref:

ž ›
.
œ
 ›[¡
Since Ref is a syllogism, Ÿ is an inconsistent set. Hence at least one sentence
of the three is false. Because Ref is a refutation of T, it is attractive to suppose that Ref
›
›
establishes that it is › that is false (for what else would Ref ’s refutation of consist
in but showing that is false?) It won’t do. Saying so is fallacious. It is the fallacy
of distributing negation through conjuction, said by some to be Aristotle’s fallacy of
Noncause as Cause. (It isn’t, but let that pass.)5 What we require is some principled
4 This
5 See
question is taken up in [Hansen and Woods, 2002] and in [Woods, 2001].
[Hansen and Woods, 2002].
CHAPTER 9. REFUTATION IN HYPOTHESIS-TESTING
180
reason to pick out ¨ , rather than © or ª , as the unique proposition refuted by Ref. In
other words, we seek a solution of the refutation specification-problem. How are we
to do this? There are two possible answers to consider. The first will prove attractive
to people who favour a broadly dialectical conception of fallacies. The second will
commend itself to those who think of fallacies as having a rather more logical character.
We examine these possibilities in turn.
First Answer (Dialectical): The first answer proposes that question-and-answer games
of the sort played by Q and A are subject to a pair of related rules. They are dialectical
rules.
Premiss-Selection Rule: In any dispute between « and ¬ in which « constructs a refutation of ¬ ’s thesis ¨ , « may use as a premiss of his refutation
any concession of ¬ provided that it is subject to the no-retraction rule.
¨
No-Retraction Rule: In any such dispute as above, no answer given by ¬
to a question of « may be given up by ¬ .
It is entirely straightforward that among the concessions germane to ¬ ’s defence of ¨ ,
itself is uniquely placed in not being subject to the No-Retraction rule. If it were,
then once conceded it could not be retracted. But if it couldn’t be retracted, it couldn’t
be refuted (or, more soberly, couldn’t be given up for having been refuted). This would
leave refutations oddly positioned. The rules of the game being what they are, a refuted proposition would be precisely what the refutation couldn’t make (or allow) ¬
to abandon. Thus it may be said that, according to answer one, if our refutation game
is a procedurally coherent enterprize, then, of all of ¬ ’s relevant concessions, ¨ alone
stands out. It is the only concession that ¬ can retract, and it thus qualifies for the
proposition which « ’s refutation Ref refutes.
It may be thought, even so, that it is unrealistically harsh to hold all of ¬ ’s other
concessions to the No-Retraction rule. A little reflection will discourage the complaint.
In the give-and-take of real life argument some latitude is given to ¬ . He is allowed to
retract some concessions some of the time. There are limits on such sufferance. If there
were not, then any thesis would be made immune to any refutation of it. If ¬ always
had the option of cancelling one of « ’s premisses, rather than giving up his own ¨ ,
then ¨ would be made strictly irrefutable by any « whose opponent were prepared to
exercise the option. Thus the No-Retraction rule can be considered an idealization of
this limit.
Second Answer (Logical): According to our second answer, that ¨ is the uniquely
determined proposition that « ’s Ref refutes can be explained with greater economy as
follows. Looking again at Ref,
©
ª
­ ¨
we see that ¨ is distinguished as the proposition refuted by Ref precisely because it
satisfies a certain condition. Before stating the condition, it is necessary to introduce a
9.2. DIALECTICAL REFUTATION
181
further fact about syllogisms. Aristotle requires that syllogisms always have consistent
premiss-sets. For suppose that they did not, and let
®
¯ ®
°
be a syllogism. Then since syllogisms obey an operation called argument-conversion,
and since conversion preserves syllogisity, our imaginary syllogism converts to
®
¯ °
®
in which the conclusion repeats a premiss. This is explicitly forbidden by Aristotle’s
definition:
A syllogismos rests on certain statements such that they involve necessarily
the assertion of something other than what has been stated, through what
has been stated (Sophistical Refutations, 1, 165a 1–3).
Whereupon, since the second argument is not a syllogism and yet is an argumentconverse of the first, neither is the first a syllogism. (This ends the aside.)
The condition proposed by the second answer to our question is:
Since
Ref is a syllogism, its premiss-set is consistent. Let Con be the set of
®
’s concessions with respect
to Ref, including the original thesis T. Thus
°Qµ
Con is the set ± ²³ ´:³
. Since Ref is valid, Con is inconsistent. It is
easy to see that Con possesses exactly three maximal consistent subsets
°Qµ
µ °
µ
: ± ´:³
³ ± ´:³ ² ³ ± ³ ² . We will say that a maximal consistent subset
of Con is excluded by Ref if and onlyµ if it does
not syllogistically imply
¯ ²:· . Thus ± ´:³ ² and ± ° ³ ² µ are excluded by Ref,
¶
Ref’s conclusion,
µ
and ± ²
is their intersection. Thus ² is the proposition refuted by Ref
precisely because it is the sole member of the intersection of all maximal
consistent subsets of Con excluded by Ref.
It is worth noting that the requirement that syllogisms be consistently-premissed bears
directly on our present question. ² is dignified as the proposition which Ref refutes
by virtue also of° the requirement
that Ref have°Qconsistent
premisses. Since Ref isµ
µ
µ
±
:
´
³
³
²
±
:
´
³
a syllogism,
is
inconsistent
and
is
consistent.
So too are ± ´:³ ²
°
µ
°
and ± ³ ²
since if they weren’t, ´ would entail ¶ ¯ ²:· and would entail ¶ ¯ ²:· , a
happenstance precluded, each time, by Ref’s syllogisity. Thus three ideas cohere: (1)
the idea that ² is unique in Con precisely because it is ² that cannot be a premiss of
Ref; (2) the idea that syllogisms must be consistently-premissed; and (3) the idea that
² is the intersection of all maximal consistent subsets of Con excluded by Ref.
Just how ‘logical’ is this characterization? Or, more carefully, how non-dialectical
is it? Suppose that we said that the idea of the proposition that a refutation refutes
CHAPTER 9. REFUTATION IN HYPOTHESIS-TESTING
182
could be analyzed without any reference to dialectical procedure. Then any syllogism
whatever would count as the refutation of the contradictory of its own conclusion. Every syllogism would be a refutation, and a great many refutations would be refutations
of propositions no one has ever proposed or will. There is no great harm (in fact, there
is considerable economy) in speaking this way. But it is not Aristotle’s way. Taken
his way, refutations can arise only in question-and-answer games of the sort that we
have been considering. We may say, then, that refutations have a dialectically minimal
characterization. That is, (a) their premisses must be answers given by ¸ to ¹ , and (b)
what they refute must be theses advanced by ¸ . Beyond this dialectical minimum there
is no need to go. In particular, there is no need to invoke the dialectical rules, PremissSelection and No-Retraction. Premiss-Selection is unneeded in as much as there is an
entirely non dialectical reason for excluding º as a premiss of any refutation of º . As
we have seen, no argument of the form
¸:»
¼
¼
¼
¸½
¾ ¸¿
is a syllogism.
Nor, as we have also seen, is the No-Retraction Rule needed to enable the specification of º as the unique proposition refuted by a refutation. Thus the idea of refutations
as refutations of some unique º can be specified without exceeding what we have been
calling the dialectical minimum. It is in precisely this sense that our target notion is a
‘broadly logical’ matter.
Whether we find ourselves drifting toward a dialectical explication of that which a
refutation refutes or to a more logical specification of it, it is fundamentally important
that on neither construal is a refutation of º probative for º . Refutations do not in
the general case falsify what they refute, and cannot.6 In contrast to this, reduction ad
impossibile arguments are probative, but they are not refutations in the present sense.
They lack the requisite forms.7 It is not that refutations don’t falsify something. They
always falsify the conjunction of the members of Con. And it is not that refutations are
indeterminate. They always refute some unique single proposition. What refutations
do not manage to do in the general case is to bring what they falsify into alignment with
what they refute. What they falsify and what they refute are different propositions.
We have said that refutations do not in the general case falsify what they refute. We
must ask whether there might be exceptions to this. There are. Consider the case in
which the premisses of a refutation, RefÀ , are (known to be) true. If so RefÀ constitutes a
proof of the falsity of º , when º is the proposition that RefÀ refutes. Does it not follow
6 We
return to this strong claim below.
one thing, reductio arguments admit of a special class of non-premisses as lines in the proof, i.e.,
assumptions. For another, the conclusion of a reductio is a pair Á ÂÃ ÄÂÅ , which though a contradiction, is
not the contradictory of the proof’s assumption.
7 For
9.2. DIALECTICAL REFUTATION
183
from this that although refutations do not as such demonstrate the falsity of what they
refute, sometimes they do?
Suppose that RefÆ is
ÇÈ
ÉÊ
Ç
È
È
Ç"Ì Ì :
Ê Í
As before Ë
in which and are (known to be) true.
is an inconsistent set.
But there is a difference. RefÆ falsifies Ê .
This makes for a curious dialectical symmetry between Î and Ï . Î ’s obligation
to answer honestly is the obligation to offer to Ï premisses which he (i.e. Î ) believes
to be true. Thus for any rule-compliant answerer against whom there is a successful
refutation, ÐÑ , given what Î is required to believe, he must also believe that ÐÑ
falsifies his own thesis Ê . Though this is what, in all consistency, Î is required to
believe, it does not follow that it is true, nor need Î himself believe that its truth
follows from what he is required to believe.
Ï , on the other hand, is differently positioned. Ï has no role to play in the semantic
characterization of Î ’s answers, hence in judging the truth values of his (i.e. Ï ’s)
premisses. Of course, Ï will often have his own opinions about the truth or falsity of
Î ’s replies, hence about the truth or falsity of his (Ï ’s) own premisses. Whatever such
opinions are, they have no role to play in the construction of Ï ’s refutation. There is
in this a strategic point. Since the success or failure of Ï ’s refutation of Î ’s thesis Ê is
independent of the truth-values of the refutation’s premisses, it might be thought that
the most appropriate stance for Ï to take toward those premisses is the one that shows
Î to best advantage. In fact, there is no semantic stance which Ï can take towards Î ’s
answers, because there is none which shows Î to best advantage. Consider the options.
O1: Ï may suppose that all of Î ’s answers are true. But this commits Ï
to holding that his own refutation is maximally damaging, i.e., is a falsification of Î ’s thesis Ê .
O2: Ï may suppose that at least one of Î ’s answers is false. This commits
Ï to holding that although his own refutation of Ê is not a falsification of
Ê , it fails to be maximally damaging because Î has given a false defence
of it.
O1 and O2 constitute a dilemma for Ï . If he exercises O1, he attributes to Î the
semantic virtue of acquiescing in a sound argument that utterly demolishes Î ’s thesis.
If Ï exercises O2, he makes it possible for Î to evade the heavy weather of an argument
that demolishes his thesis but only at the cost of attributing to himself a failed defence
of it. Ï ’s dilemma is that with respect to the truth values of Î ’s answers there is no
wholly benign assumption that he can make on Î ’s behalf. Hence Ï must make no
assumptions about the truth-values of his own premisses. That being so, the most that
Ï can suppose about his own refutation is that it refutes Ê but does not falsify it.
The asymmetry between Î and Ï is now sharply specifiable. For Î , given what
he must in any case believe, he must also believe that Ï ’s refutation falsifies Ê . For
184
CHAPTER 9. REFUTATION IN HYPOTHESIS-TESTING
Ò
, given what he must not believe, he must not believe that his refutation
falsifies Ó .
Ò
(Even
so,
there
is
nothing
in
the
dialectical
canon
that
precludes
from
knowing
that
Ô
Ò
Ô
must believe that ’s refutation falsifies ’s Ó .)
9.2.1
Dialectical Contexts
Aristotle’s theory of refutation draws on an intimate interaction between two of logic’s
analytical sectors. On the one hand, the theory makes indispensable use of syllogisms,
which are a kind of propositional structure; but it also makes essential use of behaviourconformable rules of argument. Refutations are propositional structures whose premisses arise from rule-governed dialectical activity between a questioner and an answerer, and whose target is a thesis volunteered by one of the dialectical partners (the
answerer). Aristotle’s refutations are therefore social structures of a kind the Greeks
called ‘dialectical’.
When Popper and other abductivists wrote about the method of conjecture and refutation, they are not thinking of refutation as dialectical or social entities. Refutations in
this second sense are not produced by the interrogation of a respondent. Rather they are
produced by the way the world is, by Nature Herself. Popper’s refutations are therefore
natural structures. It is true that the enquirer’s examination of the ways of nature can
be seen as the interrogation of Nature Herself [Hintikka and Bachman, 1991], but this
is a metaphor.
Can the theory of social refutation be extended in a plausible way to a theory of
natural refutation? Might they turn out to have isomorphic structures? We may think
that the metaphor of the scientist as an interrogator of Nature Herself suggests an affirmative answer, but it does no such thing. In the social theory the party whose thesis
is refuted and the party who answers the interrogator’s question are one and the same
player. But if we say that in natural refutations nature’s hard facts are answers to an
interrogator-scientist’s enquiries, then nature counts as the answerer in the refutation
exercise, yet it is not nature’s thesis that is discredited when a refutation is brought off
successfully, but rather a thesis of one or other of the interrogators of nature. This is
dissimilarity enough to discourage any serious idea that social and natural refutations
share enough intrinsic structure to warrant our thinking that a theory of natural refutation is a structure-preserving reinterpretation of the predicates of the theory of social
refutation, or some such thing.
An even greater dissimilarity arises from the conditions under which a social refutation drives Õ -revision to a mechanical solution of the specification-problem, the problem of deciding which of the answerer’s concessions have to be given up, his thesis
or one or more of the statements given in answer to the other party’s questions. The
social theory solves the specification-problem in two ways, dialectically and logically.
The dialectical solution employed a premiss-selection and a no-retraction rule, neither
of which have any counterpart in natural refutations. The logical solution pivots indispensably on peculiarities of syllogistic implication. But there is nothing which says
that a thesis contradicted by one of nature’s hard facts requires that the hard fact follow
by syllogistic logic from the conditions that brought it about.
We conclude that in seeking for a theory of natural refutations, there is little to learn
from social refutations, important as they are in dialectical contexts . So we will need
9.2. DIALECTICAL REFUTATION
to look elsewhere. (See below, chapter 13.)
185
186
CHAPTER 9. REFUTATION IN HYPOTHESIS-TESTING
Chapter 10
Interpretation Abduction
10.1 Hermeneutics
The next number of cases fall into a general category that we call interpretation abduction problems, or abductive problems whose solution requires the abducer to interpret
some linguistic data. Among logicians, the best-known interpretation abduction problem is the enthymeme-resolution problem, also called (with less than perfect accuracy)
the problem of missing-premisses. Although enthymemes will occupy the greater part
of our attention for the next several pages, it is advisable to point out that enthymemes
far from exhaust the class of hermeneutical abduction problems. Also prominent by
way of frequency and importance alike are translation abduction problems, which can
be seen as a generalization of enthymeme problems.
Enthymeme resolution is a particular case of a more general task. In its most comprehensive form it is the problem of interpreting a signal in ways that achieve for it
some contextually indicated or otherwise specified property. In enthymeme resolution,
the signal is an argument and the target property is validity. It is well to note how
narrowly circumscribed enthymeme problems are. They are restricted to signals of a
given type (arguments) and they seek the ‘repair’ of the signal in respect of a single
designated property (validity). In real-life situations most articulate utterance is incomplete in one way or another, and yet argumentative utterance is comparatively rare.
Yet even when a transmission does take on the features of an argument, the problem
to the resolver of the enthymeme is to find a way to make it valid. The first point reflects the fact that most human signalling is non-argumental; and the second gives us
occasion to notice that even when making arguments, participants more often than not
strive for standards other than validity. This, in turn, may reflect the fact that for individuals validity is an expensive commodity, that guarantees of truth-preservation do
not in general come cheap. In our discussion here the cases we consider are elementary. Validity is taken as classical and we concentrate on a type of abduction solution
known as conditionalization. It is perhaps surprising that such cases turn out to be
rather complex.
Even in this classical form, enthymeme problems harbour an ambiguity. What
187
CHAPTER 10. INTERPRETATION ABDUCTION
188
precisely is the enthymeme-resolver’s task? Is he to simply find a valid extension of
his interlocuter’s argument? Is he to furnish an interpretation of that argument which
he thinks it plausible to attribute to his interlocuter? Or is he somehow to combine the
two tasks — to try to give an interpretation of what the interlocuter means to say on
which his argument would turn out to be valid? Is the task, then,
Ö
to abduce a valid extension of his interlocuter’s argument
Ö
to abduce the parts that the interlocuter has left unexpressed
Ö
to abduce a validating such interpretation?
As is obvious, the first task is often completable in ways that stand little chance of
completing the second task, which makes of the third task something rather dubious.
Adding the negation of an original premiss or disjoining the conclusions with its own
negation, or adding as premiss the original argument’s corresponding conditional will
produce a (classically) valid extension each time. But there is no cause to suppose these
to be reasonable interpretations of what the arguer originally intended. It is unlikely
that he intended to root his conclusion in an inconsistent premiss-set; and it is unlikely
that he intended his conclusion to be the logical truths got from the disjunction of his
conclusion as originally stated with its own negation. And while we can certainly take
it that the arguer is committed to the conditionalization of his original argument (and
may indeed believe it), it is not in general a good enthymeme-resolution strategy to add
it as a premiss, since doing so produces a valid argument out of any arbitrarily selected
premiss-set in relation to any arbitrarily selected conclusion.
Framers of this class of problems see enthymemes as constituting a proper subset of
the set of invalid arguments. Members of the subset can be described as enthymematically valid. Let us say that
an argument is enthymematically valid iff
1. it is invalid
2. there is a valid extension of it under constraints ×Ø
Ù Ú Ú Ú Ù ×Û .
With this schematic definition at hand, we can be more specific in detailing the abduction tasks that enthymeme resolvers are required to perform. The enthymeme resolver
is required to specify the ×Û in ways that preserve the fact that enthymematically valid
arguments are a proper subclass of invalid arguments. Another way of saying this is
that the abducer’s central task is that of determining whether the other party’s original
argument is, as stated, enthymematically valid. Bearing in mind that not every invalid
argument is enthymematically valid, the abducer is precluded from selecting a resolution strategy that obliterates this distinction. It is for this reason that although he has
reason to believe that the original arguer believes the conditionalization of his own argument, the abducer is not at liberty to use it as a premiss in the validating extension
of the original. For we may take it that, in the absence of particular indications to the
contrary, the original arguer forwards his argument sincerely. This being so, it would
be defeasibly correct to say that every invalid argument whatever is enthymematically
valid.
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189
Raw conditionalization is not available as an enthymeme resolution device. Raw
conditionalization can be understood as unqualified conditionalization. A finding of
enthymematic validity requires the abducer to impose conditions ÜÝ Þ ß ß ß Þ Üà . One possibility is that those conditions will be constraints on conditionalization itself. Four of
these possibilities will repay consideration. 1
1. Possibility one. The abducer attributes to the arguer the conditionalization of the original argument if it is true.
2. Possibility two. The abducer attributes to the arguer the modal conditionalization of the original argument; that is, he selects a premiss
to the effect that the premisses of the original imply the conclusion.
3. Possibility three. The abducer attributes to the arguer the universal
quantification of the conditionalization of original argument.
4. Possibility four. The abducer attributes to the arguer the genericization of the conditionalization of the original argument.
Option one bids the abducer to attribute the conditionalization unless he thinks it false.
If he does think it false he must also think that the original argument is invalid in a particular way which we will call unregenerate. An argument is invalid if it is possible that
its premisses are all true and yet its conclusion false. An argument is unregenerately
invalid it its premisses are actually true and its conclusion actually false. We may now
take it that, save for unregenerate invalidity, every invalid argument is enthymematically valid, and is made so by addition of the material conditionalization of the original
argument.
There is something to be said for option one. The conditional premiss attributed
to the original arguer is effectively recognizable from any original argument having at
least one expressed premiss and an expressed conclusion. Attribution of the conditional
is also well-motivated in as much as it is a proposition to which the arguer has already
committed himself; and its use is subject to a tractable constraint in as much as the
option bids the abducer to deploy the conditional except where he (the abducer) thinks
that it is (materially) false.
Option one respects the requirement that enthymematically valid arguments be a
proper subset of invalid arguments. Even so, it has the consequence that the set of
enthymematically valid arguments is very large, and that it offers its hospitality to vast
classes of utterly bone-headed arguments. This is not, however, any reason to give
up on option one. It counsels an abductive strategy that is wholly consonant with the
abducer’s target, which is not to convert a bad argument into a good argument, but an
invalid argument into a valid argument. Validity is something, but not everything. Validity is only a particular way among the myriad others in respect of which an argument
can be defective, which is an important fact that the strategy of option one recognizes.
If there is something not to like here, it is not option one, but rather the abduction
problem to which the option provides a resolution device.
1 A further possibility is that the conditionalization attributed to the original arguer involves a conditional
sentence weaker than the material conditional. For ease of exposition, we leave this (important) possibility
undeveloped.
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Option two requires little discussion. It bids the abducer to attribute to the original
arguer the premiss which asserts that the premisses of the original imply the conclusion
of the original. If the premisses are áâ ã ä ä ä ã áå and the conclusion is æ , then the
new premiss is the statement that the á,ç jointly imply æ . There are three difficulties
to take note of. One is that the new premiss is incompatible with the invalidity of
the original argument. Another is that if the premiss is true, the original argument
is valid, hence not enthymematically valid. The third is that if the new premiss is
false, then the original argument is invalid and its extension is valid. But the cost
of this desirable circumstance is the disastrous one in which the distinction between
enthymematically valid and invalid arguments again collapses. Every invalid argument
is now enthymematically valid.
Option two is hopeless. Option one has something to be said for it, provided the
validity-objective of the enthymeme resolution project is allowed to pass muster. One
of the virtues of option one is also a weakness. The conditional it bids the abducer to
attribute to the original arguer is one to which the arguer is already pledged. And this
is something that the enthymeme-solver already knows. The attribution therefore is not
solely a consequence of the fact that, if made, it would solve the enthymeme problem,
and to that extent can be said to be an abductively modest manoeuvre.
Option three evades this difficulty while seeking to preserve the benefits of the first.
In the classic example of [Grice, 1989], the original argument is
1. John is an Englishman
2. Therefore, John is brave.
Option three calls for the extension of the argument by way of the premiss ‘All Englishman are brave’. Here we see that there is a further difficulty with option one
which option two manages to evade. If we were to respond to Grice’s sample argument
in the manner of option one, we would add the premiss, ‘If John is an Englishman, the
John is brave’, which, upon reflection, puts the abducer in an odd position. We said
earlier that this is a proposition to which the original arguer is already pledged. But
if he is already pledged to it, what is the occasion of the pledge? If the arguer thinks
that the original argument is valid, he cannot think that his argument is in need of premissory supplementation. If he acknowledges that the original argument is invalid, he
concedes that it might be true that John is an Englishman yet false that he is brave.
What reason, therefore, is there to attribute to him the proposition that, even so, it isn’t
the case that John is an Englishman and yet is not brave? When we were reflecting
on option one, we thought that the abducer was well-motivated in attributing this to
the original arguer, since in making the original argument, the arguer already pledges
himself to that conditional. But if he now grants that his argument is properly subject
to an enthymeme solution exercise, he has much less clear title to this new premiss.
Option three solves this difficulty. The premiss, ‘If John is an Englishman, then
John is brave’, is something that the arguer can hold wholly independently of his affection for his original argument. A further attraction of the present alternative is that
universal quantification mitigates, if not outright cancels, the implausibility of certain classes of singular conditionals. John’s Englishness and his bravery are a case in
point. A not uncommon reaction to the singular conditional is that John’s nationality
10.1. HERMENEUTICS
191
is irrelevant to whether he is brave or cowardly. But, as often happens in the case of
scientific laws, universal quantification tends to override the appearance of irrelevance.
It is therefore, a better candidate for a manoeuvre that deserves the name of abduction,
since it is not a proposition to which the original argument commits the arguer; and it is
a better abduction in the sense that it attributes to the arguer a proposition the holding
of which explains to some extent his subscription to a singular conditional that appears
to have been defaced by a fairly considerable degree of irrelevance.
On the other hand, the present alternative is far from problem-free. In comparison
with the first option, it is subject to some serious motivation problems. If the original
argument were
1. Sarah is hungry
2. Therefore, Harry will feed her
there is little to be said as a validating extender for
‘All hungry women are fed by Harry’
or
‘All hungry women are fed by men’
and the like. Here the likelier explanation is that Harry is a cook in domestic service
to Leslie, or that Harry and Sarah are husband and wife operating under a particular
distibution of household labour.
Option four is one way of handling a problem that arises with universally quantified
conditionals. As with the attribution of bravery to Englishmen, one thing counting
against the construction of valid extensions by way of such conditionals is their fragility
— they are sentences made false by a single true negative instance. It may therefore
be supposed that in a very large number of cases option three is a strategy in which
one exchanges original validity for subsequent unsoundness. This does no harm to
the designated objective of enthymeme abduction exercises, which is to find a valid
argument to attribute to the original arguer; but motivation again is a problem. The
strategy of option three is to attribute propositions with a good chance of being false
to people who stand a good chance of knowing that they are false (which is a reason
counting against the attribution).
Generic statements are exemplified by propositions such as, ‘Tigers are four-legged’.
When they are true, they remain true even in the face of certain kinds of true negative
instances. As we said in section 2.1 where universal quantified conditionals are fragile,
generic propositions are robust. In the case at hand, there isn’t the slightest chance of
it being true that for all è , if è is an Englishman, then è is brave. There isn’t much
chance either of such a proposition being believed by anyone whose original argument
we would have any supportable interest in rescuing by premissory augmentation. On
the other hand, stereotypical thinking being what it is, it is much more likely that some
people actually do believe (generally) that Englishman are brave. And it is a practical
certainty that anyone does indeed believe it who makes the argument that since John is
an Englishman, he is brave.
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The main difficulty with the fourth option is that premissory addition of the genericized conditionalization of the original argument does not produce a classically valid
argument, which is further reason to doubt the wisdom of reserving validity as the
target property which enthymeme resolution must achieve.2
10.1.1
Abductive Inarticulacy
Inarticulacy as Economics
In what remains of this section we shall concentrate on the inarticulateness that
attends interpersonal argumentative discourse. Much of what we will say in this regard
can without much effort be transposed to the domain of solo reasoning, but we are
inclined to think that there are inarticulacies in solo reasoning which do not easily
make the transition to interactive discourse. We shall not attempt to demonstrate this
intuition here.
The inarticulacies that we intend to track fall into two groups, circumstantial and
constitutional. In a rough and ready way, a circumstantial inarticulacy is something
omitted by a speaker under the press of conventions or contextual cues, which the
speaker is at liberty in principle to violate or ignore. Constitutional inarticulacies are
omissions made by a speaker owing to the way he is structured as a cognitive agent. In
the general case, these are omissions which it is impossible, or anyhow very difficult,
for a speaker to repair.
Our remarks remarks here draw on the model of cognitive agency developed in
chapter 2. We shall attempt to show that the dominant motivation for a speaker’s
inarticulacies is his need as an individual cognitive agent to proceed economically in
thought and speech. As we have said, the individual is a cognitive agent who must
prosecute his agendas on the cheap. Things left implicit are an essential factor in his
cognitive economy.
Circumstantial Inarticulacy. Enthymemes.
As understood by the evolving tradition in logic, an enthymeme is an argument in
which one or more premisses is not explicitly stated, as we have said, or the conclusion
is omitted. In interactive settings, the enthymeme-poser addresses an argument of this
description to an interlocuter, who is the enthymeme-solver. Resolution need not be
explicit. Rather it is supposed that in interpreting the utterance of the enthymememaker the enthymeme-solver implicitly supplies what is missing. In short, enthymemeresolution just is understanding correctly the argument made by an enthymeme-poser.
While the enthymeme-solver normally lacks occasion to voice details of his solutions, this may be something he is able to do at will in principle (and, on the traditional
account, can do). For his part, although the enthymeme-poser leaves certain things
unsaid, he could in principle, on the traditional account, have completed his argument.
Whether or not to proceed enthymematically, is left to the individual’s choice. So,
enthymeme-posing and enthymeme-resolution seem to fall into the category of circumstantial inarticulacy. We note in passing that enthymemes are just a special case
2 Of course, we don’t doubt that for a certain sort of non-classical common sense validity, genericized
conditionalization will turn the trick.
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193
of a much large richer class of circumstantial inarticulacies, which we might think of
generically and collectively as understatements.
Speaking and understanding enthymemes is utterly common in human discourse
and is something we seem to manage with ease in the general case. The commonness and ease suggest the operation of comparatively primitive skills. The suggestion
is borne out by empirical studies of language acquisition, according to which understatement is a feature of speech from the earliest manifestations of a child’s literacy.
Understatement, then, appears to be an intrinsic feature of linguistic competence. To
state is to understate.
What would a wholly general theory of enthymeme-solution look like? On the face
of it, it would come close to a general theory of understatement, which in turn would
come close to a general theory of linguistic competence. Speaking of which, it is a
vast understatement (in a slightly different sense) that there is no such theory in sight.
As if in appreciation of this, logicians and argumentation theorists have approached
the problem of circumstantial inarticulacies in a piecemeal fashion. Two of the most
prominent of these fragmentary approaches are
1. Investigations of how to validate enthymemes.
2. Discussions of Grice’s theory of conversation, and the Principle of
Charity.
Concerning (1), see the discussion in 10.1. Concerning (2) see 10.2.
It is important to see the investigation of enthymeme-validation as a very significant restriction of the general problem of enthymeme-resolution. There is little doubt
that some enthymematic utterances are produced in contexts that make it apparent that
validity is the appropriate and perhaps intended outcome of enthymeme-solution. But
we say again that it is a huge mistake to think that validity-producing solutions are
anything but a small subclass of correct solutions.
We saw that one of the attractions of the validity approach to enthymemes is that
for any missing-premiss enthymeme there is a guaranteed and mechanical way of completing the validity. The rule is:
If the enthymeme omits a premiss, then add to the premiss-set the corresponding conditional of the original argument.
Missing-conclusion enthymemes are not mechanically solvable in any realistic way.
For example, consider the rule
If the enthymeme omits the conclusion, then pick the conclusion (if any)
that strikes you as implicit in the stated premisses, and add to the premissset the conditional corresponding to the original premisses and the postulated conclusion. If a conclusion doesn’t suggest itself to you, pick one
arbitrarily and repeat the conditionalization process.
The inadequacy of such a rule speaks for itself (enthymematically, so to speak).
Even if we concentrate on what appear to be the easier cases of missing-premiss
enthymemes serious trouble lies in wait. In traditional accounts, an enthymeme is valid
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194
if it has a valid expansion (i.e., a solution that produces a valid argument). This makes
every enthymeme valid, which clearly is too latitudinarian a result.
This requires us to consider two kinds of cases.
Case one. The maker of the enthymeme does not intend it as a valid argument.
This is troublesome. It saddles the enthymeme-maker with a valid enthymeme, contrary to his intentions. This suggests the need to revise the characterization of enthymematic validity, perhaps by reversing the flow of the original conditional, thus:
If an enthymeme is valid it is a valid expansion.
But given that every argument whatever has a valid expansion,3 the present conditional
tells us essentially nothing about the validity of enthymemes. For what it tells us about
valid enthymemes holds for invalid enthymemes as well, whatever they are.
The one conditional is absurdly over-generous. The other is vacuous. Perhaps,
then, should take it that validity is not definable for enthymemes. If that were our
decision, it would fit the present case well enough, but it would run into difficulty with
our second kind of case.
Case two. The maker of the enthymeme intends it as a valid argument.
It may strike us as a violation of the Principle of Charity (see section 10.2) to ascribe to
the enthymeme-maker an intention, which is so utterly and manifestly unrealizable. It
is obvious that his enthymeme is invalid (otherwise it would not need valid expansion).
How, then, can he, with any sense at all utter a manifestly invalid argument, intending
it to be valid?
If we say, more carefully, that the enthymeme-maker in uttering his argument did so
with the intention of its being taken as having a valid expansion then, again, we attribute
to the utterer a vacuous intention. Every argument obviously has a valid expansion. So
what would it be to take an invalid argument as not having a valid expansion? This
obliges us to consider a third kind of case.
Case three. The enthymeme-maker intends his argument as having a valid
expansion of the right (i.e., non-vacuous) kind.
One not unnatural way of interpreting the difference between vacuous and nonvacuous
valid expansions is to suppose that, in the latter case, the supplementary premiss is true,
or anyhow that the utterer takes it to be true. It won’t work. In the general case, when
a speaker utters an enthymeme, it is for the interpreter a default that the enthymememaker intends his stated premisses as true and his conclusion also as true. By these
lights the utterer will also take for true the conditionalization of those premises on that
conclusion. Hence the default position is that, in effect, that the right valid expansion
of the arguer’s enthymeme is virtually vacuous. What is more, if we impose the truepremiss requirement condition stringently, and independently of what the speaker takes
for true or intends as true, then any invalid argument satisfying the following conditions
will yield the right valid expansion.
3 Either
by way of conditionalization, or addition of a new premises inconsistent with an original.
10.1. HERMENEUTICS
195
1. Original premisses are true and conclusion is true
2. Original premisses are false and conclusion is true
3. Original premisses are false and conclusion is false.
But once again, what these conditions yield is far too much to make a serious claim to
being right expansions. This suggests the following constraint.
In producing a valid expansion of an enthymeme produced with the intention of its having a nonvacuous valid expansion, it is impermissible to
deploy (true) material conditionals as the missing premiss.
There is something to be said for such a prohibition. We are here speaking of the resolution of enthymematic utterance as produced by actual agents in real-life situations.
Since the typical speaker of English has never heard of the material conditional, it is
questionable to attribute to him a desire that would be met by deployment of it (and
perhaps only by it). It bears on this point that, the vocabulary of English contain no
obvious resources for asserting a material conditional. ‘If . . . then’ does not express it,
and neither does ‘It is not the case that — and not — ’ in as much as ‘and’ seems not
to be truth functional in English.4 All the same if ‘It is not the case that — and not — ’
did pass muster as English, it is not what a speaker of English would offer untutoredly
as a means of expressing the conditional link of one sentence with another.
The conditionals that are within the expressive power of English are themselves
problematic. Consider the modal conditional ‘If . . . then necessarily . . . ’. If the requirement is that enthymemes for which the utterer intends a non-vacuous valid expansion
are to be secured by provision of extra premisses which are true propositions of this
form, then for any argument whose expansion honours this requirement we have an
argument which is already valid, hence not an enthymeme. (Assuming the appropriate
kind of Deduction Methatheorem for English). If, on the other hand, we invoke conditionals stronger than the material conditional and weaker than the modal conditional
just considered, we call for a conditional of a type whose truth conditions are seriously
in dispute. This is no problem for the native speaker of English; that is, he has no difficulty in giving utterance to such conditionals in a natural way. But it should be counted
something of a difficulty for the theorist for having placed the whole weight of nonvacuous valid expansion on an idiom of which he lacks requisite theoretical command. 5
An Abductive Approach
We have been concentrating the discussion on an even narrower issue than the resolution of enthymeme problems by valid expansion. So far, all our cases have involved
the deployment of a conditional proposition as the new premiss, or as the explicitization of the premiss attributed to the original argument as implicit in it. It takes little
reflection to see that there are many more ways of producing valid expansions of invalid
arguments than conditionalizing the stated premisses and the conclusion. The conditionalization cases appear to be the easiest to say something about. But, as we have
4 This
is disputed. See [Woods, 1967].
will Adam’s Thesis help. Adam’s Thesis states truth conditions for the conditional probability (not
the truth) of indicative conditionals. Accordingly Pr(If A then B) = Pr(A and B) divided by Pr(A). See
[Adams, 1975, p. 3].
5 Nor
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been at pains to show, getting a decent theoretical grip on such cases is inordinately
difficult.
This suggests that we should consider telling a very different kind of story about
enthymeme resolution. Also desirable would a determination to throw the theoretical
net more broadly and to cease our preoccupation with an unrealistically narrow band
of cases in which the target of the solution is validity.
The attraction of conditionalization solutions was affection for semantico-syntactic
manipulation. (It was thraldom to the Can Do Principle.) But it is a form of manipulation that seems to get us nowhere, even with respect to the unrepresentatively crimped
type of cases we have considered. This suggests abandonment of the semanticosyntactic approach, in favour of something more generally applicable.
Like all problems of interpreting understatements, enthymeme resolution is itself
an exercise in abduction. In the case of enthymemes, we have an argument, which is in
some way or other unsuccessful on its face. The first task disposed of the interpreter is
to decide whether his interlocuter has just made a bad argument. He does this as follows. He sees that the argument is bad on its face, but he is (or is not) disposed to think
that his interlocuter has just made a bad argument, period. It is hugely problematic to
state in a theoretically disciplined way conditions, which regulate this distinction, but
there is little doubt that the distinction is evident in the actual behaviour of real-life
discussants. One possibility is that the interpreter makes this decision as follows.
He decides whether his interlocuter’s argument is just bad or not bad even
though bad on its face, by assuming the latter and then reconstructing it.
If he is able to do the reconstruction, his assumption is upheld, and he also
has a solution to his enthymeme problem. If not, then he won’t be able,
without supplementary information, to make the distinction on which his
enthymeme solution depends. But the point is academic, since we already
have it that he can’t produce the enthymeme’s solution.6
The general form of abductive resolution of understatement problems is this.
1. The interlocuter makes an utterance
respects is not explicit.
é
, which in a certain respect or
2. The way in which é itself is not explicit is such as to move the interpreter to think that é is not what the utterer intended to communicate, or intended his interpreter to take from U.
3. The interpreter then tries to put himself in the shoes of the utterer.
He seeks empathy with him. He asks himself, “If I were to say what
the utterer did say explicitly, and if I made this utterance from the
perspective I believe him to have brought to bear on the making of
that utterance,7 what would I have likely intended to have had my
interpreter take me as having said?”
6 There are, of course, cases in which the interpreter will have reason to attribute a good argument to his
interlocuter even though he makes no headway in reconstructing it. Think of an enthymematic argument
about topology presented by a world famous topologist to one of his First Year students.
7 This clause leaves it open to an interpreter to interpret an argument successfully even though he himself
from his own point of view would not be making the argument be attributes to interlocuter, but rather would
10.1. HERMENEUTICS
197
In its simplest terms the abduction under review is this: What would I in his place have
intended ê to be taken for? That, defeasibly, is what the utterer of ê also intended and
what he said implicitly.
It remains to say why the interpretation of understatements is an exercise in abduction. The problem is triggered by a phenomenon ë in which the interpreter takes an
utterance ê of an interlocuter to be implicit or to contain an inarticulacy, knowledge of
which is necessary to understand what the interlocuter was intending to say. Thus the
phenomenon ë is one in which the utterer intends to say something that doesn’t get
said by ê , and yet the utterance of ê is not defective on that account.
What needs explaining is how an utterance of ê fails to say what its utterer intended
to say, and yet in saying ê the interlocuter is not simply guilty of not saying what he
intended to say.
The general conjectural explanation is that implicit in the utterance of ê there is an
inarticulacy which, once repaired, would produce an utterance ê * which would, just
so, say what the utterer intended. The specific conjectural explanation is the expansion
of ê produced by the empathetic thought-experiment that we have been discussing.
Thus what explains that ê fails to say what the utterer of ê intended to say, and yet
ê is not a defective utterance in that very regard, is that the utterance that explicitly is
ê is also implicity ê:ì , and ê:ì does, just so, say what the utterer intended. Abductive
Logic
The abduction we have been describing is subject to the regulatory control of our
three logics. The logics of generation and engagement attempt to say how conjectures
which explicitize inarticulacies get chosen. In the example we have been discussing
the selection test is by way of our empathetic thought-experiment. It then falls to the
logic of hypthesis-discharge to set forth conditions, which would determine whether
the explicitization of the inarticulacy had been done accurately or reasonably. For the
kind of case we have been considering, we can give a rough indication of how this
works. In his conversation with the utterer of ê , the interpreter has interpreted ê as
ê:ì . To test for accuracy, it suffices in what remains of their discussion or in subsequent
additional discussions, for the interpreter to guide his future remarks by the assumption
that the utterer’s position is in fact ê:ì . The straight way of determining this is by the
interpreter’s explicitly attributing ê:ì and waiting to see what happens. (“Given that
you hold that ê:ì , don’t you think that so-and-so?”) If there is no move to reject the
attribution, then, as Quine says, there is evidence of “successful negotiation and smooth
conversation,” which is evidence of communicative success, and hence is evidence
of accuracy of interpretation [Quine, 1992, 46–47]. Less directly, the interpreter can
introduce into the conversation propositions which the utterer could be expected to
accede to were it indeed true that ê:ì is a correct interpretation. These propositions
could be immediate or fairly obvious consequences of ê ì but not of ê .
have intended something else. Thus an evolutionist might be able to give a creationist spin to an argumental
utterance, which he had himself uttered it would have carried a Darwinian spin. So the factor of empathy
is essential to this account. For an appreciation of the centrality of empathy in analogical reasoning, see
[Holyoak and Thagard, 1995].
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198
Constitutional Inarticulacies
Perhaps we should be struck by how difficult it is to shed light on how even rather
simple-looking understatement problems are solved. In the approach we have sketched
the interpreter is an abductive agent. But in proposing that account we ourselves are abductive agents. Our account is highly conjectural, and it is offered for the explanatory
contribution it makes to some genuinely puzzling phenomena. Of central importance
is our ability to distinguish between genuinely defective utterances and those that are
defective merely on their face and not otherwise. The other puzzling phenomenon is
the comparative ease and accuracy with which individual cognitive agents are able to
interpret understatements. Our conjecture is that the interpreter puts himself in the utterer’s place and determines, so-placed, how he himself would intend to be understood
in uttering the original speaker’s utterance. Underlying this conjecture are numerous
further assumptions, which we have not had the time to explore. One is that the idea
of a thinking agent’s place is legitimate and theoretically load-bearing. Another is that
empathy permits, within limits, the occupancy of alien places, of places that aren’t the
agent’s own. A third is that one’s own articulacies are self-interpretable in a fairly direct way, involving neither change of place nor empathic engagement. Inarticulate
Understanding
It is easy to see that each of these assumptions could run into the heavy weather
of harsh challenge, especially the third. It is an assumption that tends to conflate understanding with articulacy. More promising is the idea that even in one’s own case,
understanding inarticulacy is not giving an articulation of it, but rather of displaying it
and responding to it in ways appropriate to an articulation of it were one available. This
leaves the door open for a notion of understanding inarticulacy in ways that themselves
are inarticulate. If so, such inarticulacies are constitutional rather than circumstantial.
They are inarticulacies, which we can’t articulate yet which we manage to understand.
Upon reflection, even if our conjectural sketch of how interpreters handle understatement is true as far as it goes, it is unlikely that the interpreter, as he negotiates the
routines of placement and empathetic engagement, is able to articulate what he is doing. In these respects, the interpreter of understatements is like the maker of them. No
matter how explicit their utterances, discussants do not in general have any command
of the precise conditions under which their utterances were given form and expression.8
This suggests constitutional inarticulancy on a rather grand scale. What would account for it? It has to do with constraints imposed by our cognitive economy, especially
by the hostility of consciousness toward informational abundance. Or so we conjecture.
10.1.2
Achilles and the Tortoise
We note that our discussion to date provides the wherewithal to dispose of two classical
difficulties; one advanced in Lewis Carroll’s Paradox, [Carroll, 1895], and the other
originating with Sextus Empiricus’ accusation that every syllogism commits the fallacy
of petitio principii Sextus Empiricus [1933]. The paradox of Lewis Carroll bears on
8 We might mention in passing visual factors in hypothesis formation in contexts of visual inference in
archaeology. As detailed by [Thagard, 1995], such hypothesis can have highly complex structures, which
are nonsentential. They are nonlinguistic and subconscious.
10.1. HERMENEUTICS
199
the discussion of option two, and Sextus’ challenge pertains to on the discussion of
option three.
In option two (section 10.1), the abducer extends the original argument by adding
as a new premiss the modal conditionalization of the original argument. His target is to
produce a valid argument from an invalid one by adding premisses which, apart form
the fact that they achieve validity, there is reason to think that the original arguer would
be prepared to concede (or, to be more exact, to concede that this very premiss was
tacitly at work in the original argument). We found option two to be open to criticism
in a number of ways; but it never occurred to us that the new argument wasn’t valid.
It is this, or something like, that Lewis Carroll’s paradox calls into question. If left
standing, it would appear to show that no argument is enthymematically valid, hence
that no invalid argument has a valid extension.
In Carroll’s note, Achilles and Tortoise are chatting about a proof from Euclid,
symbolized as
í
1.
2.
3.
î
ï
Their discussion proceeds in stages. In stage one, Tortoise accepts í and î , but he
does not accept ï , and does not accept the argument’s modal conditionalization, ð , ‘If
í and î are true, ï must be true.’
At stage two, Tortoise does something odd. He retains his commitment to í and î
and he persists in his non-acceptance of ï ; but he now accepts ð , the modal conditional
which he previously rejected. Tortoise’s position now is that even though í and î
are true and that ï follows logically from í and î , this does not logically compel
accession to ï .
At stage three, Tortoise repeats his basic position, except that he now agrees to a
conditionalization of the argument of stage two
1.
2.
3.
4.
í
ð
î
ï
namely, ‘If í[ñ î and ð are true, ï must be true.’ It is hard to know who is the greater
inadept, Tortoise or Achilles. Their disagreement is settled unawares at stage two. At
stage one, Achilles was right to reject the conditional C, precisely because he accepted
its antecedent and rejected its consequent. At stage two, he loses his purchase entirely.
He denies that the original argument is valid and he asserts that its modal conditionalization is true. What he (and Achilles, too) fails to discern is that the thing denied is
equivalent to the thing asserted. By Tortoise’s own admission of ð , the original argument was valid. So, while it is perfectly true that this argument is not answerable to
enthymeme-resolution treatment, this is because the argument is valid, not invalid, and
is therefore no general indictment of enthymeme-resolution procedures.
This is not to overlook that sometimes it is helpful to a valid argument a premiss
whose truth is equivalent to the claim of validity. We say that a semantically valid
CHAPTER 10. INTERPRETATION ABDUCTION
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argument of English is one in which it is in no sense possible for the premisses to be true
and the conclusion false. A formally valid argument is an argument that is semantically
valid in virtue of its form. Not every semantically valid argument is formally valid, as
witness the following argument, Geom:
1. The figure is a square
2. Therefore, the figure is a quadralat.
Let ò óô õö be any semantically valid but formally invalid argument. An argument is
valid iff its premisses entail its conclusion. So if we add to Geom the premiss that that
a figure is a square entails that it is a quadralat, we add a premiss which says what
Geom already is. If it were in doubt that Geom is valid, addition of the premiss would
be question begging. If it weren’t in doubt that Geom is valid, addition of the premiss
would be redundant. Even so, some good can come of adding the premiss. It converts
an argument that is semantically valid and formally invalid into an argument that is
semantically and formally valid. But, of course, this rationale for adding the premiss is
not available in the general case.
A further flaw in the Tortoise’s line of argument can now be identified. Consider
any valid argument ò óô õö from a set of premisses ó . The argument isn’t valid unless
ò óN÷Nõö , i.e., the premisses semantically entail the conclusion. But this is not the
same as saying, nor does it follow, that every entailment required by the validity of an
argument is required as a premiss for the argument to be valid.9
10.1.3
Sextus Empiricus
Option three of section 10.1 bids the abducer to attribute to the arguer the universal
quantification of the (material) conditionalization of the original argument. By this
device, the extension of the original argument is
1. All Englishmen are brave
2. John is an Englishman
3. John is brave.
Sextus charged that all such arguments beg the question, that John’s bravery is already
somehow in the premisses. If Sextus is right, the strategy of option three may seem to
fail. It doesn’t; the objective is to produce a valid extension of the original, not a nonquestion-begging extension. Even so, it cannot be an encouragement to the enthymeme
resolving abducer that his success is guaranteed to be blighted by fallacy.
9 One way of making out Tortoise to be less dim than he seems actually to have been, is to see him as
holding (but not announcing) that there is no Deduction Metatheorem provable for natural languages; in
which case there might be valid English arguments whose modal conditionalizations are false. Let ø be an
argument with inconsistent premisses and an arbitrarily selected conclusion. Then ø meets the conditions
on validity, notwithstanding that native speakers of English may reject its conditionalization, which is just an
instance of ex falso quodlibet. Relevant logicians persist with the Deduction Metatheorem, which commits
them to reject the validity of ø . But another possibility can be considered: Accept the validity of ø , deny
the modal conditionalization of ø , and (therefore) reject the Deduction Metatheorem for English.
Doubtless, however, such were not matters that occurred to Tortoise.
10.1. HERMENEUTICS
201
Mill is also said to have pressed the same complaint. In fact he rejected the complaint [1961]. It was sound, he said, only on a faulty analysis of general propositions, a
view which anticipated Wittgenstein’s position in the Tractatus Logico-Philosophicus.
If the general proposition that all Englishmen are brave is the conjunction ù of all conjunctions of the form ‘a is an Englishman and a is brave’, then the proposition ‘John
is an Englishman and John is brave’ will be a conjunct of ù . But the argument’s conclusion ‘John is brave’ occurs expressly in ù as a conjunct of that conjunct. So the
argument is circular.
Mill drew the opposite conclusion. The argument would be circular if this were the
correct view of general propositions. But, by reductio ad absurdum, it isn’t valid; so
the received view of generality is mistaken [Woods, 1999c].
10.1.4 Some Virtual Guidelines
It is necessary to emphasize that in our discussion so far, we have concentrated exclusively on a simple kind of resolution device, conditionalization in various forms.
We intend no monopoly for it, even allowing for it’s various particular deficiencies.
Some enthymemes offer it no purchase, as for example with the ‘missing conclusion’
enthymeme,
1.
2.
3.
All men are cads
And John is a man
or, ‘missing minor premiss’ enthymemes
1.
2.
3.
All men are cads
So John is a cad
or even, given the right context, a combination of the two:
1.
2.
3.
All men are cads
We have said why we think the classical preoccupation of enthymeme-theorists
with validity is unfortunate. For one thing, it makes the class of enthymeme problems an unrepresentative sample of hermeneutic abduction problems. Still, there is
something positive to be learned from considering such cases. Even though narrowly
constrained, and therefore to a degree artificially tricked out, it is surprisingly difficult to specify the right virtual rules even for the subclass of enthymeme problems
in which conditionalization is a possible option. Once the constraints improved by
the validity-only target are lifted, rules become even more complicated to get right.
Notwithstanding, we now suggest some general guidelines, which of course are virtual guidelines (most interpretation abduction takes place ‘down below’). To make our
exposition manageable we shall, however, continue to focus on the interpretation of
arguments.
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1. Arguments are often incompletely formulated. In their express form it is not
strictly true that they satisfy certain properties of arguments in which the arguer
can be supposed to have an interest.
2. The job of the abducer is to attempt to supplement the argument as stated in ways
that secure the desired property or properties.
In so doing, the abducer is under two general obligations in whose joint fulfillment
there can be, as we have said, no antecedent guarantee of consistent compliance. The
abducer must, on the hand, interpret what his interlocuter meant to say, and on the
other, must try to find in such an interpretation factors that will, once assumed, achieve
for the argument its desired objective.
What properties an argument is aiming for are usually contextually available to the
interpreter. What interpretation to consider is in turn a matter of common knowledge
and context.
Common knowledge is knowledge had by a community, K. The community may
be as big as the entire population of the world or as small as the spousal unit created
by marriage. Knowledge is common when and to the extent that it can be presumed
that members of K have it and that being members of K suffices for their having it,
other things being equal. What passes for common knowledge is not knowledge in a
strict epistemological sense, since what a community knows by common knowledge
sometimes turns out to be untrue. Common knowledge is therefore what a community
confidently takes to be knowledge and what it confidently expects its members to claim
to know.
This last makes for considerable economy. If ú is a proposition that Harry claims
to know by common knowledge, then by its commoness it is also what Harry would
expect that the others would claim to know; and this he can know without having to
ask, so to speak.
A further economical feature of common knowledge is its tacitness. It is entirely
explicable that this should be so. If ú is a proposition of common knowledge, then it
may be presumed that Harry knows it, and that if he knows it as a matter of common
knowledge that he knows that you know it, too. Then, since we are spared the nuisance
and cost of publicizing ú , there is less occasion to give it expression than other things
we know.
Context furnishes those whose actions and interactions are influenced by shared
knowledge that arises from particularities of their interaction. Context makes common
knowledge possible for pairs of interacting participants. So contextual knowledge is a
species of common knowledge. The two are also connected in a further way. What is
common knowledge for me is a matter of the epistemic community to which I belong.
Knowing that these, but not those, are the epistemic communities to which I belong is
itself a matter of context. It is a matter of the particularities of my interaction with my
fellow beings that answers that question.
Like common knowledge of all stripes, contextual knowledge is also often tacit.
The same is true of the background knowledge of a scientific theory. In some respects,
a theory’s background is the knowledge common to fellow scientists, and in other respects, a theory’s background is a matter of, say, details of the functioning of the field
10.1. HERMENEUTICS
203
team’s apparatus. What is common to background knowledge in scientific theories is
the following.
1. If û is a proposition of background knowledge for a theory üý û is not likely to
be used expressly as premises in formal derivations of the laws and theorems of
ü .
2. Notwithstanding, û ’s truth affects positively the epistemic context in which it
was possible for theorists to derive ü ’s laws and theorems.
3. In the event that ü is met with counterevidence, the likelihood increases that it is
û that will be forwarded expressly, and considered for express rejection, rather
than some expressly derived law or theorem of ü .
Feature (1) suggests that scientific theories, like enthymemes in a certain way, are
expressively incomplete, and that their correct understanding requires an abducerinterpreter to marshall the appropriate û s of background knowledge, and further that
this be done for the most part tacitly.
Characteristic (2) suggests the need to understand with care the distinction between
a theory’s heuristics and its laws and theorems. Once way for a proposition û to
contribute positively to the epistemic context in which laws and theorems are derived
is to influence the theorist in figuring out how to make his proof. Another way is for û
to function as a tacit premiss in these formal derivations.
Property (3) bears on refutation. A refutation-problem with respect to a theory
ü is the problem of determining with optimal specificity that aspect of ü which an
apparent counterexample to one of its laws or theorems makes necessary. It is a matter
of figuring out where to ‘pin the blame’. In those cases in which it is judged best to pin
the blame on û , a proposition of the theory’s background, there are two consequences
of note. One is that refutation-management is a standard way of moving û out of its
former anonymity, or bring û out of the closet, so to speak. The other is that it lies
in the very nature of pinning the blame on û that we settle the question of whether
û was a tacit premiss in ü ’s formal derivations, or whether the tacit recognition of û
facilitated — causally assisted — the theorist in thinking up the ways and means of
deriving a law or theorem of ü .
We said that the task of the abducer-interpreter of an incompletely expressed argument is to furnish an interpretation that can plausibly be attributed to the other party and
which strengthens the argument’s claim on some contextually specified target property.
Doing this, we said, was a matter of common knowledge and of context. How?
We postulate the following virtual rules.
The Common Knowledge Rule. For reasons of economy, and of plausibility as well, the hermeneutical abducer should try to draw his supplementations of the original argument from what is common knowledge for
both interpreter and arguer.
The rule pivots on the commoness of common knowledge. If û is common knowledge
for the abducer and there are contextual indications that both parties inhabit an epistemic community of a sort appropriate to discussion of the original argument, then û is
CHAPTER 10. INTERPRETATION ABDUCTION
204
knowledge for the other part, the abducer knows this; and thus can attribute it without
incurring the costs of further enquiry, and can attribute it plausibly.
The Contextual Knowledge Rule If a candidate, P, for interpretation of
the original argument is not upheld by the prior rule, the abducer should
seek to make his selection from the contextual knowledge furnished by
particularities of his interaction with the original arguer.
The rationale is the same as before: economy and plausibility.
Our cases admit of some straightforward formalization. If the original argument is
þ
1.
2.
ÿ
Then the judgement that it is an enthymematically valid argument is conveyed by acknowledging that
:
þ
ÿ
where
is background knowledge accessible in principle to utterer and intepreter
alike, and is read:
gives that since þ
ÿ
.
Then the interpreter’s abduction task is to specify a subset of
such that
þ
ÿ
is classically valid.
Seen this way, enthymeme resolution is a specification-task, itself a kind of revisiontask. We note here that refutation-management has a similar structure. A theory always has a background . If something contradicts (as is said loosely and conversationally), the theorist has the option of ‘blaming’ or ‘blaming’ . But either way
his further task is to specify the source of the error, whether in proper or in .
As has been said, the two fundamental tasks of the hermeneutical abducer may
prove not to be consistently co-performable. It is one thing to figure what the original
arguer was meaning to say; it is another thing—and sometimes a thing incompatible
with the first—to spin the original in ways that make it a better argument in some
contextually indicated way. How are such tensions to be avoided or minimized? Can
they be avoided or minimized? Enter the Principle of Charity.
10.2
Charity
The Principle of Charity found its place in the common currency of argumentation
theory by way of two sentences from a contemporary First Year logic text book.
10.2. CHARITY
205
When you encounter a discourse containing no inference indicators which
you think may nevertheless contain an argument, stop and consider very
carefully whether such interpretation is really justifiable . A good rule
is ‘the Principle of Charity’: If a passage contains to inference indicators
or other explicit signs of reasoning and the only possible argument(s) you
can locate in it would involve obviously bad reasoning, then characterize
the discourse as ‘non-argument’[Thomas, 1977, p. 9].
Three years later, Michael Scriven provided the principle with what has become a
widely attended to and well-received characterization.
The principle of Charity requires that we make the best, rather than the
worst, possible interpretation of the material we are studying [Scriven,
1976, p. 71].10 The Principle of Charity is more than a mere ethical principle, but it is at least that. It requires that you be fair or just in your
criticisms. They can be expressed in heated terms, if that is appropriate;
they may involve conclusions about the competence, intellectual level, or
conscientiousness of the person putting forward the argument, all of which
may well be justified in certain cases. But your criticisms shouldn’t be unfair; they shouldn’t take advantage of a mere slip of the tongue or make a
big point out of some irrelevant point that wasn’t put quite right.
Charity requires that we not be unfair in our disputes. Scriven’s principle also implies
(CorrectInt): Interpret your opponent or your interlocutor correctly.
It is perhaps a bit odd that in virtually none of the discussions of the Charity Principle in
theories of argumentation is there any recognition that something called the Principle
of Charity is at the centre of a brisk contention and a huge literature in the adjacent
precincts of philosophy of language and cultural anthropology.11
Given its earlier ancestry, it is appropriate to make a detour and to set up a problem
in the theory of translation for whose solution a Charity Principle is invoked. It is convenient to secure this motivation by examining briefly the principal thesis of Chapter
Two of Quine’s Word and Object [Quine, 1960]. But a second reason is that if Quine
is right about translation, this will affect importantly the logic of translation-abduction
problems.
10.2.1 Indeterminacy of Translation
‘Chapter Two of Quine’s Word and Object contains what may well be the most fascinating and most discussed philosophical argument since Kant’s Transcendental Deduction
10 See also [Baum, 1975; Johnson and Blair, 1983; Hitchcock, 1983; Govier, 1988; Johnson, 1981;
Rescher, 1964; Parret, 1978].
11 Among the exceptions that come to mind would seem to be Rescher and Parret. According to F.H. van
Eemeren and Rob Grootendorst, Rescher acknowledges a deviation in his own treatment of charity from
Davidson’s, whereas Parret’s is a Davidsonian notion [van Eemeren and Grootendorst, 1983, p. 189, nt. 47].
Van Eemeren’s and Grootendorst’s own discussion of charity is a farther exception, to which we shall recur
in due course.
CHAPTER 10. INTERPRETATION ABDUCTION
206
of the Categories’; so says Hilary Putnam [1975b].12 and he is right. Quine is talking
there about ‘radical translation and, as he proposes,
There can be no doubt that rival systems of analytical hypotheses can fit
the totality of dispositions to speech behaviour as well, and still specify
mutually incompatible transactions of countless sentences insusceptible
of independent control.
What does this mean?
Imagine that a field linguist is making the first attempt to translate into English the
language of a people deemed to be culturally very different from the translator’s own.
The linguist will prepare a translation manual which Quine calls an ‘analytical hypothesis’. Preparing such a manual in such circumstances is what Quine dubs ‘radical
translation’.
Quine assumes that the linguist knows how to recognize assent and dissent as
speech acts in the host community. If this is right, he can locate truth functions in
the host language and he can also discriminate occasion sentences, such as ‘Here’s
a cup’, for which certain stimulations will produce assent and others dissent. Standing sentences, such as ‘Mayfair is in London’, are such that once a speaker has been
prompted to assent (or dissent) he prevails in it without ‘further immediate stimulation
from the environment’ [Putnam, 1975b, p. 159].
Quine identifies the stimulus meaning of a sentence with the set of stimulations
of a host’s nerve endings which would induce assent to that sentence. Two sentences
will have the same stimulus meaning if the same stimulations would induce assent to
each. A sentence is stimulus analytic for a speaker if he assents to it no matter what
the stimulation, and a sentence is stimulus contradictory for a speaker if he dissents
from it no matter what the stimulation. Observation sentences that have intersubjective
stimulus meaning.
Central to Quine’s thesis is the notion of an analytical hypothesis. An analytical
hypothesis is a general recursive function, , from the set of all sentences of the host
language to a subset of the set of all sentences of the visiting language. The function is
constrained by three conditions.
(i) if is an observation sentence of the host language, is a visiting observation
sentence and has the same stimulus meaning for visiting speakers as does s
for host speakers;
(ii)
commutes with the truth functions, i.e., on;
equals , and so
is stimulus analytic (or stimulus contradictory) in the host language, then
is stimulus analytic (or stimulus contradictory) in the visiting language.
(iii) if
If the linguist happens to be bilingual in the host language, condition (i) is replaced by
12 We follow Putnam’s exposition, which is the best three-page account of the indeterminacy thesis we
know of.
10.2. CHARITY
207
(i*) if is an occasion sentence of the host language, then is an occasion sentence in the visiting language, and the stimulus meaning of for the linguist is
the same as the stimulus meaning of for the linguist.
Conditions (i)–(iii) or (i*)–(iii) are Putnam’s paraphrase of conditions (1)–(4) or (1*)–
(4) of chapter two of Word and Object. Translations made in conformity with these
conditions are made in circumstances that collectively exhaust the evidence for the
translation in question.
The thesis of chapter two is this. Suppose that our linguist entertains two rival analytical hypotheses and . There is, says Quine, no ‘fact of the matter’ as to whether
the translations afforded by are correct or as to whether the translations afforded by
are correct. Given that conditions (1)–(4) or (1*)–(4) exhaust all possible evidence
for any translation, and could each fulfill (1)–(4) or (1*)–(4) and still ‘specify
mutually incompatible translations of countless sentences insusceptible of independent
control [i.e., beyond the sway of evidence]’. So there is no such thing as an objectively correct translation. If this is right, it bears directly on the translator’s abductive
task. In conjecturing about how to interpret an interlocuter’s utterances, the abducer is
at liberty to conjecture that the meaning of the interlocuter’s utterance is the same as
some utterance of the abducer’s language. But it is not, and cannot, be a condition of
abductive success in such a case that the two utterances acutally mean the same.
But couldn’t the translator become bilingual and from that position check to see
which of or (or some other manual) is objectively correct? He can do the one,
says Quine, but not the other. Becoming bilingual just means that the linguist’s analytical hypothesis will be governed by (1*)–(4) rather than (1)–(4); but fulfilling (1*)–(4)
will not block the specification of mutually incompatible translations of countless sentences beyond the sway of (1*)–(4). The fate of the bilingualist suggests to Quine that
indeterminacy considerations apply equally to intralinguistic translation and interpretation.
How is the translator (or interpreter) to choose among rival analytical hypotheses?
Suppose he specifies some further condition (5), and that conformity with conditions
(1)–(5) produced a unique translation, that is, that there were one and only one translation that fulfilled those conditions. Such a translation would be rivalless, but it could
not be said to be objectively correct, for there is no fact of the matter as to whether
there is a fact of the matter that (5) is an objectively correct constraint.
Charity is one way of specifying condition (5). It is a way of discriminating rationally among rival analytical hypotheses, never mind that none could he said to be
objectively correct. If we take Charity in Davidson’s way we will hold that ‘no sense
can be made of the idea that the conceptual resources of different languages differ
dramatically’ [Davidson, 1984]. And
Charity is forced on us; whether we like it or not, if we want to understand
others, we must count them right in most matters.
Davidson concedes that this is a ‘confused ideal’.
The aim of interpretation is not agreement but understanding. My point
has always been that understanding can be secured only by interpreting in a
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CHAPTER 10. INTERPRETATION ABDUCTION
way that makes for the right sort of agreement. The ’right sort’, however, is
no easier to specify than to say what constitutes a good reason for holding
a particular belief [Davidson, 1984, p. xvii].
The heart of Davidson’s approach is that in interpreting another person it is necessary,
if we are to make sense of him at all, to make an assumption of the following sort.
[Charity]: He is not noticeably deficient in holding true those things which
we ourselves are rather adept at holding true. (‘Oh, look, it’s raining’).
But, by parity of reasoning, a corollary should be acknowledged.
Concerning those things that we notice we ourselves are not particularly
adept at holding intersubjectively-true (e.g. a theory in sharp contention),
we should likewise suppose that the persons whom we are interpreting may
not be significantly more adept about such matters than we. Where there
is intersubjective disagreement at home it is not unreasonable to predict it
elsewhere.
This carries interesting consequences. It permits us to preserve disagreement in our
interpretations when two conditions are met: first, when the interpreter takes heed
of all available evidence and, thus, complies with (1)–(4) or (1*)–(4), and second,
when he has reason to believe that the subject matter of the interpretation has had an
intersubjectively bad dialectical track record here at home. (The theory of being qua
being; or the morality of euthanasia).
There is a great deal to puzzle over in the modern history of, as we shall now
say, Radical Charity (the Principle of Charity in the context of radical translation),
ensuing from Wilson [1959]. We shall not persist with those puzzles here, for our
prior question is what, if anything, does Radical Charity have to do with Charity as a
dialogical maxim? The answer would appear to be ‘Not much’.
If we consider the sorts of cases mentioned by Thomas and Scriven, it is plain that
Dialogical Charity (as we shall now call it) is not the principle that fills for condition
(5) on radical translation.
For if it does follow from Scriven’s principle that the interpreter should furnish
objectively correct interpretations, it is a presupposition of Radical Charity that it is
precisely this that an interpreter cannot do for any sentence or discourse that outreaches
the provisions of conditions (1)–(4) or (1*) to (4). Scriven’s principle is not only not
the same as Radical Charity but, if the implication we’ve been discussing holds, it is
incompatible with it.
Suppose now that we have been radicalized by Quine’s thesis about the possibility
of objectively correct interpretation. In that case, we will say that our implication,
CorrectInt (‘Interpret your opponent or your interlocuter correctly’), does not hold.
Will Radical Charity now serve as an explication of Dialogical Charity? We think not.
Here is why.
I. Radical Charity has nothing to do with fairness. It does nothing to disenjoin the attribution of embarrassing or loony views to natives or others
on grounds that this is a nasty, culturocentric thing to do.
10.2. CHARITY
209
II. Radical Charity, applied at home, holds us to the view that the community of our co-linguists is not in general to be counted as massively
in error on those matters concerning which we are in relative semantic
serenity, that is, intellectually sincere and reflectively untroubled serenity.
So Radical Charity gives no instructions for the individual case, except as
noted below.
III. Radical Charity does have a kind of negative implication for various
kinds of case, contextually specifiable in individual situations. But here,
often as not, the Principle is suspended, not applied. So, for example, the
very fact that we are locked in a dispute with a co-linguist counts as some
evidence that he is wrong, and our interpretation of what he says must
preserve this fact or make a nonsense of our disagreement.
Perhaps the charitable dialectician will object that we got things muddled. He might
point out that neither Radical Charity nor Dialogical Charity is a carte blanche, and that
neither requires the suppression of disagreement in individual cases. This is quite right.
Radical Charity posits uniformities in human nature that would justify the reflection of
agreement in interpretation in the general case. But we want to suggest that not only
does Dialogical Charity permit disagreement-preserving interpretations in particular
cases, it also permits them and counsels them in the general case as well or, more
carefully, for the general run of cases for which the Dialogical Principle has apparently
been designed. If this is right, the two Principles remain significantly different. The
question is: Is this right? (We return to this below).
10.2.2 Dialogical Charity
We now turn to a discussion of Dialogical Charity which stands out for its careful attention to detail. It is in other ways a limited account, but there is a reason for this, as
we will see. In van Eemeren and Grootendorst [1983] a charity principle is invoked in
the course of trying to solve a problem that might be said to be the classical motivation of all dialectical invocations of charity. (Henceforth charity is Dialogical Charity
unless otherwise indicated.) The problem is that of reconstructing an interlocutor’s
argument in a way that captures a premiss heretofore unexpressed. So we are back
to the problem of enthymeme-resolution. Van Eemeren and Grootendorst locate the
problem in a theory of the cooperative resolution of disputes, for the procedural canons
of which they rely heavily on features of Gricean co-operation. 13 They propose that
13 The
features of Gricean co-operation are given in the following maxims.
1. Quantity. I expect your contribution to be neither more nor less than is required.
2. Quality. I expect your contributions to be genuine and not spurious.
3. Relation. I expect a partner’s contribution to be appropriate to the immediate needs at
each stage of the transaction.
4. Manner. I expect a partner to make it clear what contribution he is making and to
execute his performance with reasonable dispatch.
See [Grice, 1989, p. 28].
CHAPTER 10. INTERPRETATION ABDUCTION
210
the unexpressed premiss be associated with a conversational implicature, for whose
specification they provide essentially Gricean instructions. We suppose that Harry and
Sarah are in the process of co-operative conflict resolution. Harry produces what van
Eemeren and Grootendorst call a ‘single-argumentation’, that is (roughly), an argument
with just one conclusion. The argument is classically invalid as it stands, is recognized
to be so by Sarah, and may be recognized to be so by Harry as well. The task for Sarah
is to attribute to X a belief or a commitment which makes the expansion of the original
argument valid (the expansion just is the result of adding the new or unexpressed commitment to the premiss-set of the original argument), and, moreover, which is held to
the condition that expansion remains a single argumentation (more of this anon). Their
account requires the following assumptions:
1. The conflict resolution mechanisms available to the disputants are understood by
each to be Grice-co-operative.
2. It is a violation of the procedural canon to attribute violation of the procedural
canon unless (counterfactually, here) there is reason to do so [from (1)].
3. It
understood that with regard
to Harry’s utterance of his argument,
,isitmutually
is a condition of adequacy that be valid.
4. Sarah knows that is not valid and, moreover, that taken at face value it violates
the Relation maxim; that is, at face value
there is reason to question the relevance
of the premiss-set to the conclusion of .
5. To take
at face value would be a violation of procedure for Sarah [from (2)].
6. To avoid violation, Sarah must attribute to Harry some commitment which
validates an expansion of [van Eemeren and Grootendorst, 1983, 134–215 ].
Since
any number, in fact an infinity, of sentences will serve to validate an expansion
of , the question now is for which value of ’ ’ will a Grice co-operator
produce a
valid expansion of Harry’s argument ? For ease of exposition let be the previously
discussed argument,
1. John is English.
2. Therefore, John is brave.
Earlier writers, Scriven, for example, propose that the specification of the missing premiss might be held to a condition of minimality or a stronger condition of optimality
[Scriven, 1976, p. 43].
By the minimality requirement, Sarah would opt for ‘If John is English then John
is brave’,
which has two significant features. One is that it does validate an expansion
of and the other is that it is not ‘independently well-supported’ [Scriven, 1976]. This
can be understood
that it produces a valid
in the following way: although Sarah will seethat
expansion of , she may doubt that it is true. If she supposes
Harry tacitly holds
this premiss as true on no grounds other than the validity of , the original argument,
she attributes to Harry a serious error and has placed herself at risk of noncooperation.
10.2. CHARITY
211
Or if he attributes to X tacit resort to this premiss just because Harry holds-true the
premiss and conclusion of the original argument, then although this makes ‘If John
is English then John is brave’ truth functionally true and although, so taken, this will
generate a valid expansion of , Sarah nevertheless attributes a question-begging manoeuvre to Harry, in violation of procedural assumption (2), above. We may say, then,
that fulfilment of the minimality condition can’t be an option for Sarah without procedural default. Optimality can now be understood as the attribution to Harry of a
commitment which (a) produces a valid expansion of , and yet (b) does not land
Sarah in procedural defection.
Can we give instructions to Sarah as to how he is to fulfil the optimality requirement? Van Eemeren and Grootendorst introduce a pair of terms which preserve the
distinction, as we have explained it, between Scriven’s minimality and optimality.
They speak of producing a premiss which does ‘the logical minimum’ for the repair
of Harry’s defective argument, and a premiss which will do the ‘conversational minimum’ for it. They propose that the condition of conversational minimality will turn the
trick and that it does a better job than optimality.
Van Eemeren and Grootendorst are grappling with the problem of how to produce a
valid expansion of in the most Grice-cooperative way. They point out that inasmuch
as their ‘analytical reconstruction’ (as they call it) of the interplay between Harry and
Sarah is Gricean, anyone who Grice-cooperates is one ‘who observes the cooperative
principle [and who] must also observe the conversational maxims.’ So the cooperator
is held to the maxim of Relation (which they mistakenly say commits her to produce a
valid expansion of 14 ) and also to the maxims of Quantity and of Quality. Recall that
the Quality maxim holds a cooperator to providing just the information needed, and no
more, for the task at hand; and that the Quality maxim holds her to say what is true
and, negatively, not to say what she doesn’t believe to be true, or that for which she
(realizes she) lacks adequate evidence.
We are now at the nub of the van Eemeren and Grootendorst solution of the missing
premiss problem. Various candidates are now easily dismissed:
* If John is English then John is brave
is dismissed because it gives only the logical minimum, which is insufficient, for
the reason already sketched. There is in fact an algorithm that stands in wait. If ! s" # ’, the corresponding conditional of A is ‘If
is any argument ‘ $%& $' $%! , then # ’, and the conditional expansion of is the argument that
results from adding the corresponding conditional of to the premiss-set of , and
adding no other premiss. In a conditional expansion of is what is produced by exercising possibility one of the previous section. The mechanical rule is: Do not solve the
problem of the specification of the missing premiss of an argument in such a way as
to generate the conditional expansion of .
A second candidate,
* All island dwellers are brave
14 Sarah
could just as well attribute to Harry the generic sentence, ‘The English are brave’ which, while it
won’t produce a valid expansion, certainly takes care of any worry about the relevance of the premiss to the
conclusion in the original argument
CHAPTER 10. INTERPRETATION ABDUCTION
212
is excluded since it doesn’t produce a valid expansion of ( .
A third candidate,
* Since all island dwellers are brave and all Englishmen are island dwellers,
then all English are brave.
won’t do since it violates the maxim of Quantity; it says more than is needed for the
task at hand. It is easy to see that any proferred premiss which itself has the structure
of an argument will violate the maxim. So another algorithm suggests itself.
MRule: reject any premiss that produces an expansion of ( that is not itself a singleargumentation, that is, an argument with one conclusion only.
A fourth candidate,
* All Englishmen are brave and island dwellers and subjects of the Queen
and eligible for the National Health Service
while true, is also disqualified by the Quantity maxim. Here too another rule drops out.
Rule:
admit as a premiss no conjunction containing one or more conjuncts validating
the expansion and at least one that does not. Call such conjunctions ‘distributively inert’.
We can see that the tight and admittedly dialectically unrealistic constraints that van
Eemeren and Grootendorst impose on their Grice-cooperators make possible a close to
mechanical solution of the problem of the missing premiss, as follows:
)'*
iff
is an admissible choice in the generation of a classically valid expansion of (
(a)
*
(c)
*
generates a valid expansion;
(b) the valid expansion produced by * is not the conditional expansion of ( ;
does not produce a valid expansion of ( which fails to be a singleargumentation (or as van Eemeren and Grootendorst say, is a compound
argumentation);
(d) if * is a conjunction fulfilling (a),
conjunction.
*
must not be a distributively inert
Applying these contrasts will give just what van Eemeren and Grootendorst are after:
* All Englishmen are brave.
The example here discussed is naively simple and unrealistic, and the cooperative constraints imposed, for all their benign and collegial attractiveness, are as untrue to real
life as a Bowdlerized fairy tale. The story told throughout about Charity is descriptively inadequate, and earlier admonitions about careless attributions of ‘normativity’
to ‘analytical reconstructions’ are so many chickens now come home to roost. All of
this is true but explicable.
10.2. CHARITY
213
The great charm of the van Eemeren and Grootendorst solution bears an undeniable
ancestral link to the solution of the classical problem of which their own is close kin.
The classical problem is that of converting a fragmentary argument into a categorical
syllogism. Given the syntactic devices for the specification of propositional type and
syllogistic form then, the theory of the syllogism provides the solution of the problem
of the enthymeme as it there occurs. It is not for nothing that the classical problem
endures to this day in logic textbooks: it is solvable mechanically. However, once we
start dismantling the structural assumptions of the case, once we abandon the unrealistic requirement that the expanded argument be classically valid, those mappings break
down, and the Gricean presumptions are left to do their work all but unguided. But
they do seem to work for the restricted kind of case.
10.2.3 Radical Charity Again
A word about Charity, now ‘radical’. It is possible to be a charitist about radical translation without embracing either Quine’s indeterminacy thesis or Davidson’s defence
of Radical Charity. David Lewis is one such theorist [Lewis, 1983, pp. 108–115 ].15
Lewis, too, considers Charity in the context of radical translation. We suppose that our
linguist wants to make sense of Karl, his host. Lewis’s Charity is restricted to attributions we should be prepared make to Karl, knowing only his physical situation and
his physical history, but not knowing anything whatever about his psychology and not
having (for now) any linguistic access to him. In other words, we imagine impediments to the exercise of our rules of Common Knowledge and Contextual Knowledge.
Lewis’ Charity directs us to attribute to Karl the beliefs and desires that we ourselves
would have in his physical situation. Karl holds his hand out and upwards. Facing the
sun, Karl now stops squinting. Charity instructs the linguistic to assume that Karl had
the sun in his eyes and wanted to shield his eyes and did. Lewis’ Charity directs us
to make uniformity assumptions. The human animal is approximately the same wherever we find him. Charity is thus a basic abductive strategy regarding the doxastic and
conative properties of human beings. Charity for Lewis is but one of six principles of
which the linguist is invited to avail himself. The Principle of Rationality extends the
inductive strategy to Karl’s inferences. Karl is taken to be an inferer rather like the
linguist. The Principle of Truthfulness attributes to Karl (tacit) subscription to a convention of truthfulness in his language just as the linguist so subscribes in his language
community.
The further three principles tell the Lewisian linguist how to approach Karl’s language. The Principle of Generativity posits that Karl’s language is much like the visitor’s own and, in particular, that the truth conditions of sentences in Karl’s language
are simply and finitely specifiable and uniformly so. How else could Karl learn his language? The Manifestation Principle provides that the linguist should assume that the
beliefs expressible in Karl’s language should ordinarily be manifest in his dispositionsto-utter: So he assumes that Karl is not a radical concealer of his beliefs. And, finally,
the Triangle Principle bids the linguist to constrain his translations of Karl’s language
in such a way that sentences expressing beliefs and desires express the same beliefs
15 The
chapter is called ‘Radical Interpretation.’
CHAPTER 10. INTERPRETATION ABDUCTION
214
and desires.
Lewis believes that, armed with his six principles, a linguist can overcome indeterminacy latencies in radical translation. But the more important point, for our current
purposes, is that Lewis’ principles constitute a certain view about uniformities in human nature. Each licenses a generic sentence in the form ‘Human beings are or do
so-and-so.’ The principles can be linked to what Marvin Minsky has called a framesystem [Minsky, 1977].
A frame system is intended to be a structured piece of knowledge about
some typical entity or state affairs. It contains variables onto which go
specific values + + + [Johnson-Laird, 1988, p. 107].
Following Putnam’s well-known example, [1975], if we possess a frame system for
tigers and a tiger that we are inspecting is three-legged, then the variable NUMBEROF-LEGS will take the value THREE. But in the absence of such information, the
frame-system will specify the characteristic value FOUR. FOUR is said to be the ‘default value’ for this variable. A characteristic property of a thing or event is the default
value with regard to a given variable furnished by, or which would be furnished by,
its frame-system in the absence of contrary information. Of course, the notion of a
frame-system is a partial elucidation of the concept of a prototype, for a prototype of
something just is a frame-system in which all default values are specified. It also enshrines something of what it is to be characteristic. for it is characteristic of lions to be
four-legged, never mind that Pussy the tiger might be three-legged.
Lewis’ broadly charitist heuristic comes to this. The linguist should treat Karl as
exemplifying the prototype human being, and unless he does, radical interpretation of
him is impossible. Lewisean charity (broadly speaking — we now mean all six principles), posits a dialectical demeanour not greatly unlike that of a Grice-cooperator.
Even so, Radical Charity, now taken in Lewis’ sense, diverges from Dialogical Charity
in a fundamental way. For although we may approach a co-lingual disputant prototypically, the whole point of disputation is the systematic displacement of default values
with contextual determined ‘real’ ones. Our fellow disputant is measured against prototype with regard to his rationality and his truthfulness. Dialogical Charity commands
no more than that we not attribute to our co-linguist defections from rationality and
truthfulness that are not manifested in his dialogical performance and evidenced by our
beliefs about the world.
10.3
Semiotics
Semiotics is the science of semiosis, which defines a relation of signification on a triple
of relata , sign, object, mind - . Although semiosis received significant theoretical attention in the Middle Ages, especially in the logic of John of Saint Thomas, Peirce is
rightly considered its modern originator. On Peirce’s account, perception and cognition
alike are inherently inferential. In any act of awareness, a sign is presented to the knowing agent, who then makes an inference to the relatum of the sign. On its face, semiosis
presents the abductive theorist with a very large class of data for analysis and of problems for resolution. If all awareness, whether perceptual or propositional, is semiotic,
10.3. SEMIOTICS
215
then all awareness involves the interpretation of signs, and all such interpretation is
inferential. This would appear to justify our subsuming the class of hermeneutic abduction problems in the more capacious class of semiotic abductive problems. What
makes the latter a superset of the former is that the former involves the interpretation
of linguistic signs, whereas semiosis also includes the interpretation of non-linguistic
signs. What makes it justified to conceive of semiotic inference as abductive is that this
is how Peirce himself conceived of it, notwithstanding his insistence elsewhere that
abduction belongs to science and not to practical affairs. Our view is that semiotic interpretation often, if not always, takes the form of abduction-resolution and, therefore,
that a comprehensive theory of abduction must embrace (much of) semiosis. This is a
daunting task which we shall not attempt to perform here. But we will take note of a
flaw in Peirce’s universalist presumptions for semiosis, for the claim that all awareness
is semiotic.
basic0 form, awareness is defined for a sign . , a mind / , and an0 object
0 . In/ itsis most
aware of when he attends to . and infers that its designation is . But
what is it for / to attend to . itself? It is to be aware of . as a sign. For this to be
so, there must be some further sign .21 to which / attends, from which M infers that
the relatum of .21 is . . Either all awareness is semiotic or not. If the former is true,
then semiotics harbours a vicious infinite regress. If the latter is true, it is necessary
to say in a principled way what the difference is between semiotic awareness and nonsemiotic awareness. There has not yet been any settled consensus as to how to handle
this problem. In our view, the difficulty arises from the lack of success to date in
producing a wholly general theory of signs — linguistic and non-linguistic, natural
and conventional, and so on.
216
CHAPTER 10. INTERPRETATION ABDUCTION
Chapter 11
Relevance
11.1 Agenda Relevance
Under the category of RELATION, I place a single maxim, namely,
‘Be relevant’. 3 3 3 [I]t’s formulation conceals a number of questions that
exercise me a good deal: questions about what different kinds and focuses
of relevance there may be, how these shift in the course of a talk exchange,
how to allow for the fact that subjects of conversation are legitimately exchanged, and so on. I find the treatment of such questions exceedingly
difficult 3 3 3 .
Paul Grice, Studies in the Way of Words
The logic of discovery has two parts, the logic of hypothesis-generation and the
logic of hypothesis-engagement. In chapter 1, we saw the abductive agent as operating
in the following schematic way. The agent’s circumstances at the time constitute an
abductive trigger with respect to some target 4 . The solution of the agent’s abductive
problem requires that he find an H for 4 . It is not typically the case that H is unique
for 4 ; accordingly, H in general will be one of up to indeterminately many alternatives. If we suppose that H is indeed the solution required by 4 , and if we suppose
that 5 is a space of possibilities to which the agent has prior access then the agent
must execute an ordered pair of fundamental tasks, each of which corresponds to the
set-theoretic operation of non-conservative restriction. Jointly they are the Cut Down
Problem discussed in chapter 2.The first operation find the relevant possibilities in 5 ,
a proper subset of 5 , which we designate as 6 . 6 is what, in chapter 2, we called the
candidate space. A candidate space is a space of relevant possibilities. Drawing out the
logic of this operation is the principal business of the generational sublogic, the logic
of hypothesis-formation. Next comes the operation of finding in 6 the more plausible
candidates, a proper subset of 6 , which we designated 7 . Specifying 7 is the principal
task of the second sublogic, the logic of hypothesis-engagement. Thus 5 itself is input
to 6 , whose output in turn is input to 7 .
217
218
CHAPTER 11. RELEVANCE
It is open to dispute as to where to lodge the next step in the process, in which,
so too speak, a unit subset of 8 is formed, which in the case before us is 9 H : . Since
9 H: has H as it sole member; accordingly H itself may be dubbed the (uniquely) best
hypothesis for ; . The disputed function as we say, is whether to see the selection of H
as part of the engagement process or as a part of the discharge process.
An hypothesis qualifies for discharge when it meets the appropriate justification
conditions. This might lead us to suppose that since the justification conditions will
favour the most plausible hypothesis, there is a considerable economy to be achieved
in trying to find the most plausible candidate as early as possible in the abductive process, and, so, to make it a matter for the engagement-sublogic rather than the logic of
retroduction. This, however, is not our view of the matter. We are convinced by actual
cases that justification does not always favour the most plausible, and that on occasion
justification actually trumps plausibility. This is a point that goes well beyond the issue
of the logic in which to specify the most plausible; it also provides for the overriding
of plausibility in any degree by particular factors of an hypothesis is justification. This,
we take it, is the moral to draw from Hanson’s insight that one way of arriving at a candidate for conjecture, is to pay attention to the contribution it would make if deployed
in requisite ways. It would be surprising if such were not the case, given the variety
of things that the justification of an hypothesis could end up justifying. Planck again
comes to mind. The predictively rich unification of the Rayleigh-Jeans and Wein’s radiation laws comes at the cost of admitting quanta into the theory’s ontology. Doing so
moved physics ahead momentously, but Planck did not for a moment think that the existence of quanta was in the least plausible. By these lights, what justifies the discharge
of an hypothesis can override its total want of plausibility. It is advisable therefore to
sound the admonition early rather than late that there are classes of abduction solutions
that leave little room for considerations of plausibility, if any at all.
It would be natural to think that much the same is also true of the relevance factor.
Certainly there are accounts of relevance in which the relevance or irrelevance of an H
to a ; is prior to whatever facts of the matter apply to H in a justifactory way. But, as we
will see, ours is a conception of relevance on which the relevance of an H to a ; could
be established only by the justification conditions that H chances to satisfy. In such
cases we might say that the relevance of H had been constituted by those justificational
circumstances, in the absence of which there would have been little or nothing to be
said for its relevance.
The moral is that antecedent determinations of neither plausibility nor relevance is
intrinsic to the consummation of a retroduction. This does not change the fact that,
for many cases of various kinds, plausibility and relevance are indeed indispensable
parts of the conceptual map that attends full-bore abduction, whose explications are the
proper business, respectively, of the logic of generation and the logic of engagement.
The main task of the present chapter is an explication of relevance.
Cognitive agents bring to their sundry tasks the net of their cognitive capacities
together with what they can recall of previous cognitions to which they have retained
requisite access. This is a substantial resource — the sum total of what the agent knows
and know how to do at that time — and is too much for indiscriminate deployment for
the performance of any given cognitive task. In chapter 1 we speculated on the influence of consciousness on the cognitive economy. Consciousness, we said, is a radical
11.2. ALTERNATIVE ACCOUNTS
219
suppressor of information. It is a small step to the supposition that on virtually any
occasion of an agent’s performance of a cognitive task, the information that is suppressed by virtue of constraints that flow from the very structure of consciousness will
be information irrelevant to the task at hand. This is not to say that the information that
is preserved in consciousness on any such occasion is guaranteed to be relevant to it.
For the relevant information, unconscious or conscious, might simply not be available
to the tasker on that occasion. But where the task is satisfactorily performed then and
there, it can safely be supposed that the informational cutdown that consciousness itself
provides has left a residue of information relevant to the task that the agent seeks, or is
disposed, to perform or to the agenda that he wishes to advance and close.
It hardly needs saying that for large classes of cases simply being conscious of the
agenda you wish to perform and of whatever else is concurrently in your awareness is
a long way from having performed the task. There is abundant evidence that in such
cases human beings are capable of attending to proper subclasses of what they are
concurrently aware of, and that what distinguishes those subclasses from the others is
the greater relevance of the information they contain to the agent’s present task. Either
way, conscious task-solving appears to involve consideration of relevance at two stages,
each of which represents the restriction of a larger body of information to a smaller. In
place one it is evident that the dynamic of the restriction is automatic and unconscious.
In phase two the dynamic of the restriction is less clear.
In Gabbay and Woods [2002a] we propose an explication of relevance along the
following lines. Relevance is defined for ordered quadruples < = > ?2> @> = A B , where I is
information, S is a cognitive agent, A is an agenda of S’s and I* is background information. Seen this way, information I is relevant for subject S as regards his cognitive
agenda A, and in the light of background information I*. Relevance is a causal relation. It is processed by S in ways that bring about changes in S that advance or close his
agenda A. If S’s agenda is to construct a proof of a proposition P, information I might
play on S in ways that induce the belief that such-and-such a lemma would facilitate
the proof of P; and the belief might be correct. If so, information I was relevant for S
because it got him to recognize a way of moving the proof along. If, on the other hand,
S’s agenda were to determine whether Sarah has already come home from her work at
the office, he might open the front door and call out “Hi, Sarah. Are you here yet?”
Hearing no answer provides him with information that makes him believe that Sarah
has not yet arrived. This is information relevant for S in this matter; it affected him in
ways that closed his Sarah-agenda.
11.2 Alternative Accounts
Before continuing with agenda relevance, it would be well to pause and take note of
the fact that among logicians relevance is customarily taken as a propositional relation,
either semantic or probabilistic. Since ours is a considerable departure, it is nothing if
not prudent to try to take the measure of the leading propositional relation-accounts,
especially in the context of the abductive sublogic of hypothesis-generation. In this
section we shall attempt to show two things. One is that such treatments of relevance
have internal difficulties apart from any consideration of their role in their explication
CHAPTER 11. RELEVANCE
220
of abduction. The other is that, irrespective of their internal difficulties, they don’t adapt
well to the demands on the concept of relevance imposed by the sublogic of hypothesisgeneration. It is not our intention to make a detailed examination of all going accounts
of relevance (not even [Gabbay and Woods, 2002a] does that). It will suffice, we
think, to give the flavour of the various kinds of approaches that have attained a certain
standing in the research community. We have no interest in showing that such accounts
are no good, but rather that the agenda relevance approach has advantages that serve us
well in the explication of abduction.
11.2.1
Relevant Logic
We begin with relevant logic, the product of a huge research programme since the late
1950s, and dignified by results of considerable technical sophistication. (An excellent
overview is [Dunn, 1986].) The earliest of what became standard systems of relevant
logic are due to Anderson and Belnap [1975] and [1992]. Such systems embed two
core ideas about relevance, content-sharing and full use of hypotheses. According to
the first idea two statements are relevant to one another when they overlap in meaning.
The second idea is different. It represents relevance as a relation on the hypotheses of
a proof and its conclusion.
The first idea is given (weak) formal representation as follows.
Content containment (CC): Where S and SC are statements, S is relevant to
SC (and it to S) only if S and SC share a propositional letter.
The second idea of full-use of hypothesis is given by this condition.
Full-use of hypotheses (FUH): Let D be a proof of a formula S from a set
of hypotheses E H F , G G G , HH I . Then D is a relevant proof of S from E H F ,
G G G , HHI if and only if each HJ is used in D .
The intent of FUH is likewise rather weak. It is that a relevant proof is one in which
all hypotheses are in fact used, not that they must be used. There is in this contrast the
difference between a relevant logic and a linear logic. In a linear logic a valid proof
can have no valid subproofs of the same conclusion of the same conclusion. Relevant
proofs are not required to meet this condition.
One of the attractions of CC is that it suggests a natural-seeming constraint on the
consequence relation. Thus
Conseq: Where SC is a statement and K is a set of statements SC is a logical
consequence of K only if K is relevant to S; i.e., for all S K K, S is relevant
to SC .
On the face of it, Conseq disarms the classical theorem known as ex falso quodlibet,
in which negation-inconsistency implies absolute inconsistency. Clearly, the conditional statement L S MN S O Q P fails CC for arbitrary Q; its antecedent and consequent
contain no sentence letters in common. It comes as something of a surprise therefore
that there is a proof of Q from L S MN S P in which at no step is CC breached. The proof
in its modern form is due to Lewis and Langford [1932].
11.2. ALTERNATIVE ACCOUNTS
1. SQ2R S
2. S
3. SS Q
4.
R
S
5. Q
221
hypothesis
1, simplification
2, addition
1, simplification
3, 4, disjunctive syllogism
It is clear that at every step, CC is honoured. Allowing for a sixth step given by conditionalization on (1) and (5), we have it that the conditional conclusion T SQR S U Q V
fails CC. What this appears to show if relevant logicians are right is that the Deduction
Theorem fails for the consequence relation, since what we appear to have here is a valid
deduction of Q from T SQR S V but a corresponding conditional T SQ2R S U Q V that fails
owing to its noncompliance with CC.
Of course, this would be right if CC were a sufficient condition on relevance. In
fact it is a necessary condition, and, being so, gives the relevant logician some purchase
against the Lewis-Langford proof. For they can say that compliance with CC at each
step of the proof of Q from T SQR S V doesn’t give relevance, a fact that if underscored
dramatically by the real, not just apparent, irrelevance of the sixth line, the conditional
proposition T SQR S U Q V . The Deduction Theorem recurs with interest, as we see from
the following argument.
1. The Deduction Theorem is sound.
2. Conditional proposition # 6 is false.
3. It is false owing to irrelevance.
4. Accordingly, the Lewis-Langford proof is defective.
5. Even though the derivations of lines 2 to 5 satisfy CC, this is compatible with the irrelevance of one or more of those derivations.
6. Therefore, it is reasonable to suppose that the proof of Q fails owing
to irrelevance.
As far as we are aware, no one has pressed the present argument against the LewisLangford proof. The standard move is to condemn disjunctive syllogism outright as an
obviously invalid rule [Anderson and Belnap, 1975, need page number here]. Stephen
Read sounds a similar theme in proposing that both addition and disjunctive syllogism
are valid rules but for different senses of the connective ‘or’. This means that if both
addition and disjunctive syllogism are given valid deployment in the Lewis-Langford
proof, then the proof equivocates on ‘or’. On the other hand ‘or’ if treated univocally
in the proof, one of those two rules is invalid. Either way, the proof is invalid.
The criticisms of Anderson and Belnap and Read converge on the claim that disjunctive syllogism is invalid. Anderson and Belnap hold that disjunctive syllogism
commits a fallacy of relevance [Anderson and Belnap, 1975, pp. 164–165 ]. And according to Read, disjunctive syllogism has instantiations that equivocate on ‘or’. We
ourselves are unconvinced. If, as Anderson and Belnap aver, disjunctive syllogism
commit a fallacy of relevance (i.e., is a rule in which the antecedent fails to be relevant
to the conclusion), we must ask, “For which conception of relevance is this so?”. It
222
CHAPTER 11. RELEVANCE
bears on this that disjunctive syllogism satisfies CC. This does not guarantee relevance,
but it leaves it wholly undetermined as to what the irrelevance it is alleged to commit
might actually be. Nor is it full-use irrelevance in as much as any proof of W from hypotheses X Y[Z\W^] _2Y` does make use of both hypotheses. Accordingly, there is nothing
in what Anderson and Belnap have told us about the concept of relevance that gives
any reason to suppose that this is what disjunctive syllogism lacks.
Nor are we able to see in disjunctive syllogism any occasion of equivocation. Let a
be any contradiction in the form b Y[c[_2Yd in which, whatever its other details, c obeys
simplification and _ functions as standard negation. Then in any language having these
resources in addition to truth-functional disjunction will guarantee a truth-preserving
route from b Yec_2Yd to arbitrary W . For this to be the case, it is unnecessary that Z
preserve the meaning of any ordinary language ‘or’; nor does it matter whether the f
of the conditional sentence b Ygcg_YhfiWjd has the same meaning as any ordinary
language ‘if’ or expresses anything that native speakers would be prepared to call ‘entailment’ in any ordinary sense. What counts is that the derivation is proof-preserving
and that there is a requisite Deduction Theorem. Thus for any such contradiction there
is a truth-preserving derivation of arbitrary W such that for the requisite ‘if’, it is true
that if Yece_Y , then W . We may say, then, that the accounts of relevance offered by
Anderson and Belnap and by Read fail in one of their fundamental purposes, namely,
the disarming of ex falso quodlibet. This attests to what is in any event perfectly evident, that in offering only a necessary condition on relevance, CC gives a weak and
incomplete explication of it. So, in requiring that relevance be a necessary condition
on logical consequence, proponents of content containment relevance have no more to
offer than a necessary condition on a necessary condition on logical consequence. We
note again that the desired outcome of CC-relevance constraint on logical consequence
cannot be delivered by a FUH conception of it. The Lewis-Langford proof meets the
requirements of FUH.
There is much more to relevant logic than its discontent with ex falso quodlibet. Substantive work has been done in advancing the cause of relevance modeltheoretically and proof-theoretically beyond the early systems of over forty years ago.
Relevant logic remains an active research programme and it continues to attract logicians of considerable imagination and technical acumen. Also discernible in this
sprawling work is a tendency toward logical pluralism. (See here [Restall and Beall,
2001; Priest, 2001; Woods, 2002b; Woods, 2001h].) One of its attractions is the appearance, if not the reality, that relevant implication is just one of many kinds of implication,
along with classical, intuitionistic, paraconsistent, and so on. It is a view that makes a
virtue of toleration, obviating the need to argue that “my” implication is the one and
only real implication. In a sense, then, logical pluralism pacifies the earlier quarrels
between relevant and modal logicians about which approach delivers the real goods for
implication. But not every difficulty is dealt with in this way. Complexity is a case in
point; so is intensionality. We touch on these points briefly, in reverse order.
One of the insights that has persisted since earlier times is that a relevant logic is
at least partially intensional. It has been known for some time that the implication
fragment kmf l of the system R of Anderson and Belnap gives a reasonable seeming
treatment of relevant proof, deductibility and entailment. knf l is decidable. It also
11.2. ALTERNATIVE ACCOUNTS
223
r
excludes ex falso quodlibet. Suppose that we were to add to the language of oqp the
extensional connectives s and t . In doing so we lose each of these attractive features
[Avron, 1992]. our p cannot be extended to its own supertheory R without R failing to
be an extensional logic. What is more,
To complete the picture, we add that in Routley-Meyer semantics t , for
itself, has a very simple and intuitive interpretation. The price is, however,
that implication has a very complicated (and in our opinion) unintuitive
semantics. Again — it is the attempt to have together both a relevant
implication and an extensional conjunction which is incoherent and causes
difficulties. Each of them is separately OK! ([Avron, 1992, p. 280, n. 26];
emphasis added in the second instance.)
Quine’s resistance to relevant logic turns on the same point. He has repeatedly insisted
that any deviation from an extensional connective entails deviating from them all. Since
standard systems of relevant logic intensionalize a connective, Quine’s point is that the
whole logic goes intensional. An intensional logic is a modal logic, and for Quine
modal logic is a contradictio in adjectivo.
One of things modern logicians look for in a system of logic is its adaptability to
arithmetic. R# is the system of Peano arithmetic that arises from R. The Gramma Rule
[Friedman and Meyer, 1988]. Gamma decrees that from theorems S and
fails
v S w Sinx y R#
, Sx is also a theorem. This is misfortune for R#, as it significantly complicates
its proof-finding capacity. Similarly, R itself has complexity problems all its own. R is
undecidable, as mentioned earlier [Urquhart, 1984]. Decidability can be restored if we
surrender the distributivity axiom. This produces the system LR; but LR
has an awesomely complex decision-problem: at best ESPACE hard, at
worst space-hard, in a function that is primitive recursive in the generalized Ackermann exponential. From NP to ESPACE or worse, courtesy of
relevance! — so much then, for this brand of relevance [Tennant, 1993, p.
7].
The decision-problem for LR is intractable. They have no practical solutions.
Whether this matters is a function of what systems such as R, R# and LR are wanted
for. We ourselves haven’t the slightest inclination to condemn them outright, nor are
we disposed to join in disputations about what logic really is and really isn’t, without
regard for its presumed service obligations. With two possible exceptions, reluctantly
conceded, Quine regards all systems of nonclassical logic as not logic at all. The
possible exceptions are intuitionistic logic and quantum logic [Quine, 1970]. This is
not the place to pursue these matters. (But see [Woods and Peacock, 2002].) Our
task is to ascertain whether relevant logics do the business required by the relevance
component in the conceptual map of abduction.
There are two questions which we might properly put to the relevance logician. One
is whether relevance logic explicates the concept of relevance in a way that elucidates
the conceptual map of abduction in the requisite way. In putting it that the abducer
seeks relevant proper subsets of sets possible hypotheses, we put it that the soughtafter subsets z instantiate a concept of relevance. Which concept is it, and does the
account given by relevant logicians explicate it?
224
CHAPTER 11. RELEVANCE
The other matter that arises is whether the rules (or virtual rules) of relevant logic
serve as protocols which enable the abducer to cutdown sets of possibilities to subsets
of relevant possibilities. Does the abducer implement a relevant logic in executing the
routines of the logic of discovery?
It may strike us that the last question is easily handled. The abducer in whom we
are here interested is the individual agent. Individual agents are practical agents. They
execute their cognitive tasks with limited cognitive resources. The standard systems
of relevant logic are intractable. The tasks that they are purpose-built to solve have no
practical solution. They cannot be solved by practical agents. This being so, must it
not be said that, except at the unconscious level, in seeking for relevant subsets, the
abducer is not executing the virtual rules of relevant logic? (We are not so sure; see
just below.)
Down below is another matter; down below, the home of what Douglas Walton has
called our “darkside commitments” [Walton, 1996, pp. 203, 251]. We want to take
care to make of down below what Molière’s médicin malgré lui made of the dormitive
virtue. Not much is known directly of cognitive operations down below. Circumspect
indirection is usually the best that can be done. It is here that the question of conceptual
adequacy re-enters the picture. It does so suggestively rather than decisively; but it is
grist for our mill all the same.
We said that relevant logics have sought to take note of two basic conceptions of
relevance, that of content-containment and that of full-use of hypotheses. In the case
of CC-relevance, the most that we have by way of explication is a necessary condition.
In one way, then, it is a very weak explication, but even so, it is already too strong for
the conceptual map we are in process of developing. CC-relevance is a dyadic relation
on statements. What the abducer wants at the end of the process is a statement that
he is justified in categorializing, i.e., his abductive target { . Then the CC-relevance of
the relevant possibilities in | requires that for each Se| , S and { not fail to share a
sentence letter.
Accordingly since the virtual rules of some of the basic relevance logics seek to
unpack the only partially analyzed concept of CC-relevance, and since CC-relevance is
not the concept of relevance instantiated in the construction of | , then it is implausible
to think that, if the | -forming exercise is transacted down below, this is achieved via
the virtual rules of such a logic.
Readers may agree with us that FUH-relevance is the more fully realized notion.
But here, too, it is not obvious, to say the least, that this is the concept instantiated by
the relevance of | is a set of possible hypotheses for { . It is not a proof of { (not in
general anyhow), hence is not a proof which deploys all its members as steps.
We now come back to the promised point about complexity. The logic of generation
presumes that possible hypotheses are initially generated by an operation that cuts
} . The
complexity results in relevant logic establish that the decision problem is either
unsolvable at all or is unsolvable practically (i.e., is intractable). One way of characterizing the decision problem is to say that it is the task of determining mechanically
finitely and infallibly with respect to any pair of wffs whether the one relevantly implies the other. The problem’s intractability guarantees that no practical agent has the
resources to solve it.
This would be a serious setback were it the case that the only way in which an
11.2. ALTERNATIVE ACCOUNTS
225
abducer could form ~ from  would be by solving the decision problem. In fact,
however, requiring him to do so would be asking too much. It is well to note that
an essential objective of relevant logics is the characterization of relevant implication.
Relevant implication is grist for the abducer’s mill. It leads very naturally to the idea
that ~ is populated by those sentences S that relevantly imply € , where relevance is
content containment-relevance. Even where relevance is the full use of hypothesesvariety, it too gives rise to the intuitive idea that 2‚ ~ if it is a line in any relevant proof
of € .
There are two questions to sort out. One is, what are the actual protocols that
human abducers execute in forming ~ ? The other is, how is ~ specified by a relevant
logic, independently of any assumption that the virtual rules are actually deployed by
real-life abducers? The answers we are presently entertaining are possible answers to
this second question; it is less certain that they are answers to the first. It is entirely
possible that for wide ranges of case, ~ is formed automatically and subconsciously
in part, and deliberatively in part. In such cases, it is reasonable to think that the parts
of ~ might have different etiologies. If, as we suggested, unconscious agency can be
seen as having a PDP architecture, then parts of ~ could be fixed in ways that tolerate
considerable levels of complexity, short of running the solution, if there is one, to the
decision problem. The other part of ~ might still be derived from implications from
 to € that the conscious abducer is able to make. In so doing, it is possible that he
is deploying something like (some of) the rules of relevant implication without being
aware of it.
How, then, is complexity a problem for relevant logic once they are shot of the
ludicrous assumption that the solution of their intractable decision problem by actual
agents in real time is a condition of their logical adequacy? The answer is that such
systems are complex rather than simple, and that their complexity is unnecessary. It
has nothing to do with the fact the actual reasoners cannot be organic realizations of
the decision solution. A theory is unnecessarily complex if a less complex alternative exists which delivers what the former does as well or well enough as to make the
marginal improvements in delivery too costly for the benefit it achieves. The complexity and simplicity controversy is important in logic and science alike. But it isn’t the
question we are interested in here, except to say that whether a system is unnecessarily complex pivots on what the system aims to do and what its alternatives are. We
said in chapter 2, that one of our tasks was to make progress with what we called the
Complexity Problem. We are now positioned to redeem part of that pledge. Even if
we were to grant that relevant logic is unnecessarily complex as an account, say, of the
entailment-relation, it wouldn’t follow that none of its routines could be run locally in
the context of a generation-sublogic that is implementable by a practical agent. Further, if some such routines were practically implementable, it would not follow that the
implementing agent has the means (or the interest) in running them globally, i.e., that
in ways that sort out every arbitrarily selected case in just the right way.
The intractability displayed by decidable relevant logics is a problem only on the
assumption that those logics are models of what human beings actually do when they
think and, hence, that those agents actually run (or ought to run) the decision-solution.
But this is no discouragement of the selective deployment of the various implication
rules to solve discrete or local problems. It also bears on the present issue that relevant
CHAPTER 11. RELEVANCE
226
logics give principled accounts of relevant implication and relevant proof. Both answer
a very basic idea about how hypothesis are generated. It is an idea that we are prepared
to propose
The Fundamental Conjecture (FC) of the Logic of Hypotheses-Generation:
In carving out ƒ from „ , an abducer is disposed to select sentences that do
or might be considered to generate … .
Here the generation-relation is loosely conceived so as to cover any broadly implicational relation from an †2‡ „ to … , whether strict, relevant, causal, explanatory, or whatever else. It is a modest partial response to the demands of the Consequence Problem
discussed in chapter ????.
Thus the concept of generation enters our story in a second way. The principal
task of the present sublogic of discovery is to give an account of how hypotheses are
generated. According to FC this is done by way of a generation-relation between
members of „ and … .
The second major assumption of the present sublogic is
The Relevance Condition (RC): In carving out ƒ from „ , an abducer seeks
for generation relations on … that, as it happens, satisfy conditions that
make it plausible that ƒ count as a set of relevant possibilities from „ .
We note in passing that RC doesn’t require that the abducing agent even raise the question of the relevance of the members of ƒ . Whether the generation-relation contributes
to the relevance of the members of ƒ is something for the logic to take note of, irrespective of whether that is also something in the agent’s mind. The same is true of the
generation relation of FC. The logic proposes that an ƒ is formed for an agent with
respect to a target … , when its members bear the generation-relation to … and the agent
is disposed to recognize that this is so (e.g., would answer affirmatively though not
necessarily intuitively if asked).
What makes relevant logic a candidate for absorption into the logic of hypothesisgeneration is that in both its CC- and FUH- versions, it instantiates our two fundamental assumptions, FC and RC. Thus in the requisite respects, they are the right kind of
logics for this aspect of abduction, never mind that very clearly their implication and
proof relations aren’t general enough, that the concepts of relevance they seek to honour are too restrictive, and that in various respects they run into internal difficulties.
Even so, an admonition should be sounded. Relevant implication on H and … is too
strong a condition. For whatever is in ƒ , and thence in ˆ , it is necessary for its contribution to the abduction problem in hand that it not imply … directly. What abduction
problems require in the general case are Hs which together with some antecedently
available database or background knowledge generate the target in question.
11.2.2
Probabilistic Relevance
Intuitively speaking, there is something to be said for the idea a sentence, S is relevant
to a sentence S‰ when S raises or lowers the probability of S‰ . On this view, relevance
is a matter of conditional probability.
11.2. ALTERNATIVE ACCOUNTS
227
Cond: S is relevant to SŠ iff Pr(SŠ /S) Œ ‹ 0.5.1
Cond evokes the Indifference Principle of the classical construal of probability.
How good a definition is Cond? Consider the three mutually exclusive propositions
This apple is red.
This apple is green.
This apple is yellow.
But the restricted disjunction axiom of the probability calculus in which S, SŠ and SŠ Š
are mutually exclusive that
(1)
\Ž  ‘2Š ‘2Š Š ’ “”\Œ ‹ [Ž  ’ “”•'\Ž  2Š ’ “”•'\Ž  2Š Š ’ “”
Let Q be “This apple costs 25 cents.” Then putting our trio of sentences as S, SŠ , and
SŠ Š , the probability of each on Q is 0.5. But from (1), Pr(S‘ SŠ ‘ SŠ Š /Q) sums to 1.5,
which is impossible.
The objection presupposes that the cost of an apple is irrelevant to its colour. The
same presupposition is at work in a second objection, also due to [Keynes, 1971, p.
47]. By Cond we have it that Pr(x is red/x costs 25 cents) = 0.5, and also that Pr(X is
red – x is an apple/x costs 25 cents) = 0.5. It is provable in the calculus of probability
that if Pr(SŠ /S) = Pr (SŠ – SŠ Š /S), then S deductively implies that SŠ—˜2Š Š . But given the
present interpretation, this shows that “x is red” entails “x is an apple”, which is untrue.
[Bowles, 1990] is an attempt to rescue Cond from these embarrassments.
I suggest that we modify [Cond] by adding a restriction ™ ™ ™ : when we
say something like ‘p’ is relevant or irrelevant to ‘Q’ if and only if the
probability of ‘Q’ is some value, n, conditional on ‘p’, our determination
of ‘n’ is based on a consideration of ‘p’ and ‘Q’ alone [Bowles, 1990, p.
69].
Bowles’ restriction avoids the first objection in the following way. The conditional
probability given that “This apple costs 25 cents” is 0.5 for each of “This apple is red”,
“This apple is green” and “This apple is yellow”. If we reckon the probability on the
same condition of their disjunction, we run foul of the restriction the calculation in
question takes into account not just the disjunction and the condition, but also each of
the three disjuncts. The second counterexample is bundled in the same sort of way.
Bowles’ restriction raises a number of points, only two of which we shall take up
here. Consider again the example of the apples. Intuitively the molecular proposition
“This apple is red or this apple is green” is irrelevant to the proposition that the apple
costs 25 cents. A probability relevantist will understand this untuitive irrelevance as a
matter of not mattering for likelihood. But that fact that the disjunction is irrelevant to
the proposition about cost turns on semantic relations affecting the components of the
1 This
(a)
(b)
resembles what Peter Gärdenfors calls the “traditional” (and redundant) definition
›œ  ž š Ÿ ž[ \¡2¢¤ £ ›œ  ž š Ÿ ¡2¢ .
S is irrelevant to Sš on E iff ›œ  ž š Ÿ ž\ [¡2¢¤ £ ›œ  ž Ÿ ¡2¢ .
š
S is relevant to S on evidence E iff
Cited in [Schlesinger, 1986, p. 58].
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CHAPTER 11. RELEVANCE
alteration. The irrelevance of the apple’s price to the disjunctions must pertain to its
irrelevance to the disjunction. Relevance therefore is at least to some degree responsive
to some closure conditions on some logical operations. This presents Bowles with
Hobson’s Choice. If he rigs Cond so as to ignore these elementary facts of closure,
the ensuing account leaves relevance significantly undetermined. If he removes the
chokehold on closure by removing the restriction on Cond, he is back in the embrace
of the Keynesian absurdities.
A further feature of Bowles’ approach to relevance is that having jettisoned the
classical notion of conditional probability in favour of his restricted version, he leaves
the ensuing conditional probability undefined.
The apparatus of conditional probability is involved by probability relevantists to
elucidate the idea that relevance is a matter of raising or lowering likelihood. It doesn’t
work for the classical notion of conditional probability. It saddled the analysis of relevance with absurdities; so, in that sense, it overdetermined the target concept. Bowles’
is drawn to the same basic idea, but without the Keynesian excesses. These he averts by
constraining conditional probability. But the constraint is too aggressive. It leaves the
concepts of relevance and irrelevance underdetermined in quite counterintuitive ways.
A third difficulty is that, in Bowles’ approach, every statement is relevant to every
valid sentence. The guiding intuition here is that the relevance of a proposition consists in its mattering for the likelihood of any proposition to which it is relevant. This
is a further respect in which the restricted conditional rule overdetermines the target
concept that it is intended to elucidate.
Doubtless there are other ways of deploying the concept of probability in the explication of relevance that avoid the internal difficulties that crop up in the classical
treatment of probability and in variations of it such as that of Bowles. We shall not
press the question of the internal adequacy of such accounts, but rather will turn briefly
to the other two criteria that have a bearing here. One is the conceptual adequacy of a
probabilistic explication of relevance. The other is whether the Cut Down question is
adequately answered by the manipulation of those very probabilities that constitute the
conceptual explication of relevance.
The issue of conceptual adequacy requires us to consider whether the relevant possibilities that make their way into an abducer’s candidate space are not carved out by
the condition that any proposition whatever meets provided only that it changes the
probability of ¥ , the abducer’s target. It is also necessary to reflect on whether the idea
of mattering for probability serves to guide the explication of relevance in general, not
just in its application to the determination of candidate spaces ¦ .
For probabilistic relevance to have any chance of collecting an affirmative answer
to the first of this pair of sub-questions, it is necessary to replace the basic intuition of
mattering for probability with that of mattering for it positively. It is of no use to the
abducer to have this ¦ stacked with propositions that make his target ¥ less probable
than it would be independently of those propositions. So the question is whether it is
reasonable to think that these spaces are constituted by all propositions that increase
the probability of ¥ .
It won’t do. There are wide ranges of cases in which the winning H for a ¥ is a
significant depressor of its prior probability. A case in point is the caloric theory of
heat which provides an attractive explanation of the fluid characteristics of heat flow.
11.2. ALTERNATIVE ACCOUNTS
229
But the calorie does nothing whatever to increase the probability of any § representing
the fluidity of heat. Because this seems also to answer, negatively, the question of the
adequacy of increasing probability as the explication of a wholly general concept of
relevance, we shall not press the point further.
This leaves the issue of how ¨ s are actually constituted. Here, too, it is necessary
to consider two possibilities. One is that ¨ s are constituted automatically and without
regard for any effort the abducer to stock his ¨ by conscious inference or calculation.
The other is that an abducer’s ¨ is at least partly a matter of relevant possibilities that
he seeks in thinking up as such.
If the response to the first possibility is affirmative, then for any abducer and for
any target, there is a set of relevant possibilities H © , ..., Hª for that target which is
the extension of the relation of probability increase, with the H« as domain and § the
counterdomain. Given the defectiveness of classical and Bowlesian conditionalization,
it might be supposed that a confident answer awaits a firm specification of the requisite conditionalization. It doesn’t, never mind that not having it is a liability for the
probabilistic relevantist. Even when conditionalization is left as an intuitive primitive,
we see in what was lately discussed, it will deny membership in ¨ to possibilities that
should be included, and will include propositions that should be rejected.
Finally, it is reasonable to suppose that an abducer stocks his ¨ s by conscious
application of whatever probability rules suffice for increasing the probability of a §
in face of a H. It would seem not, not only for reasons we have already canvassed but
also for reasons of complexity.
The last time we considered the Complexity Problem (section ??? above), we distinguished between the mathematical property of intractability and the methodological
property of lack of simplicity. In the case of the going systems of relevant logic we
pointed out that the decision problem, where there is one, is intractable and that, in
comparison with classical systems, relevant logics are methodologically complex. But
these facts did not preclude the correct and useful use of various particular rules of
relevant implication or relevant proof in local contexts.
In the instance of probabilistic relevance things are different. Whatever its precise
details it fails to the conditionalization rule to deliver the good for probabilistic relevance, and whatever its details, if it is a credible conditionalization rule, it will prove
locally intractable, as Harman and others have noted. Let any proposed H imply just
ten statements conjoined. Then to be able to put H in ¨ an abducer will need to make
nearly a thousand probability calculations; and if H involves thirty statements, the abducer’s computations must run to a billion [Harman, 1986, Ch. 2].
11.2.3 Dialectical Relevance
Hintikka and others have emphasized that enquiry is a mode of interrogation, not only
of research databases but also, in the case of experimental investigations, of Nature
[Hintikka, 1979]. Hintikka is also of the view that logic itself is the investigation of
interrogative structures [Hintikka et al., 2001]. Implicit in these suggestions is the possibility that the relevance-relation might fruitfully be given a dialectical explication.
We use the word “dialectic” in one of its dominant senses among present-day argumentation theorists, as broadly interchangeable with “dialogical”, as indicated by this
CHAPTER 11. RELEVANCE
230
remark by Douglas Walton [2000].
The kind of relevance defined
[in this approach] can be called dialectical relevance, meaning that an
argument, a question, or other type of speech act, is judged to be relevant
insofar as it plays a part or has a function in goal-directed conversations,
that is, a dialogue exchange between two participants who are aware of
each other’s moves in the exchange.
Walton goes on to say that “the ultimate aim of the theory is to be useful in judging
cases of material relevance” [Walton, 2000, p. 197]. A sufficient condition on the
material relevance of an argument (or other speech act), is that
it bears directly or strongly on the issue [in question] so that it is worth
prolonged or detailed consideration in relation to the specific problem or
issue that the conversation is supposed to resolve [Walton, 2000, 142].
What is it to be dialectically relevant?
[A]n argument, or other move in a dialogue, is dialectically relevant (in
that type of dialogue) if and only if it can be chained forward using either
the method of profile reconstruction or argument extrapolation (whichever
is more applicable to the given case) so that the line of argumentation
reaches one or other of the pair of propositions that defines the issue of the
dialogue [Walton, 2000, 219].
Walton derives his notion of forward chaining from AI. Let K be a class of statements and C a statement. There is a forward chaining from K to C if C is validly
deducible from K. Backwards chaining, on the other hand, is as we have described it
in chapter 1. Suppose that C is not deducible from K and that for a certain statement
S, ¬ S ­ ® K does imply C. Then one forward chains to S if one accepts S on the basis of
the contribution it makes to this successful deduction of C.
It is not necessary, in order to take the measure of Walton’s dialectical relevance, to
expound what its author means by argument-extrapolation and profile reconstruction. It
is enough to say that Walton Understands argument-extrapolation to be a generalization
of procedures for the solution of enthymeme-problems; and by profile reconstruction
he understands a generalization of rules for conversational coherence as developed,
for example, by Jacobs and Jackson and others [Jacobs and Jackson, 1983; Schegloff,
1988; Levinson, 1981; Walton and Krabbe, 1995]. (We note in passing, but will not
otherwise develop the point, that coherence embodies relevance considerations of the
kind that Walton is trying to capture in the definition at hand.)
The basic idea, then, is that a move M in a dialogue D with respect to a target ¯ is
dialectically relevant if either
1. There is an extrapolation that produces a set of moves S such that
¬ M­ ® S forward chains to ¯ ; or
2. There is a coherent conversation C arising from M such that ¬ M ­
forward chains to ¯ .
®
C
11.2. ALTERNATIVE ACCOUNTS
231
Walton leaves the structure of forward chaining underspecified in ways that weaken his
project. If deduction is monotonic in forward chaining, then every move is relevant in
any dialogue whose target is any ° . For every ° there is an S that implies it; hence for
every ° there is an arbitrary move M such that ± M ² ³ S deductively implies ° . On the
other hand, we may have that forward chaining-deducibility is non-monotonic. Even
so, the definition of dialectical relevance is satisfied if for any move M in D there is an S
such that ± M ² ³ S ´µ° . In particular, let S be the propositional content of the negation of
move M, where M is the move of proposing a proposition P. The ± M ² ³ S ´µ° for any M
whatever. Of course, what’s wanted is that the extrapolation to S involve considerations
that are relevant to ° or to the question of how to get to ° .
As Walton says, the ultimate aim of the theory of dialectical relevance is to determine the presence or absence of material relevance, which in turn is present when
it bears “directly” or “strongly” on ° in ways that make it worth prolonged or detailed consideration. But, as we currently have it, the definition of dialectical relevance
overdetermines its concept of relevance in ways lately noted. In that state dialectical relevance can hardly be expected to play its adjudicational function in relation to
material relevance.
Perhaps these are internal difficulties that could be straightened out with a little
effort. Desirable though such an outcome would be, it would not be decisive for our
purposes, concerning which we must re-visit the questions of the conceptual adequacy
of dialectical relevance to the demands of ¶ , as well as the real-life implementability
of its rules.
The theory of dialectical relevance has some attractive conceptual features. It sees
relevance as contextualized by the factor of agent-goals. The main idea comports well
with our conception of ¶ . A sentence is in ¶ if it makes some contribution to the
generation of ° . Also welcome is the dynamic structure of Walton’s analysis. Something is relevant if it helps produce a contextually indicated desire for change. Thus it
answers loosely to our generation-generation model of the constitution of ¶ . ¶ is the
set of possibilities generated for possible consideration by the abducer, and in doing so
we postulate that the abducer is disposed (if not consciously so) to select those propositions that do, or might appear to him to bear a basic generation-relation from each
member of ¶ to ° .
Even so, Walton’s approach is over-narrow for our purposes especially in two particular respects. Taken his way, relevance is an intrinsically dialogical notion, whereas
(unless we embrace the fiction that all thought is a conversation, if only with oneself or with Nature) no such condition is intrinsic to abduction. Relatedly, Walton’s
generation-relation is deduction, whereas our proposal is for something much more
general.
It might strike the reader that in plumping for a forward chaining condition, Walton
has got things exactly backward for abduction. This would be true if Walton were trying
to elucidate the retroductive or quasi-retroductive component of abduction by way of
relations that forward-chain. Of course, relevance is Walton’s target, not abductions.
For him, like us, the relevance of a sentence is a matter of how it forward-chains.
By Walton’s lights, the forward-chaining is deductive; by ours, it is something more
general.
Finally, is it plausible to suppose that ¶ is constituted by conformity to the rules
CHAPTER 11. RELEVANCE
232
of dialectical relevance? Once again, the question subdivides into two. If · is specified simply as the counter-domain of the forward-chaining relation that Walton makes
intrinsic to relevance, or is · constituted by actions taken by the abducer, whether
unconsciously, consciously, or a mixture of the two.
Given the misgivings already noted, the answer to all these questions must be in
the negative. Apart from this, there is also again the question of complexity. There is
no indication that Walton thinks that his system is decidable; almost certainly he thinks
it is not. Even so, were there a bona fide decision-problem for dialectical relevance,
it would be too much for any real-life individual to run the requisite algorithm. On
the other hand, if the question is whether the account of dialectical relevance is simple
enough, our answer would be that it is too simple, by virtue of the deductivism of
forward chaining.
In a somewhat different approach to Walton’s, [Krabbe, 1992] develops an account
of relevance criticism in persuasion-dialogues. In some ways, as we shall shortly see,
Krabbe’s account is of greater service to our interest in · than other dialectical approaches. [TBA]
11.2.4
Contextual Effects
Arguably the most cited work on relevance written by non-logicians is Relevance, by
the linguists Dan Sperber and Deirdre Wilson. Like Walton’s account of dialectical
relevance, these authors give contextuality a central place in their account.
Sperber and Wilson define relevance for ordered pairs, ¸ ¹%º »¼ where ¹ is an assumption or belief and » a context, itself a conjunction of beliefs.2 The principal claim
of their account of relevance is:
Relevance. A belief is relevant in a context if and only if it has some
contextual effect in that context.3
An assumption or belief has a contextual effect in a context when it strengthens or
reinforces a belief contained in that context, when it contradicts a belief contained in
that context and thus forces an “erasure”, or when it licenses implications. Contextual
effects can in each case be likened to changes of mind. Degrees of confidence are
raised or lowered, beliefs are contradicted and erased, or new beliefs are derived.
Contextual implication is defined in the following way. Where ¹ is a belief and »
a context and ½ a further belief, then:
Contextual implication. ¹ contextually implies ½ in context » iff (i)
¾ ¹º »¿ non-trivially implies ½ ; (ii) ¹ does not non-trivially imply ½ ; and
(iii) » does not non-trivially imply ½ .
The central idea is that, upon placing new information ¹ into a given inventory
of beliefs » , an implication is sanctioned of some further assumption ½ , where ½
couldn’t be got either from » alone or from À ¹jÁ alone.
2 “Assumption” is their preferred term, but it is clear that it is sufficiently interchangeable with “belief”
for our purposes here. The same holds for their “thesis” which sometimes does duty for “assumption”.
3 [Sperber and Wilson, 1995, p. 122].
11.2. ALTERNATIVE ACCOUNTS
233
It is necessary to say something about the idea of non-trivial implication, which
drives the definition of contextual implication.
Non-trivial implication.  logically and non-trivially implies à iff when
 is the set of initial theses in a derivation involving only elimination rules,
à belongs to the set of final theses [Sperber and Wilson, 1995, p. 97].
For their part, “[e]limination rules ... are genuinely interpretative: the output assumptions explicate or analyse the content of the input assumptions” [Sperber and
Wilson, 1995, p. 97]. Further, “[o]ur hypothesis is that the human deductive device
has access only to elimination rules, and yields only non-trivial conclusions ...” [Sperber and Wilson, 1995, p. 97].
Elimination rules contrast with introduction rules. Consider the rule of Ä -introduction,
[A]
1.
2.
P
ÅeÂÆÄÃ
1, Ä -introduction
This can be compared with the Ç -elimination,
[B]
1.
2.
Â'ÇÃ
ÅeÂ
1, Ç -elimination.
It becomes clear at once that the difference between the sort of rule reflected in [A]
and the sort of rule reflected in [B] lays unconvincing claim on the distinction between
triviality and non-triviality. This might lead us to think that the account of contextual
implication is ruined. But this would be over-hasty, and in fact quite wrong — the
upgrading of a semantical quibble to a substantival complaint. For it is open to Sperber
and Wilson to suppress all reference to triviality and non-triviality and to re-write the
definition of contextual implication as follows.
Revised contextual implication.  contextually implies à in context È iff
(i) É ÂÊ ÈË implies à ; (ii)  does not imply à ; (iii) È does not imply à ;
and (iv) the implications cited in (i) to (iii) employ elimination rules only,
that is, they are implications whose consequents (purport to) explicate the
content of their antecedents.4
Contextual implications also involve synthetic implications [Sperber and Wilson,
1995, p. 109]. A synthetic implication is one using at least one synthetic rules of
derivation,[Sperber and Wilson, 1995, p. 104] and a synthetic rule takes not one but
“two separate assumptions as input”. So, for example, the rule of Ç -elimination “which
takes a single conjoined assumption as input, is an analytic rule, and modus ponendo
ponens, which takes a conditional assumption and its antecedent as input, is a synthetic
rule” [Sperber and Wilson, 1995, p. 97]. There are difficulties with this approach
perhaps the most damaging of which is the following.
Let à be any proposition that is a contextually independent member of such a
context È . That is, we put it that no proper subset of È contradicts à or non-trivially
implies it. Since ÃÍÌÈ , we have it that C Î'Ã . But È just is the union of È'ÏeÐ ÃÑ and
Ò
4 In particular, condition (ii), that
does not imply
tion rule sanctioning the inference of from .
Ó
Ò
Ó
comes to this: that there is no sequence of elimina-
CHAPTER 11. RELEVANCE
234
Ô ÕÖ .
Õ
Õ
×ÍØ , by the contextual
Now cannot
by nontrivial
Õ inbe× ;got
Õ be got implication
Õ alonefrom
independence
of
nor
can
from
by
nontrivial
implication. But
×ÚÙ Õ ; so Õ is contextually implied by ×ÛØ'Ü ÕjÝ in × . CHECK DEF ON P. 200 THE
EXAMPLE APPEARS TO CONTRADICT IT. S EE COND . #3.
Let × be any set of assumptions which, intuitively speaking, are pair-wise irrelevant. × might be a set such as Ü “2+3=5”,
dog is in the manger”, “Venus is a
Ý , in which“The
Þ
Þ
Þ
planet”, “Mr. Blair is British”,
,
all
member-sentences
are atomic. Pick
Õ
any member of × and call it . If we putÕ it further that all members of × are pairwise
contextually
in × . Even so,
logically
inert, then it is immediateÕj
that
Ý in × is. ×Æ
ÕjÝ is theindependent
Õ is contextually
implied
by
Æ
×
e
Ø
Ü
g
Ø
Ü
maximal
proper
subset of
× with respect to Õ . For every distinct member of × , there is a distinct such maximal
proper subset. A maximal
a largest subset of a set of mutually irreleÕjsubset
Ý is theis largest
Õ aside,proper
ß
×
e
Ø
Ü
vant assumptions.
set of irrelevancies in × Õj
compatible
ÕjÝ being a proper subset of Õ . What is more,
Ý , Õ itself
with ×Ø^Ü
for
any
such
×
^
Ø
Ü
j
Õ
Ý
is intuitively irrelevant to every member of ×àØ'Ü
which nevertheless is relevant to
it. Every such context is a set of contextually independent assumptions. But there is no
set of contextually independent assumptions whose every member is such that there is
a maximal subcontext relevant to it in that context. So, again, if
× = Ü “2+3=5”,
Ý “The dog is in the manger”, “Venus is a planet”, “Mr. Blair
is
English”
Õ = “Venus is a planet”
then ÕjÝ
×ØjÜ = Ü “2+3=5”, “The dog is in the manger”, “Mr. Blair is English” Ý .
ÕjÝ
Nothing in ×ÛØ'Ü
is relevant
THIS BE “ CONTEXTUALLY IMPLIED ”?
Õ . Yet ’×ßT Ø'
ÕjÝ , norStoHOULDN
to anything else in ×ßØ'Ü
Ü ÕjÝ is relevant to Õ in × .
We saw in our discussion of dialectical relevance that logical consequence is not the
preferred tool for carving out relevance. Even implication constrained by a syntheticity
constraint won’t evade the excessiveness produced by ex falso quodlibet; and further
constraints in the manner of relevant and paraconsistent logic are something we are
opposed to on general methodological grounds. (We prefer to use labels which are
more natural. We might here add that such constraints without labels complicate the
accommodating logic momentously.) What the present difficulty suggest is that just as
the structure of inference is too fine-grained to be caught by the implication relation,
no matter how informally quantified, so too, is relevance too fine-grained to be caught
by the likes of implication.
Õ
Let us now relax the constraint of contextual Õ independence. Put it that is contextually implied by some proper subset of × and is consistent in × . We have it then
for every proposition consistently a member of any context that that proposition is contextually implied in × . Since this suffices for relevance, every proposition is relevant
in any consistent context that contains it. But this is an excessive result.
The Sperber-Wilson account shares some of the virtues of a dialectical approach
to relevance, without having to encumber itself with the latter’s over-narrow dialogical
contraints. On both approaches relevance is tied to belief-revision and update processes, concerning which there is a considerable independent literature, which in turn,
is of obvious application to the structure of abduction. The central idea, therefore, is
11.2. ALTERNATIVE ACCOUNTS
235
that something is relevant for an agent when it charges his mind in certain ways. This,
too, is the dominant intuition that the theory of agenda relevance tries to elucidate and
make something of. Yet, as with dialectical relevance, the unpacking of the figure of
change in view is too narrow, especially as regards changes driven by the relation of
contextual implication. It is also, unwittingly, too liberal an account, as we have just
seen. Thus, for much the same reasons advanced in our discussion of Walton, the
Sperber-Wilson approach to relevance does not do especially well either on the score
of conceptual adequacy or the rules for the constitution of á .
11.2.5 Topical Relevance
One of the conceptions of relevance which Anderson and Belnap attempt to exploit
is that of content-containment. Content-containment is a case of topical relevance.
Intuitively, sentences are relevant to one another when they are about some same thing.
We saw that there were difficulties with CC-relevance. But since there is more to
topical relevance than variable sharing, we shouldn’t be over-ready to give short shrift
to topical relevance. Whatever we will say about á in the end, it does seem that any
successful H for a â will require that in some sense H and â share a subject.
In this subsection our stalking horse again is Walton in his earlier work Topical
Relevance in Argumentation. Walton’s chief analytical tool is an adaptation of the
consequence relation of related logics in the manner of Epstein [1979]. A proposition
S is said to imply a proposition Sã just in case S classically implies Sã , and S and Sã
share a topic. Topical relevance in turn is shared subject-matter. Thus,
(TR): ä is topically relevant to Sã if
matter in common.
ä
and Sã have at least one subject
Though TR gives only a sufficient condition, we see no reason not to strengthen it.
Thus
(TR*):
matter.
ä
is topically relevant to Sã iff
ä
and Sã share a topic or subject
Relatedness logic imposes a relevance condition upon certain of the classical operations. It introduces a propositional relation, r, definable in the first instance over
atomic sentences but easily generalizable to molecular ones as well. The idea of a
subject matter is handled set theoretically. Let T be a set of topics—roughly all of the
things that the totality of the sentences of our given language å are about. The idea of
T is not far off the idea of a non-empty universe of discourse for å or å ’s domain of
interpretation. Now let P be the subject matter of of æ and Q of ç . Both P and Q are
é ë , that is, just in case
subsets of T. P and Q share a subject matter just in case P è Q êß
there exists a non-empty intersection of P and Q. Equivalently, ì í î æï ç ð ñ holds just in
case P è Q êß
é ë .5
The topical account provides a rather coarse-grained treatment of relevance. Like
the relevance logics of Pittsburgh and beyond, relevance is offered as a constraint on
5 See
for example, [Epstein, 1979] and [Walton, 1979].
CHAPTER 11. RELEVANCE
236
implication. Relevance is needed to filter out impurities that afflict classical implication. It is quite true that Walton also puts topical relevance to other uses. He proposes
that topical relevance will assist in the construction of expert systems devoted principally to classification. But topical relevance is too crude to serve the interests of
propositional relevance. For example, given that Sue, Bill and Peter are members of
the set of humans then we will have it that
(*) “Sue hit Bill” is relevant to “Peter plays the cello”.
Similar cases abound.
It is perhaps unsurprising that the theory of topical relevance should lapse into such
promiscuity. It employs a set theoretic apparatus which is too crude, too undiscriminating for relevance. The appropriation of set theory into relevance theory makes no
more strategic sense than the attempted reconstruction of the part-whole relation set
theoretically.
Even so, it is, we think, fair to see Walton’s account as standing on the continuum a
fair distance from commonness. Here is a notion of relevance, Walton is saying, that we
can work up in a relatedness logic. The relevance that relatedness logic contrives will
be helpful for the specification of expert systems in which documentary classification
is a principal task.
This gives a certain shape to our interest in promiscuity. Whether it is possible to
judge “Sue hit Bill” as relevant to “Peter plays the Cello” turns out to turn on whether it
is promiscuous to judge that the two sentences share a subject matter. This judgement
can’t be made independently of knowing whether a classification program for an expert
system is abetted by having it so. We might discover that the cataloguing devices of
an expert system need to be contrived so as to recognize a common subject matter
here. If so, the charge of promiscuity would be blunted. It would succeed luxuriantly
had Topical Relevance been oriented towards a commoner analysis of relevance. But
Walton says that it is not so, and we believe him. So we will say what, in effect, Walton
himself says. Topical relevance will not do as a general analysis of relevance.
Deep in the heart of topical relevance is the idea of “aboutness”. Aboutness is
contextually sensitive.6 Whether “The Pope has two wives” and “There are just two
states in the U.S.A.” are about some same thing, for example, about the number two
is fixed only by context. It turns out that in the theory of topical relevance they are
relevant context be hanged. (See [Iseminger, 1986, p. 7].) This is embarrassing on its
face. It lumbers us, as Gary Iseminger points out, with the true relatedness conditional,
“If the Pope has two wives then there are just two states in the U.S.A.” Its constituent
sentences both false, the theory declares them to be topically relevant.
Topicality in AI
...The depiction of one’s knowledge as an immense set of individually
stored “sentences” raises a severe problem concerning the relevant retrieval or application of...internal representations. How is it one is able
6 Concerning his own probabilistic definition, Schlesinger allows that it may be necessary to contextualize
it by talking “about the relevance of to on evidence in the context of q1 and so on” ([Schlesinger, 1986,
p. 65]; emphasis in the original).
ò ó
ô
11.2. ALTERNATIVE ACCOUNTS
237
to retrieve, from the millions of sentences stored, exactly the handful that
is relevant to one’s predictive or explanatory problem, and how is it one
is generally able to do this in a few tenths of a second? This is known as
the “frame problem” in AI [sic], and it arises because, from the point of
view of fast and relevant retrieval, a long list of sentences is an appallingly
inefficient way to store information [Churchland, 1989, pp. 155–156 ].7
In fact, however, what philosophers call the frame problem, AI theorists call the relevance problem. As characterized by AI researchers, the frame problem is that of
knowing how in detail to adjust the knowledge base to accommodate the flow through
of new information.
A computer is a universal symbol system, also known as a general-purpose storedprogram computer. Any such system is subject to three classes of operation: general
operations, such as identify by which the system’s symbols can be identified at any
given location; specific, such as the delete or erasure-operation; and control, such as
halt. Any general-purpose stored-program computer requires the storage of large quantities of factual information, or knowledge. This generates the knowledge-problem,
which encompasses three subproblems. One is the knowledge-organization problem.
It is the problem of deciding on the various patterns in which knowledge should be
stored. A second difficulty is the frame problem. It is the problem of determining precisely where in the system’s knowledge base to make updates or revisions when new
information is made to flow through the system. The third problem is the relevance
problem. It is the problem of determining what information in the knowledge base
would or might be of assistance in handling a given problem that has been presented
to the computer.8 Thus what philosophers call the frame problem, AI theorists call the
relevance problem
The knowledge problem is an extremely difficult one, and is taken by lots of AI
researchers as the fundamental problem of computer science. One way in which to
make the problem manageable is to constrain the knowledge base by a “microworlds”
assumption. This is common practice in computer diagnostics — MYCIN, for example
— in which the knowledge base is stocked with very little information, which is also
comparatively easy to organize (e.g., “symptom” and “disorder”) and retrieve. In such
highly restricted contexts, the relevance problem is often satisfactorily handled by way
of measures for the determination of topical relevance. In a wholly general way, this
means in its search for relevant information, for information that might help solve some
problem at hand, the computer would search for information that was “about” what the
problem was about.
For problems of any degree of real-life complexity, measures for topical relevance
don’t solve the relevance problem; topical relevance greatly underdetermines the rele7 Cf. [Haugeland, 1987, p. 204]: “...the so-called frame problem is how to ‘notice’ salient side effects
without having to eliminate all other possibilities that might conceivably have occurred...” And: “The frame
problem is superficially analogous to the knowledge-access problem. But the frame problem arises with
knowledge of the current situation, here and now—especially with keeping such knowledge up to date as
events happen and the situation evolves”.
8 Cf. Minsky: “The problem of selecting relevance from excessive variety is a key issue!... For each “fact”
one needs meta-facts about how it is to be used and when it should not be used” [Minsky, 1981, p. 124].
CHAPTER 11. RELEVANCE
238
vance required for specific problem solving.9 As computer scientists become increasingly aware of the necessity of realistic problem-solvers to have very large knowledge
bases. (Lenat and Guha [1990], and other works by Lenat’s CYC team), the central
problems of relevant search became increasingly ill-handled by way of topic-matching.
We propose to take this lesson to heart. Topical relevance is something that a theory of
relevance should attempt to elucidate, but doing so is a small part of a solution to the
general problem of relevance.
We note in passing the inadequacy of CYC’s effors to bring the more general notion
to heel,
It is well known that the performance of most inference mechanisms (especially those that operate on declaratively represented bodies of knowledge) degrades drastically as the size of the knowledge base . . . increases.
A solution of this problem needs to be found for the declarative paradigm
to yield usable systems. The main observation that suggests a solution is
that while the KB might have a large body of knowledge, only some small
portion of it is used for any given task. If this relevant portion can be a
priori identified, the size of the search space can be drastically reduced
thereby speeding up the inference ([Guha and Levy, 1990, p. 1], emphasis
added).
To this end, the à priori specification of relevant portions of the KB for any given
problem, the CYC team proposed specific and general axioms for relevance. Specific
axioms specify different sectors of the Knowledge Base “according to their relevance
to the problem-solving task at hand” [Blair et al., 1992, p. 15]. If the problem is one in
aircraft wing-design, the KB will be subject to an axiom that tells it that it is better to
look in the aeronautical engineering section rather than the biochemistry section. But
the relevance afforded by this axiom is little more than topical relevance at best.
Among the general axioms is the axiom of
Temporal proximity: It is necessary to consider only events that are temporally close to the time of the event or proposition at issue [Guha and Levy,
1990, p.7].
There are also general axioms of spatial and informational proximity [Guha and Levy,
1990, pp. 8–11], and a level of grain axiom. (In determining whether Harry is qualified
to be County Treasurer, it is unnecessary to consider the molecular make-up of his
thumb). It is easy to see, however, that the axioms often give the wrong guidance
and that, even when it is not wrong, it leaves the search space horrifically large. This
prompts a sobering thought from Jack Copeland:
Perhaps CYC will teach us that the relevance problem is intractable for
a KB organized in accordance with the data [i.e., symbol processing or
linguistic] model of knowledge [1993, p. 116].
[TBA]
9 This
is not to minimize the difficulty of producing fruitful measures for tracking topical or aboutnessrelevance. See, for example [Bruza et al., 2000].
11.3. AGENDA RELEVANCE AGAIN
11.3 Agenda Relevance Again
[TBA]
239
240
CHAPTER 11. RELEVANCE
Chapter 12
Limitation Issues
12.1 The Problem of Abduction
In section 3.2 we made brief allusion to the problem of abduction, by analogy with
Hume’s famous criticism of induction. In the present section we attempt to give this
issue its due attention.
We begin by examining briefly a controversy over the design argument for the existence of God, and, in particular, Hume’s celebrated criticism of it. What we wish to
point out is that if the design argument is taken sample-inductively, Hume’s derision
is justified; but if the argument is given an abductive interpretation, Hume’s criticisms
are unavailing. If we are right in so saying, we have a principled reason to favour a taxonomy in which induction and abduction are distinct species of ampliative reasoning;
for it is a distinction whose relata produce utterly different consequences for the design
argument.
The design argument has been advanced since antiquity by thinkers of the first rank,
although perhaps the best-known version of it is the argument of Paley [1822]. Hume’s
celebrated attack [Hume, 1947] preceded Paley’s book by twenty-six years. So Hume
was not attacking Paley. Even so, there is no reason to think that there is anything in
Paley’s actual argument which would have made Hume re-shape his criticisms. So we
shall use Paley’s version, ignoring the anachronism.
Hume takes the design argument to have the following schematic form
1. Watches are the result of intelligent design
2. Watches and organisms are similar to each other
3. Therefore, organisms are the result of intelligent design.
Hume sees the design apologist as holding that the argument at hand is inductively
strong and that its degree of strength is the same or approximately the same as the degree of similarity between watches and organisms. One of Hume’s criticisms is that the
degree of similarity between watches and organisms is extremely small. Accordingly,
the design argument is very weak at best.
241
CHAPTER 12. LIMITATION ISSUES
242
Hume marshalls a second criticism against the design argument. Although he does
not frame his objection in the terms that we shall use in formulating it, there is no
serious doubt that the formulation is wholly faithful to Hume’s intent. Hume’s point is
that the design argument commits a serious sampling error.
If we are to suppose that in the real world, in the world as we actually have it,
organisms are the product of design, we would have had to examine other worlds to
see whether the target correlation also obtained there. Since this is impossible, the
argument’s sample size is zero.
Hume has other objections to the design argument, which we won’t take time to
review here. Suffice it to say that all of Hume’s criticisms fail if the design argument is
interpreted, not inductively but abductively, as an inference to the best explanation. So
construed, the argument displays the following form.
1. The watch is complex and well-adapted for telling the time
2. The watch is the result of intelligent design
3. The watch is the result of random processes.
Paley holds (in effect) that the likelihood of (1) given (2) is substantially greater than
that of (1) given (3). And he concludes, therefore, that (2) is the more reasonable hypothesis. Likelihood is not probability. As here used, it is a measure of conjectural
strength. Paley applies the same argument to organisms. Which, he asks, is the hypothesis that better explains the fact that organisms, too, are complex and well-adapted
for survival and reproduction. Is it the hypothesis that organisms result from random
processes? Or is it the hypothesis that organisms are the product of intelligent design?
Paley’s answer is well-known. Given that the Darwinian hypothesis [1964], was not
available for another fifty-four years, it can be argued that Paley was right to chose
the design hypothesis over the sole alternative that was actually in contention. And, so
construed, Hume’s criticisms of the argument fail trivially.
This section has a three-part organization. We first present a justification of induction which, we conjecture, would have satisfied Hume had he thought to give to
abductive reasoning a place in the argument for justification. We then consider objections and replies. We conclude with some observations about skepticism.
12.1.1
Justifying Induction
If the argument of the past several paragraphs is sound, then it must be granted that
Hume’s rather spectacular failure to put paid to the design argument derives from a
blindspot about abduction. It is scarcely credible that Hume would have been able to
persist with his criticism of the design argument had he chanced to think of the abductive option. What is more, since Hume evidently missed the abductive boat in connection with the design argument, it would be interesting to consider whether he also
missed that same boat in other matters of critical moment. Did he, for example, display
this same blindspot in the reasoning that constitutes Hume’s Problem of Induction?
Hume wished to know whether there exists an epistemic justification of the reliance
we place on our standard inductive practices. Collectively this is the problem of determining whether induction is provably rational. Of course, Hume’s answer was in the
12.1. THE PROBLEM OF ABDUCTION
243
negative. He asks his readers to consider the set of conditions under which his question
could attract an affirmative answer.
I.
There exists a sound deductive argument for the proposition that our
inductive practices are rationally reliable.
II. There exists a tenable inductive argument that our inductive practices
are rationally reliable.
Hume’s attack on induction takes the form of denying that either of these conditions
is satisfied and that jointly they exhaust the relevant justificatory options. In what
must surely be one of the first things philosophy undergraduates come to learn about
Hume, he claims that condition II is unsatisfiable, owing to the intrinsic circularity of
an inductive argument that sought to establish the justification of, among other things,
inductive arguments.
The trouble with condition I is not as well-known or as easy to fathom. It is a
condition that sustains two readings. One is that no deductive argument could possibly
turn the trick for induction. The other is that there is no known deductive justification
of induction. It may be that Hume leans toward the first of these readings, and does
so on the strength an autoepistemic argument, itself a benign variant of the modern
conception of the argumentum ad ignorantiam.
1. If there were a sound non-question-begging defence of induction, we
would know of it by now.
2. But we don’t.
3. So there isn’t.
Counting against the present conjecture is that, although the autoepistemic argument
is deductively valid, and sound if premiss (1) is true, it can be argued with some plausibility that, in pleading (1), Hume would have committed the same offence that he
so emphatically asserts would be committed by any defence of induction that satisfies
condition II. For how else would premiss (1) of the autoepistemic argument be sustained, if not on some purportedly inductive principle that realities (of certain kinds)
have a disposition to reveal themselves in a timely fashion?
Our principal purpose here is not to assess Hume’s argument against the rationality
of induction (although we return to this issue briefly toward the end of this section),
but rather to raise the question whether Hume is right in supposing that there are just
two types of argument that could be marshalled in favour of induction (or of anything
else, for that matter); and whether should Hume’s assumption of exhaustiveness prove
incorrect, there might be some new hope for the justification of induction.
We say that in attacking the design argument, Hume took it to be an argument of
a certain type, and in so doing quite overlooked the possibility (indeed the fact) that
it is an argument of a different type, of a type known as abductive. This was Hume’s
blindspot towards the design argument. Is it possible that this same blindspot affected
(and spoiled), his treatment of the Problem of Induction?
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12.1.2
Justifying Abduction
There is reason to think so. If we grant the distinction between sample-induction and
abduction, we see that there is a third condition under which it may turn out that
the problem of induction is solved, where induction here is generic induction or that
form of ampliative reasoning exhausted by the distinction between its species, sampleinduction and abduction. 1 The third condition turns out to be a pair of conditions that
vary systematically depending on which species of (generic) induction is up for justificatory consideration.
Here is the defence of generic induction that Hume’s blindspot for abduction caused
him to miss. For ease of reference we shall call it “Hume’s Lost Chance for Induction”
or LCI, for short.
1. Induction is either sample-induction or abduction and nothing else.
2. Consider, first, sample-induction.
a. It is not in doubt that on the whole our sample-inductive practices to date have been reliable.
b. The reliability-to-date in those practices is best explained by
their general rationality.
c. Accordingly, sample-induction has a tenable abductive justification.
3. Consider now abduction.
a. It is uncontested that on the whole our abductive practices to
date have been reliable.
b. Their continued reliability can be inferred from this sample.
c. Accordingly, abduction has a tenable sample-inductive justification.
4. Thus, sample-induction has a tenable, non-circular abductive justification, and abduction has a tenable, non-circular sample-inductive
justification. Since this is all there is to generic induction, generic
abduction has a tenable, non-circular justification.
It takes little effort to see that the form in which we have constructed LCI omits some
important qualifications and nuances. So, for example, we haven’t made it explicit that
the proposed justification is not a strict proof, but is a proof in a looser non-deductive
sense, and that the claim of inductive justification is a claim whose instances are sustainable only defeasibly. Nor have we sought to make explicit the role in these matters
played by the so-called Principle of Uniformity. These omissions have not been made
out of carelessness, but rather in the belief that, for purposes of appraisal, enough of
the structure and detail of the defence of induction is present in our present schematic
version of it, LCI. Perhaps we are wrong in so thinking. After all, there are different
1 Perhaps sample-induction and abduction do not in fact exhaust the class of ampliative modes of reasoning. If not, the case that we are attempting to make here easily extends to this more capacious inductive
universe.
12.1. THE PROBLEM OF ABDUCTION
245
ways of composing samples and, correspondingly, different ways of producing inductions from them — projectively, by generalization, statistically, and so on. Similarly,
a defeasibly rational practice is one whose precepts will occasionally let us down. But
there are practices and practices, and correspondingly different thresholds of rationally
tolerable setbacks. In saying nothing further of these things, do we not run the risk that
an appraisal of the schematic structure of the present argument would be compromised
by consideration of these other matters? After all, are we not here claiming that Hume’s
own approach to induction was too coarse-grained and that the argument he mounted
against the rationality of induction would have been compromised, indeed overturned,
had he paid attention to a distinction that even a slightly more detailed approach to
induction would have made apparent?
Of course. But appraisal must start somewhere. We shall stay with the schematic
version of the argument because even in this less than fine-grained version, it attracts
objections. If those objections proved sound, the need for more detailed scrutiny of
nuances would have lapsed. What, then, are the objections?
12.1.3 Objections
Objection 1.
The argument assumes a “fact not in evidence” (as the lawyers say),
namely, that inductive practices of the one kind are justified by an induction of the other
kind.
But what hasn’t been shown is that the type of induction that establishes the intentionality of the other type is itself well-justified.
Reply 1. On the contrary, sample-induction is justified abductively, and abduction is
justified sample-inductively.
Objection 2.
Hume’s objection against an inductive justification of induction was,
quite rightly, that it was achieved by an argument that is circular in one step. In the
argument that (we say) Hume should have considered, the circularity is achieved in
two steps, even if not in one.
Reply 2.
If in the argument that (we say) Hume should have considered, there is no
argument circular in one-step for the justification of either kind of induction, then there
are embedded in that larger argument, LCI two non-circular and apparently tenable
justifications. How can an argument made up of non-circular subarguments be made
circular by them?
Objection 3.
That’s easy to answer. What LCI proposes is that sample-induction is
justified if abduction also is, and that abduction is justified if sample-induction likewise
is. So the schematic argument is a circular argument in the form,
1. If S1 then A and if A then SI
2. Hence S1 iff A.
Reply 3. That’s not circularity. It’s a proof of the justificatory equivalence of sampleinduction and abduction. (A rather nice bonus!)
Objection 4.
Be that as it may, it surely does not follow that if an argument has
no circular subarguments, it itself is free of circularity. Consider the analogy with the
consistency of sets. It may be that on a given partitioning of a given set S all proper
subsets are consistent and yet S itself is inconsistent.
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Reply 4. Yes, care needs to be taken not to rely wholly on the noncircularity of LCI’s
proper subarguments in claiming that LCI itself is circularity-free. However, if LCI is
circular, the onus is on its critics to demonstrate it. What has been shown so far is not
LCI’s circularity but, to repeat the point, the justificatory equivalence of both forms of
generic induction.
Objection 5.
You’re missing the point. If you justify a certain kind of epistemic
practice by appeal to the justification of another kind of epistemic practice, then your
appeal is unfounded, if it itself can’t be justified, and is circular if its justification is by
appeal to practices the rationality of which is secured by appeal to it.
Reply 5.
Consider an analogy with geometry. Riemannian geometry is consistent
relative to Euclidean geometry, and Euclidian geometry is consistent relative to Riemannian geometry. There are two points to take particular note of. One is that these
proofs of relative consistency are not proofs of the consistency of the one geometry
relative to a proof of the consistency of the other. (There are no such proofs.) The
second is that the lack of a one-shot consistency proof for all of geometry or geometry
as such is insufficient grounds for the skeptical assertion that our geometric practices
cannot be supposed to have any kind of rational justification.
Taking these points in reverse order, the same would appear to be the case with
induction. That there is no one-shot proof of the rationality of induction as such is insufficient reason for skepticism across-the-board. The two varieties of induction have
relative justifications. If this sort of relativity is good enough to deter blanket skepticism about the consistency of geometry, it is hard to see why the same slack should not
be cut to the rationality of induction.
The other point tries to make something of the distinction between practices that
are rational and proofs of their rationality. The same distinction is evident in the difference between relying upon a justification furnished by a principle of such a practice
and relying upon a proof of its justification. In general, inductive reasoning is a form
of reliance in the first, not the second, of these two senses. When we infer from a stratified random sample of appropriate size in which 70% of the F are Gs, that something
close to this percentage obtains in the population at large, we are employing a justified
principle of statistical reasoning. We are not employing the proof (such as may be),
that that principle is justified.
It is the same way with justification of induction. When we justify sample-induction,
we employ rational principles of abduction. We do not employ the proof that abductive principles are rational. Similarly, when we justify abduction, we employ rational
procedures of sample-induction. We do not employ proofs of their rationality.
If, in either of these situations, we did employ proofs of the rationality of the principles we have employed, our justifications would be circular. But as we have seen
these respective justifications go through well short of any such over-arching reliance
on such proofs. So, sample-induction is justified by abductive reasoning, concerning
which there is a proof of its rationality; and abduction is justified by sample-inductive
reasoning, concerning which, too, there is a proof of its rationality. But these proofs
aren’t involved in these respective justifications.
Thus is solved the Problem of Induction.
Objection 6.
Not so fast. If the object of the present exercise is to construct an argument on Hume’s behalf, which, Hume himself would certainly have produced had he
12.1. THE PROBLEM OF ABDUCTION
247
not had a blindspot about abduction, it is not enough that the argument produced here,
LCI, gives a plausible defence of induction. It is also necessary to show that it is an argument of a sort that Hume would have found appealing. Another way of saying this is
that the mere recognition of abduction as a form of ampliative reasoning is insufficient
to make of LCI an argument to which Hume would have been drawn. In fact, even if it
had occurred to Hume to give an expressly abductive justification of sample-induction,
this alone does not show that Hume would have thought this abductive manoeuvre to
be justified.
Reply 6.
Of course. The heart of LCI turns on the distinction between a mode
of justification which draws on principles which are themselves rational and a mode of
justification which draws on a proof of the rationality of those principles. In the defence
of induction presented here, the argument exemplifies the first sense of reliance but not
the second. The question is whether there is any reason to suppose that Hume himself
would have sanctioned this modus operandi.
There is. To see that this is so, it sufficies to re-examine Hume’s original argument.
One way, he said, in which induction could be justified (if it could be justified at all),
is by way of a sound deductive argument. This would be an argument in which the
conclusion asserted or immediately implied the justification of induction by deductive
consequence from other propositions that are not in doubt. Having considered this option, Hume gave up on it. Why? As we saw, there are two possible explanations. One
is that neither Hume nor anyone else knows of such a proof. The other is that Hume
believes (without showing) that there is no such proof. But what there is no evidence
for is a third possibility in which Hume requires that a deductive justification of induction must rely directly on a proof of the justification of deduction. What is wholly
evident is that Hume believes that deduction is justified, never mind what else he may
have thought about the provability of its justification. Thus were a deductive proof of
the rationality of induction (not of its own rationality) ready to hand, Hume would have
grasped it, and that would have ended his skepticism about induction. What this tells
us is that in our choice of the principles that do justify induction, it is unnecessary (and
in some cases, question-begging) to include a proof of their own justifiability. Had
the derived deductive proof been available to Hume, he could have said that there is
a justification of induction relative to deduction. He would not have said that there
is a justification of induction relative to a proof of the justification of deduction. Accordingly, Hume’s reasoning about a deductive justification of induction has the same
structure as our two justifications of the varieties of induction set out in LCI. It is a
modus operandi of which Hume would have approved.
12.1.4 Skepticism
In all that has so far passed between my re-jigged Humean and his imaginary critic there
has been an assumption that it has not occurred to anyone on either side to question. It
is the assumption that a best-explanation inference is always a relevant consideration in
the context of an assessment of the justification of induction. Thus, even if we grant that
the best explanation of the reliability to date of our sample-inductive practices is that
those practices are reliable in the general case, and even if we grant that there are lots
of contexts in which it would be reasonable to infer the truth of this best explanation,
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it may not be an allowable inference in the context of a skeptical attack on sampleinduction.
Our question now is whether explanatory attractiveness trumps skeptical force.
There is reason to think that it does not, contrary to the claim advanced by LCI. Counting against inference to the best explanation in the context of an examination of the
bona fides of induction is that the very idea of explanation is saturated with presumptions of inductive normalcy. For large ranges of cases, what best explains something
does so only if it is inductively normal. What best explains that my pet raven Fred is
black is that ravens are black. What best explains Harry’s bleeding nose is that Hector
punched it and punching noses cause them to bleed. And so on.
Sometimes an attack on induction is an argument on behalf of skepticism about induction. In its most damaging form, a skeptical attack on the justifiability of a practice
is an argument to the effect that even if those practices seem to us to be perfectly all
right, the proposition that they are all right does not meet some contextually indicated
standard of proof.2 In offering a best-explanation defence of such a practice the defender fails to take the measure of the critic’s skepticism. He fails to see that he is
begging the question against the skeptic.
A skeptic who has an adequate appreciation of his own argumentative designs
should be willing and eager to press the following argument as a counter to the bestexplanation move:
1. Suppose that, contrary to how things appear, our inductive practices
are in fact untenable.
2. Then it cannot be the case that even if the best explanation of their
tenability to date is their tenability as such, their tenability as such is
a correct inference.
3. Given what we presently know, it can’t be ruled out that the proposition assumed in (1) will turn out to be true.
4. That fact would be wholly compatible with the other fact that our
inductive practices so far give no comfort to that supposition.
5. Consequently, the proffered best explanation cannot overturn skepticism with regard to induction.
The argument here sketched would also seem to generalize to skepticism about causality, the external world, memory, and so on. In so doing it furnishes us with a version of
the limitation theorem for abduction which we set out in seciton 3.3.
Limitation Theorem
Abductive reasoning is insufficient to overcome
skeptical claims (whether about induction, causality, the external world,
memory, etc., etc.).
Still, it would be folly to make too much of the Limitation Theorem. It is a discouragement to those who would offer an abductive answer to skepticism about induction.
2 Of course, the central question for skepticism is what is taken as the standard of proof, and what it is
that makes it an appropriate standard to invoke. In the case of skepticism about induction, the standard of
proof is irrefutable proof of future reliability. There is a huge literature about this.
12.2. MORE ON SKEPTICISM
249
It is no discouragement at all to those who would offer an abductive answer to the claim
that induction lacks a justification. It is true that if skepticism about induction holds
water then induction lacks a justification. It is also true that, if induction does indeed
lack a justification, skepticism with regard to induction holds water. But what is not
true is that any way of successfully countering the claim that induction lacks a justification must also be a way of successfully countering skepticism about induction. One
response, not uncommon to skepticism among philosophers, is to grant it and forget
about it. In the case before us, granting it would amount to a concession that the reliability of induction fails to meet a certain standard of proof. Forgetting about it would
amount to pressing the abductive justification of induction even so.
It is textually ambiguous as to whether Hume was a skeptic about induction or some
looser breed of non-justificationist. Given the weight of his skeptical leanings in other
matters, perhaps we should also give the nod to inductive skepticism as well. But as
we now see, if our own interest is merely in whether induction has a justification, then
the answer is that it does. It is the justification furnished by our argument LCI.
12.2 More On Skepticism
Consider the following pattern of argument:
1. One of the parties to the argument calls into skeptical question some presumed
fact supported by the other party. Call this presumed fact the skeptic’s target.
Prominent among such targets have been the presumed reality of the external
world and the presumed realism of scientific inquiry.
2. The critic of skepticism does not attack skepticism directly. He may even concede that all attempts to demonstrate its falsity must fail—that all such attempts
will turn out to be either non-sequiturs or question-begging. Instead, he advances
his criticism indirectly. He points out that the extraordinary success of science,
its predictive reliability, its technological adaptability, its enjoyment of vast consensus, are things that need to be explained.
3. He then presses an inference to the best explanation. What is it, he asks rhetorically, that best explains the predictive, adaptative and consensual success of
science? He answers his own question: What best explains these things is that
the disclosures of science are objectively true (or some close approximation) and
that the world which science purports to plumb is objectively and antecedently
there, independent of theory and at least a good approximation of science’s description of it.
4. Therefore, scientific realism is a philosophically plausible theory, and a rational
thing to believe even granting that this defence of realism is not a demonstration
that anti-realism is false.
Given the level of generality and abstraction at which we have sketched this abductive defence of philosophical realism, it is unavoidable that the picture be rather crude
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and coarse-grained. This happens not to matter for present purposes, since, crudeness aside, we have managed to display enough of the abductive support of realism to
discern its fatal flaw. The flaw is that the abduction argument fails to take into consideration that the concept of explanation is not a metaphysically neutral one, nor is it one
that announces its own domain of appropriate applicability.
In what way is the concept of explanation metaphysically partial? In the matter
of scientific success, the explanation that strikes us as most satisfying, most obvious,
indeed as best, is realism. We need only ask ourselves what would count as the best
explanation of the success of science if it were taken as a matter of metaphysical fact
that realism is false? Plainly, it could not consistently be the objective reality of nature
that explains the success of science on that assumption. It is far from obvious that the
success of science on the anti-realist assumption would have an explanation. This tells
us two things.
1. Our concept of explanation is already a realist concept. This is because common
sense and workaday science itself are suffused with realist presumptions.
2. The same bias affects our judgements of the necessity and possibility of explanations.
The anti-skeptic, in proposing that in the success of science there is something that
requires explanation, is so inclined because he will not countenance as an explanation
anything that fails to deliver the goods for realism.
We have in this an
Explanation Limitation Theorem: Every concept of explanation has embedded assumptions õ ö ÷ ø ø ø ÷ õ ù concerning what it is to be an explanation.
Let ú be a domain of discourse in which all or most of the õ û fail. Then
no call for an explanation embodying those assumptions can be honoured
in ú ; and if ú has not evolved a concept of explanation appropriate to its
topical peculiarities, no call for any explanation of a ú -phenomenon can
be said to have an explanation. 3
Skeptics are routinely underestimated by philosophers. Philosophers who take it as obvious that the success of science demands an explanation fail to take into account that
no skeptic is committed to any such imperative. If he has the proper strategic command
of his skeptical thrust, the anti-realist will be careful to avoid saying that this, that or
the other thing either requires or admits of explanation. So the careless anti-skeptic
begs the question against the anti-realist. Even if we did suppose that the success of
anti-realistic science admitted of an explanation, it would have to be a very different
explanation than the one endorsed by common sense. Very different and highly speculative, and not, certainly, one that would strike us in any way as obvious. This is
because human beings take the realist stance as naturally as they breathe. Explanations of anything affected by the presumption of the truth of skepticism are bound to
3 The Theorem finds formal reflection in conditions on abduce-functions. Part of a logical system is its
abduce-mechanism; and not every such mechanism is appropriate for every logical system. See chapter 12
below.
12.3. VOLTAIRE’S RIPOSTE
251
strike us as odd. Just as such explanations are hard to swallow, skepticism itself is hard
to believe. Certainly no one could succeed in conforming his behaviour to skepticism
about the external world, even if intellectually he acceded to its truth. If we were to
recognize a notion of spontaneous belief, skepticism cannot be the object of belief in
this sense. But any skeptic who knows what he is doing is his aware of this. It is no
part of his skepticism to change our behaviour, get us to live life in all its particularity
in ways that embody the truth of skepticism. The skeptic can hit his target that not ü
without occasioning any obligation to explain the facts that suggest that ü is possible.
We noted earlier that those who attempt direct refutations of skepticism run a high
risk of begging the question, something which inference to the best explanation is designed to avoid. Whatever the intention of the abducer, the fact remains that his project
is every bit as question-begging as the modes of criticism he seeks to avoid. A similar argument may be found in [Fine, 1989]. We paraphrase the key idea: People who
demand an explanation of the Bell inequalities in quantum theory are calling upon a
concept surfeited with assumptions from Newtonian mechanics. It is also reminiscent of Hanson’s doctrine of theory-ladenness, extended from observations to facts and
causes [Hanson, 1958, pp. 37–49].
12.3 Voltaire’s Riposte
The failure of inference to the best explanation to answer in a satisfactory way various
kinds of skepticism calls to mind an objection known in the literature as Voltaire’s Riposte. Voltaire’s Risposte is that inference to the best explanation makes the successes
of our inferential practice a miracle.
We are to infer that the hypothesis which would, if true, provide the loveliest explanation of our evidence, is the explanation that is likeliest to be
true. But why should we believe that we inhabit the loveliest of all possible worlds? If loveliness is subjective, it is no guide to inference; and even
if it is objective, why should it line up with truth? [Lipton, 1991, p. 123].
(A lovely explanation is one which increases understanding, or has the potential to do
so [Lipton, 1991, p. 61].)
Voltaire’s Riposte bears on our own Problem of Abduction. They are not, however,
wholly identical questions, since not every abductive inference is inference to the best
explanation. To see that this is so, it suffices to revisit Newton. The target of his famous
‘Hypotheses non fingo’ was instantaneous gravitational action over arbitrary distances,
which Newton regarded not as unlikely, not as implausible, but as a conceptual absurdity, which it certainly was by the standards of scientific commonsense. What, then,
did Newton refuse to do? Did he refuse to accept the theory of gravitational forces?
Did he disdain its extraordinary predictive power? Did he expunge this conceptual
impossibility from its indispensable place in gravitational theory? No. He refused to
dignify the absurdity with a conjecture that would explain it, for he thought that it lay
utterly beyond the reach of any explanation.
Action-at-a-distance is mathematical consequence of, as Newton and his peers
thought, a powerful and empirically successful theory. If the action at a distance the-
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orem were to be cancelled, the theory would be substantially disfigured. Left in, the
theory prospers; taken out, it is significantly hobbled. So in it stayed. Newton’s decision to hold his nose for action-at-a-distance has an unmistakably abductive texture to
it. In schematic form, his reasoning was this:
1. Let ý be the complete theory of gravity, and let þ be the action at a distance
theorem. Thus ý ÿ
þ is what remains of the theory if the action at a distance theorem is removed. Because þ is conceptually impossible, according to
Newton, ý µþ is an abduction trigger for Newton.
2. The basic facts are these. ý is an extremely good theory. It provides a coherent,
mathematically deep and predictively abundant and accurate account of gravitais a theoretical disaster, weak in
tional phenomena. On the other hand ýÍÿ þ
all respects, and more, in which ý is strong.
3. Accordingly, even though we don’t understand þ
, even though by the lights
of commonsense it is nonsense, we leave it in for the impressive theoretical yield
it abets. Without , there is no ý . And ý is a powerfully successful theory.
Therefore, it is prudent to accept for the contributions it makes to a rich and
successful theory.
This is abduction without any pretense of explanation. The abduction turns on a strategic consideration, which is to accept what is indispensable for an otherwise good theory. There is no requirement or presumption that the abducing gravitationalist will
believe or understand þ . There is no requirement that the theorist search out an explanation of þ , since he entertains þ , indeed accepts þ , not only while robustly disbelieving it, but while also believing it inexplicable. Nor is there any question then,
of adopting this inexplicable þ for the explanatory power that it confers on the theory.
What is intelligible is also without explanatory power.
A realistically-minded theorist will want to tart up the basic story, so as to downplay
the dominance of a sheerly strategic or prudential policy in regard to þ . He will likely
want to say that its role in a hugely successful theory is occasion to lighten up on the
prior indictment of þ . So he may now put it that he merely thought þ was absurd,
but now sees that it must be all right.4 Even so, its all-rightness makes a difference
that makes no difference, as long as the non fingo resolve remains in force. Say that þ
is true if we must, but if non fingo continues to call the shots, it is a truth the theorist
cannot understand, and a truth, therefore, without explanatory force for him, and is a
clear case of the non-concurrence of predictive and explanatory success.
It would be a serious mistake to leave the suggestion that practitioners of the axiomatic method in science are somehow imaginatively deficient. Any investigator must
use his creative imagination to pull off anything as daunting as a fully-fledged scientific
theory, whether axiomatically formatted or set out in a hypothetico-deductive arrangement. The intended contrast between theorists who do and those who don’t employ
techniques of the imagination bears on the distinction between theory and heuristics.
4 On the other hand, perhaps the realistically-disposed theorist has not had sufficient instruction in the
fallacy of division. For a discussion of this, the Indispensability Thesis, see [Woods, 2002b, ch. 8].
12.3. VOLTAIRE’S RIPOSTE
253
Investigators who take the axiomatic approach to science invest their axioms with
explanatory power. Investigators who take the hypothetico-deductive, or some other
abductive approach, invest their conjectures with explanatory potential. In each approach there is a role reserved for explanation. Each approach therefore is vulnerable
to the Heuristic Fallacy. But it may also be said that in his inclination to embed his
conjectures in his theory on the strength of their explanatory potential, the abductive
theorist is more at risk of the Heuristic Fallacy. This is a feature of abduction that
demands for a watching brief.
Given our view of logical agency and the scarce-resource compensation strategies
required for individual agents, we are at a juncture at which it is also worth remarking
that we do not take the heuristic-theoretic distinction in quite the hard-edged way in
which it would have been understood, say, by Reichenbach. On what we could call its
classical interpretation, a theoretical principle (or virtual rule) sets out a normatively
prescribed process of reasoning, whereas a heuristic device is a normatively subpar
but prudentially attractive means of arriving at some approximation of the same end as
that for which the theoretical principle exists [Tversky and Kahneman, 1974]. Modus
tollens is a case in point. It is a truth-preserving principle of logical theory. Experimental evidence suggests (Evans et al. [1983], Johnson-Laird and Byrne [1991]; cf.
Rips [1994]). Manktelow [1999] says that in classes of real-life contexts individual
reasoners actually employ an heuristic counterpart of the logical law, which is subpar
in the sense that it is not truth-preserving.
Our own inclination is to query the purported sharpness of the distinction and to
resist the suggestion that heuristic manoeuvres are intrinsically subpar or deficient. In
general, the goodness of a reasoning rule depends on its context of application, the
objective of the reasoning agent and the resources available to the agent at the point of
deployment.
It should be emphasised, therefore, that the heuristic-theoretical distinction embedded in our description of the Heuristic Fallacy is not the classical distinction. For the
purposes of the Fallacy, the intended distinction is a case of a pre-theoretical/theoretical
distinction. It is the distinction between what the theorist must believe pre-theoretically
in order to think his theory up, and those propositions that the theory should officially
sanction. The Fallacy is that of supposing that anything in the first category must also
be in the second.5
Newton’s situation is no rara avis in science. The quantum physics of the present
day is rife with indispensable absurdities, absurd anyhow by the lights of commonsense
and the predecessor-sciences. There is in this an echo of the partiality of the concept
of explanation. If, as Fine avers explanation in physics is awash in assumptions drawn
from Newtonian mechanics, then it must follow for example that the Bell locality inequalities have no explanation. By the same token, if what seems absurd in physics
is an idea that drinks deeply of extra-quantal assumptions, then the Bell locality inequalities are absurd, and that’s an end to it. By the same reasoning, if what is absurd
in Newtonian mechanics is so for a concept of absurdity which is inherently extra5 There is a further notion of heuristics in the context of a class of computers called General Problem
Solvers. Heuristics are that part of a GPS program which pre-sets the order in which various moves should
be made under certain given circumstances. They include the computer’s set of preconditions. Where the
option space is large, it is an enormously difficult task to specify such conditions.
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Newtonian, indeed pre-Newtonian, then the action at a distance theorem is absurd, and
that too is an end to it.6
The Problem of Abduction in the form of Voltaire’s Riposte is the problem of showing that what has explanatory power is probative or truth-enhancing. We have already
explained why we think that Voltaire’s Riposte will fail to produce a satisfactory answer under a generalized charge of skepticism—of skepticism about the external world
or the realism of science, for example. It is not our purpose to press that attack here,
beyond mentioning the Limitation Theorem of several paragraphs ago. Like Peirce,
like all working scientists, and like best-explanation theorists such as Lipton, our first
objective in developing an account of the reach of abduction is descriptive adequacy,
subject to such subsequent correction in the light of norms as we might eventually
pledge to.
There is a further question not directly posed by Voltaire’s Riposte, but since it is
prior to the question that it does directly ask, we should deal with it first. Concerning
these very practices which the Riposte calls our attention to, Lipton asks rhetorically:
‘Let us turn now to Voltaire. Why should the explanation that would provide the most
understanding if it were true be the explanation that is most likely to be true?’ [Lipton,
1990, p. 125]. Lipton’s answer, in part, is that
it just turns out that our judgements of likeliness are guided by explanatory considerations. Perhaps this does make it surprising that our inductive
methods should be successful, but the main purpose of the model of Inference to the Best Explanation has been to describe our practices, not to
justify them [Lipton, 1990, p. 125].
We note in passing that nothing in this reply to Voltaire shows or purports to show that
our best-explanation inferential practice is probative. There is nothing more in it than
the assertion that, as we actually deploy our inferential policies, we see that there is a
significant concurrence between what as a matter of fact we take to be best-explained
and what as a matter of fact we take most likely to be true.
The question at hand, however, is whether Lipton is correct in his claim that any
account of inference to the best explanation is descriptively accurate if it licenses the
inference to the existence of theoretical entities or states of affairs on the strength of
the explanatory force they would have, if they actually existed. We join van Fraassen
and others [van Fraassen, 1980, pp. 20–21] in giving a negative answer to this question
of descriptive adequacy. Van Fraassen makes two objections.
1. If we carefully examine the inferential practices of best-explanation inferers, we
see that two hypotheses fit these data at least equally well. One is the realist hypothesis which says that reasoners infer the truth of those conjectures that lend
explanatory force to the theories that invoke them. The other is the constructivist
hypothesis which says that reasoners infer the empirical adequacy of those conjectures that lend explanatory force to the theories that involve them. Assuming
6 What is more, quantum electrodynamics is inconsistent, a fact which inclines some theorists to opt for
proofs formalized in such a way that the ‘good’ derivations alone are licensed. We return to the problem of
inconsistent data-bases in Part III.
12.4. PROBATIVITY
255
the two hypotheses to be observationally equivalent, the tie is broken in favour
of the second by considerations of metaphysical parsimony.
2. It is also necessary to ask in what does the explanatory force of a conjecture
consist? What is it that makes one conjecture a better explanation of the data,
rather than some other? In answering these questions, Harman says, in part:
(ceteris paribus) of , provided:
— has
(a) a higher probability than
— bestows higher probability on (b) is a better explanation than
than
does [Harman, 1968, p. 169].
As van Fraassen points out, (a) is a Bayesian test, relying as it does on initial or a
priori probabilities, and (b) is a principle from a more traditional understanding
of statistics. But (b) does imply that
and
always assign definite probabilities to , an implication that is contradicted by simple cases. Let
be the
denial of , and suppose that
says that the probability of is 0.75. Then
the most that it falls to
(i.e., not- ) to determine is that has a probability
other than 0.75. Beyond that who’s to tell? This the problem of ‘unavailable
probabilities’. Bayesians handle the problem by subjectivizing probabilities, and
by proposing that for every proposition a subject can express, he will accord it a
subjective degree of belief. (We let pass the empirical adequacy of this assumpand , the requisite probabilities are
tion.) This means that for arbitrary
always available. But, then, asks van Fraassen, isn’t this too much subjectivity
for serious realism as regards the winning ?
We conclude, therefore, that the assumption that we take our best explanation inferential practices to be probative is not to be borne out in practice in the case of the
postulation of non-observational theoretical entities. Lipton’s assertion of descriptive
adequacy is defective in this regard.
12.4 Probativity
We see now why the question of whether we take our best-explanation inferences to
be probative is prior to the question posed by Voltaire’s Riposte. If we do not take
such postulations to be probative, if instead we think of them as empirically adequate,
is there anything left in Voltaire’s Riposte to answer? The answer is yes, twice-over.
We need to keep in mind cases in which we postulate entities which, while observable,
are technically beyond the abducer’s reach at the time the conjecture was made. And,
beyond that, it is helpful to test our abductive resources counterfactually. So it would be
helpful to know whether Voltaire’s question can be answered even on the counterfactual
assumption of the probity of best-explanation inferences.
The first kind of case is a commonplace in theoretical science. Theory postulates;
and experiment subsequently confirms or disconfirms. (The existence of neutrinos was
predicted long before experimental confirmation, to take just one example).
CHAPTER 12. LIMITATION ISSUES
256
Case two is more interesting. It asks, once the requisite probity-assumptions are
made, what would explain the fact that hypotheses that best explain the data are true?
We can put the question schematically. Let
be any datum such that if is the best
explanation of a datum , then is true. Our abduction problem is to find an
such
that
!
is true
! is best? (Let us call this Voltaire’s
Given
When
is best,
Then what is that
such that
Schema.)
to be ‘surprising’. According to Voltaire’s Riposte,
is inexplicable,
Lipton finds
like the action-at-a-distance theorem, possibly. As we saw, Peirce suggests that the best
direct evidence for
is that for particular ’s, assigning truth to
when
is
best turns out to have been the correct thing to do. Let us be precise. It turns out to
have been the correct thing to do, when
has dealt with observables only. It is less
than the correct thing to do when the entities involved in turn out not to be ‘there’.
We must take it, then, that an inductive justification of abduction offers no solace to the
epistemological realist. Peirce’s inductions must take the form that in entertaining an
hypothesis , explanatory force is a reliable indicator of accepting the in question,
which is what a constructive empiricist in the manner of van Fraassen will also say.
We have seen that induction enjoys an abductive justification. For the best explanation that we have of our inductive practices is that in general they get things right rather
than wrong. But given that deriving true propositions is just one way of getting things
right and that another way of getting things right is to adopt the propositions having the
greatest explanatory or predictive clout, there is nothing in an abductive justification of
induction that gives the nod to the thesis of the probity of explanatory success.
Let us revisit Voltaire’s Schema.
!
! is best, is true
such that ! is best.
Given
When
Our task is to find an
If
is true, then given its own generality, it is true of itself. Thus
When
!
is best, then
implies
is true.
is true.
In other words
When
(When
!
Again, given the generality of
such that
is best, then
is true) is best, then
, we must find an !
When
(When
. (When
is best, then
is true) is best, then
is true.
to explain this datum, i.e., an
is true) is best, then
The infinite regress that threatens flows from the generality of
shows that it cannot
always be true both that any abductive situation such as
will have an
, and that
is true if
is best with regard to
.
It is well to keep in mind that we are proceeding counterfactually and for the sake
of argument, assuming the truth of
. That is, we are taking explanatory success
12.4. PROBATIVITY
257
to be a reliable indicator of the truth of the explanans. What is needed is an answer
to this question: What explains this concurrence of explanatory success and truth?
The question is unanswerable. It is unanswerable not for the reason that underlies our
Explanation Limitation Theorem of pages ago. We do not say that what Voltaire’s
Riposte calls for is an explanation, the very conception of which contradicts properties
of the skeptical view which the explanation was designed to explain away.
We are asking what it is that explains the purported fact that whatever has explanatory force is true. Our provisional answer is that nothing does, even if it is true that that
explanatory success is indeed a good predictor of truth. It all turns on what we are to
mean by explanatory success. If we mean ‘what we take to be explanatorily satisfying’,
there is no serious historian of science who will take that to be a reliable marker for
truth. If, on the other hand, we understand explanatory success in some non-subjective
way, there may be such a sense in which explanatory success is a marker for truth,
but saying so in just this way leaves it wholly undisclosed as to what this sense of
explanatory force is, and how and with what reliability beings like us have access to it.
12.4.1 Transcendental Arguments
Transcendental arguments are arguments in the form
1. State of affairs
2.
"
"
is a fact.
would not be possible unless it were the case that
3. Therefore,
#
#
.
is the case.
Transcendental arguments are so-called after Kant’s employment of them in a section of
[Kant, 1933] entitled ‘Transcendental Deduction of the Pure Concepts of Understanding’. In that section, Kant produced a transcendental argument which we reconstruct
as follows:
1. It is a fact that human experience occurs.
2. Human experience would not be possible unless it were the case that the categories (i.e., the pure concepts of understanding ) were objectively valid (i.e.,
actually instantiated in reality).
3. Therefore, the categories are objectively real.
A second place where Kant deploys a transcendental argument is in the section entitled
‘Refutation of Idealism’.
1. It is a fact that we are conscious of the temporal order of certain of our mental
states.
2. It would not be possible to have such awareness unless we were also conscious
of the existence of things external to ourselves in space.
3. Therefore, there are things external to us in space (and so idealism is false).
CHAPTER 12. LIMITATION ISSUES
258
Kant’s two arguments are directed against philosophical skepticism: about causality in
the first instance, and about the external world in the second. Our claim in this section
is that a form of abductive argument known as inference to the best explanation fails
as an attack upon skepticism. There is a sense in which Kant’s arguments appear to
offer explanations of what is stated in their conclusions, and they are indeed directed
at skeptical targets. This prompts two questions
1. Are Kant’s arguments sound?
2. Are Kant’s arguments abductive?
If both questions are correctly answered in the affirmative, then we would have a class
of abductive arguments that succeed in their anti-skepticism. And given that they might
be read as explanation arguments, perhaps they would be a counterexample to our
claim that inference-to-the-best-explanation arguments cannot prevail against skeptical
targets.
We take these questions in reverse order. Are transcendental arguments abductive? We note in passing that in each case premiss two is a modalization of a barer
conditional that suffices for their respective argument’s validity. Kant appears to have
elected to put them in modalized form because he believes that they are in each case
purely conceptual truths. In any event, neither premiss two nor premiss one is a matter
of conjecture, which some readers will take as showing that Kant’s arguments aren’t
abductive and that, for our present purposes, the question of their soundness need not
be pursued.
But couldn’t there be classes of instances of
1. State of affairs
2.
$
$
is a fact
would not be possible unless it were the case that
3. Therefore,
%
%
%
is the case
&
%
in which is inconsistent with some form of skepticism , and in which is indeed
a matter of conjecture?
It won’t work. Unless the assertion of is compatible with , the argument would
be question-begging against . And unless the assertion of is not compatible with
, then the argument against must fail. Therefore, if the anti-skeptic introduces his
transcendental premiss ‘ wouldn’t be possible unless ’ as a matter of conjecture, he
invites the skeptic to abandon his skepticism on the strength of a supposition which no
confident skeptic would accept nor should.
We conclude, then, that abductive arguments do not succeed against skeptical targets. And, as we have just seen, this is sometimes the case because they are dialectically compromised by the very factors that make them abductive arguments.
It will now repay us briefly to revisit Russell’s regressivism. Consider the following
claim:
(1) Given
(i.e.,
) in which is a set of wffs, is an obviously
true sentence and is a wff. When is the fact expressed by (1) abductively significant
with respect to ? That is, when can we reasonably say that since
&
&
$
')( *+-,.
*
*
$
&
%
'/0 *21 3 .
&
'
%
.
12.4. PROBATIVITY
259
45-67
1.
7
2.
is obviously true
then
3. it is plausible to suppose that
4
holds as well.
Clearly enough, for this to be a reasonable abduction, there must have been some antitriviality constraints on the choice of . For example, if is classical deducibility, we
can’t allow
. Nor can we permit
for arbitrary
. Similarly, if
, then the proof of cannot exploit that fact.
We can generalize these constraints by saying that
must be a serious proof.
It must meet the standards of the discipline in question, whatever they are in time.
Then we must determine the strength of ’s dependency on as in the order of
increasing dependency
78>9;<29
4
4 8:92;<=9
4 8>9@?7
7
5
9AB
45-67
7
45-67
1.
2. As above, and
BD5C 7
, i.e.,
4
4
is not redundant in
7
45-67
.
3. As above, and there is no other proof of .
The dependencies at this third level, give all the wherewithal for a transcendental argument, as follows:
7
i
ii
is provable
7
4
can’t be proved without assuming
provable).
, i.e., if
< (4
is assumed) then
< (7
is
But this is a set-up for a modus tollens argument which is an exercise in deduction, not
abduction. It would appear that the point generalizes. Transcendental arguments are
special cases of modus tollens arguments, and as such are deductive. If we are to persist
in the view that abduction is disjoint from deduction then in the example before us we
need to weaken the dependencies between and . In virtual mathematical practice
(think of the role played by the axiom of infinity in set theory), the requisite kind of
dependency of upon in a discipline is something like this: plays a fruitful role
in . It asserts directly or indirectly in the derivation of families of results (and, so,
not just ) which
7
E
7
4
4
E
7
4
F
1. so far as is yet known can’t otherwise be derived
or
2. more weakly, so far as is yet known cannot be derived as efficiently or simply.
While (1) gives the stronger dependency, it doesn’t constitute a transcendental, or any
kind of, deductive tie.
This permits a clarification of a point, two paragraphs ago, about why transcendental arguments aren’t abductions. Lacking the element of conjecture, we said that
260
CHAPTER 12. LIMITATION ISSUES
some readers will take this as negatively decisive for the abduction question. No doubt
this is so for some readers; but it is not so for us. When Russell or Gödel abduced
the probable truth of a recondite principle of logic on the basis of its contribution to
the derivation of an obvious truth of arithmetic, they were not conjecturing. Just as
explanation is not essential to abduction, neither is conjecture. By these lights, that
transcendental arguments are non-conjectural cannot decide the rejection of their status as abductions. Better, then, to cite a factor that bears on this issue relevantly and
deeply. Transcendental arguments are deductive. That is why they aren’t abductive.
Part III
Formal Models of Abduction
261
Chapter 13
A Glimpse of Formality
In the previous eleven chapters, we have endeavoured to produce a conceptual explication of abduction. A conceptual explication involves a considerable clarification of
basic data about what abduction is and how it operates. It thus functions as an informal
theory . As we have already explained, the informal theory is input for a formal theory,
which latter is the business of the present Part of the book. In this chapter, we want to
help the reader orient himself.
13.1 Some Schematic Remarks
Two well-known logicians Bjoern and Johannes (B and J) have accepted a visiting
position for three months at the newly established Institute of Artificial Intelligence
Technology in the Far East. The Institute takes pride not only in its high salaries for
visitors (they negotiate their net take-home amount per month, all expenses paid!) but
also in its support facilities, which are themselves run by advanced AI programs.
Our two visitors have been located in a large lab containing computers, printers, an
especially fancy copying machine, automated windows and doors, in short, the latest
in latex technology 1
B and J are the only people in the lab. Life passed smoothly in the first two days
until B complained to J that the photocopier does not seem to be responding. ‘I keyed in
my code but nothing is happening’ he said. J advised him to switch it off and on again
and re-try. To no avail. ‘Let me try’, J says. He switched the machine off and on again
and keyed in his code. The machine worked. B happily resumed his photocopying.
In the days ahead, every now and then the photocopier would conk out again. Various attempts were made to start it, usually successful, after a number of tries.
It is now today, and whether the copier will work today (and how) has become a
focus of their attention. After a while, B and J also noticed that on rainy days, when
the photocopier is working, the door is open, and if they insist on shutting the door, the
photocopier stops working.
1 (including
an electric pencil sharpener).
263
CHAPTER 13. A GLIMPSE OF FORMALITY
264
B conjectured that there is some intelligence in charge of their lab and that there
is some kind of program coordinating the various operations. Being logicians, they
decided to build a model of the relevant operations and discover what the program
does. To this ed, they decided that three languages would be required.
G
1. State Language
capable of describing and reasoning on the state of the lab.
2. An Action language
H
I
3. A meta language
coordinating the interaction between the other two languages.
It is in the metalanguage that B and J hoped to formulate the hidden rules of the
program that was controlling the lab.
B and J understood that the state language and its logic can be any logical system
in the sense discussed below in Chapter ??. For their present purposes the language
chosen was a propositional language with atoms intended to mean:
J
K
O
LM
LN
Q
P
= B uses the photocopier
= J uses the photocopier
= The photocopier accepts B’s code
= The photocopier accepts J’s code
= The window is open
= The door is open
= It is raining outside.
Given this interpretation, there are some obvious rules. For example,
RTSU J2V K W
RTSU L M V L N W
XY
[Y
Z Y U [ Y \ ]-Y W [
_` Y
_>a=]^Y b
The action language describes actions
of the form
, where
is the
precondition and
is the post condition. If is a state an d
then it may be
that action can be executed, in which case we move to a new state
, obtained
from by the revision process :
_
XY
]^Y
a
_
U _\ ] Wcd _ea2]f
The job of the metalanguage is to give expression to conditions that regulate the intersection of the actions, revisions and the logic of the theories .
Following the example of B and J, we shall use an executable temporal logic as a
metalanguage but will write the rules in semi-English in this introduction.
The expressions of
are as follows:
_
1. Execute
dow.
Ug W\h g
I
G
a literal of . For example, Execute
UO W
means: close the win-
13.1. SOME SCHEMATIC REMARKS
2. Formulas of
i
265
are also atomic formulas of
kl
l
j
.
l
mon
3. Temporal operators
, yesterday (or some state before obtains),
(some
past state obtains); and
(from a past state at which was true, has
been true in all intermediate states.)
l
u)l
4.
pq lr s=t
l
s
l
, (always )
5. Rules of the form
Past
There are examples of
j
j
wff
vxw=y z { | } z q ~)t 
rules in action.
€  three times in the past, up to now, without any € ‚ between them, w=y z { | } z q ƒ
€  t . That is, we cannot use the machine more than three times without someone
1. If
else using it.
2. Always either the door or the window is open or the photocopier is not on.
3. If it rains, then close the window.
An abductive logic describes or mimics what an abductive agent does.2 In its most
basic sense, a formal abductive agent is faced with the following situation. There is
a database (or a belief-set or a theory) and a fact symbolized by a wff such that
. The basic abductive task is to adjust in ways that now yield , and to do
so in compliance with the appropriate constraints (e.g. explanatory power, predictive
accuracy, simplicity, etc.) The purpose of this final chapter is to explain the kind of
logics that are involved in our study of abduction. A formal treatment of such logics
is given in volume 2, in the context of formal models of abduction. This chapter will
take the reader through the fundamental concepts involved by following a few simple
examples.
Consider a simple propositional language, with atoms
and the connectives
for implication, and for falsity. We can form the set of all formulas in the
usual way.
How do we define a logical system on this language? We define the following two
notions
„
„†‡…

„
…
ˆ ‰or Š r ‹ r    Œ
Ž
(1) The notion of a theory
„
…
(database) of the logic
(2) The notion of consequence for the logic, i.e. the notion of the conditions under
which we say the proves A in the logic, i.e.
.
„
„
„^‘ ’
The usual notion of a theory is that
is a set of wffs. This notion works for
many logics but not for all. Many logics require, in view of the intended application,
additional structure in the theory. For example, a theory can be considered abstractly
as a list of wffs
„
„“Dq ’@ r    r ’)” t
2 The
discussion in this section is appropriate for any application area modelled by a logical theory. In the
philosophy of science we have explanation/abduction in the context of much richer mathematical theories
(mechanics, relativity, etc). The ‘data’ and ‘inference mechanisms’ are different.‘
CHAPTER 13. A GLIMPSE OF FORMALITY
266
• –@— ˜ ™ ™ ™ ˜ –)š ›œT
where the consequence relation holds between lists of wffs and a formula (i.e.
). We shall come back to this later. Meanwhile we examine what
our options are for defining . We begin with the options for classical logic.
œ
ž Ÿ ˜ ¡ ¢ in the traditional manner and let
x
œ – iff ¤-¥ • ¤ ›¦ £ • ¥ • ›§D¡ ž¥ • –)›§D¡ › ™
£ Option 1. Tableaux.
Define truth functions h:Wff
The proof theory is constructed from semantic tableaux.
–‡¨>
• –©°­@›©±
• –ª©«• ¬©®­)› ›:©®­
–²
–©³­
¨˜ ¯
£ œ–
²
–¯:
£e´ Ÿ ² –@¢
Option 2. Resolution.
Let
be defined as
. Let
be defined as
. Let
be defined as
. Write every wff in conjunctive
normal form using
and . To check whether
consider
written
in conjunctive normal form as clauses, and define and apply resolution.
Option 3.
Use a goal directed proof mechanism (see Definition ??)
£ œ –
• –>©x›2©x–>œ-µ –
These are three different ways of telling us when
ence let us check the example of
.
. To illustrate the differ-
Option 1.
Try to find a falsifying , using tableaux:
The proof goes as follows:
¥
1
¶ ·¸±¹Tº¸±·
·
·
0
·
· ¸±¹
·
·
¹
The tableaux is closed. So a countermodel
• • –e©¼›©¼–)› ˜ –@¢
² ˜ ² –@¢§ Ÿ – Ÿ ˜ – ¯ ² ˜ ² –@¢ . ²
Option 2.
To take the set
·
¥-» does not exist.
and rewrite it in clausal form as
Ÿ –T¯–˜ –T¯
13.1. SOME SCHEMATIC REMARKS
½¾ ¿=½
267
give the empty clause.
Option 3
To deploy the goal directed computation of Definition ??
Success
if
Success
if
Success
if (by restart)
Success
if Success!
It is for us an important point of departure that a logical system is not just (the
consequence relation) but also the proof option chosen for . Accordingly, the above
discussion yields three logical systems, ( , Option 1), ( , Option 2) and ( , Option 3).
They share the same consequence relation, but the proof theory is different.
The abductive mechanism which we will develop in detail in Chapter 3 of volume
2 will depend on the proof theory, and so as far as abduction is concerned, the three
logics may give different results. Here is an example of an abductive problem:
½ ÂxÃÄ2Âx½@Å ¾ ½)ÄÆDÇ
ÀÁ À:
À Á À ½:ÂxÃÄ2Âx½@Å ¾ ½>ÂxÃÄÆDÇ
À Á À ½:ÂxÃÄ2Âx½¾ ½@Å ¾ ÃÄÆDÇ
À Á À ½:ÂxÃÄ2Âx½¾ ½@Å ¾ ½)Ä2ÆDÇ
ÈÈÈ
É
É
É
É
É
½>Âxà É-Ê Ã
We abduce by applying the proof theory and adding assumptions at key moments of
failure.
Option 1.
1
ËÌ°Í
Í
Ë
Í
0
Í
Í
The left box is not closed. To close it, we can abduce
We run a resolution on
¿2½
½
or abduce
Ã
.3 Option 2.
¿2½ Îþ ¿=Ã
obtaining
. From this we are not able to obtain the empty clause. So our obvious
abductive option is to add A.
In fact, we can add the negation of any formula in the set, but it is preferable to
follow our proof procedure as far as we can go, and abduce only at the last moment,
when we are stuck.
Ï ÐѐÒ2Ó-ÑÔ
3 We can also abduce their disjunction
, which is weaker. However, our policy in these
examples is to enable success immediately at the point at which we are stuck.
CHAPTER 13. A GLIMPSE OF FORMALITY
268
Option 3.
Success(A B, B)=1
if
Success(A B, A)=1.
For this to succeed we must add A to the data.
We saw above that applying the proof mechanism and trying to succeed with
gives rise to several candidates for abduction. The next step is to choose which
candidate to take. Do we add ? Do we add ? The choice actually requires a
second logic, which we can call the background logic of discovery. We may include
the abduction algoirthm itself (as defined using ) as part of the logic of discovery.
We mentioned the possibility that the database may be a structure, say a list. Why
do we need structure? The need arose from the interpretation of the logic and its
applications. If, for example,
is strict S4 temporal implication, then
means
that whenever is true (in the future) then is true. We might think of
as
an insurance policy (whenever you are disabled you get $100,000). To apply modus
ponens, we must have come after
not before. So
Õ
Õ
ׇØ-Ù ×
Ö>Õ
×
Ö
Ú
Õ
Ö
Ö
but
Ö>Õ × ×
ÖÕ
×
Ö Õ ×
>
Û Ö:Õ ×Ü Ö)Ý Ø×
Û Ö Ü Ö>Õ × Ý^Þ ×ß
à Õ Û
Ö Õ × × Ý means whenever C holds then we get the payout of the insurance
policy
Û Ö>Ö:Õ Õ × Ý2Õ . à means whenever we have a policy then we get à .
Let us now reconsider the abduction problem
Ö:Õ ×‡Ø-Ù ×
for our strict implication. The available proof theories require modification because the
options so far considered yield classical logic. Indeed in the literature there are tableaux
systems (option 1) for strict implication, and there are resolution systems (option 2) for
strict implication; but they are too complex to describe here, and fortunately there is no
need. It is sufficient for the reader to realize that the abduction options depend on proof
procedures. For option 3 (goal directed) it is easy to see intuitively what needs to be
abduced in our example. We need to abduce , but we must be careful. We abduce
to be positioned after
(not before); otherwise we cannot prove . Again, think
of
as the insurance policy. I want to get ($100,000). How do I do it? I need
to have an accident after I get the policy, not before. To write this properly, we need to
adopt labelling as part of our logic and to abduce with a label indicating its position.
To achieve this uniformly we need to label the data
and label
and then abduce
. Our new database is
Ö:Õ ×
Ö>Õ ×
Ö
×
×
Ö
áâ Ö>Õ ×
ãâ Ö
áä:ã .
×
Since á)â ÖeÕ
ãâ Ö
ãâ Ö
is the original datum and
to refutation) we may double label the data as:
áâ Ö>Õ ×
Ö
ãâ ×Ü áä:ã
is abduced (and hence possibly open
13.1. SOME SCHEMATIC REMARKS
269
å :
æ çxè
é êæ
(original, ):
(abduced, )
.
åë:é
We see how easy it is to motivate moving into deductive systems involving structured labeled databases!
We can now take our insurance example a bit further. We hinted above that in order
for to obtain we need to assume that obtained after
.
Suppose we have an insurance assessor checking the truth of A. Is there a disability?, he asks. The assessor can take actions to try and refute —may he tempt
the policy holder to a football game and video it? The actions taken by the insurance
assessor will create an
such that
and our
and
yield a contradiction. The action has the form action=
.
is the precondition
for taking the action. Preconditions are required to be reasonable because we need to
have reasonable grounds for suspicion before we can be allowed to ‘trip-up’ the policyholder. If the precondition can be obtained then we take the action and if successful we
get
. Thus our system must be a logic with actions, labels, inconsistency and
abduction—and one more thing: a refutation/revision process.
goes into the database.
Our inconsistency component tells us it is inconsistent. We must know how to deal
with it. One possibility is to take
out! If so, we must be able to recognize this
option from the labels.
We summarize the logic components that are needed to deal with our simple insurance example:
è
æ
æ>çxè
æ
ìê í
éê æ
ìê í
é ë ì
î ì ïê íï ð ì ñê íñ ò ì ïê í ï
ì ñê^íñ
ì ñ@ê íñ
éê-æ
1. Data are labelled and structured, forming databases
2. A proof mechanism
åê æ
ô
ó
.4
is available for proving formulae
æ
å óöõ^÷
with labels .
3. A notion of inconsistency.
óø÷>åê=æ
óù
4. An abduction mechanism. Given
we can abduce several
such
that
. Note that we need to know how to add (input)
to
. (If and
are structured how do we combine them?) Note that in nonmonotonic logic and in general labelled deductive systems we may get
to
be provable by deleting from . So we must equally allow for some
such that
or a combination of add and delete. Thus ‘input’ could mean a
combination of add and delete.
ó
óú>óù)õ ÷Dåê=æ
ó
óù
ó
ó‡û óùoõå)ê æ
óù
ó ù
å@ê^æ
5. A notion of action along preconditions and postconditions which can be taken to
check/refute an abduced
óù
6. A revision process is needed to identify and take out refuted wffs in case of
inconsistency.
4 The most general notion of database allows data items to have procedural ‘patches’ which affect the
proof procedure once the item is used.
itself can operate in several modes and the ‘patches’ can shift
the mode. Component (8) below contains action as a mode.
ü
ü
CHAPTER 13. A GLIMPSE OF FORMALITY
270
7. A proof theory that takes account of actions.
8. We need one more component. Imagine in our example that the manager of the
insurance company asks the assessor whether a $100,000 ( ) is to be paid out
(i.e. he asks, does
?). Of course, the answer is yes (from the
above considerations). But what the manager is really asking is Have you (or
can you) take action to check/refute ? The assessor can say ‘Yes, my action
is available to refute (i.e. show
)’. We write the above as
,
reading: After action is taken,
follows.
ý
þÿ°ý Dý
13.2
þ
þ
2 þ
2þ
2þ
Case Study: Defeasible Logic
We briefly revisit the account of individual cognitive agency. An individual agent manages his affairs with straited resources: information, time and computational power.
Slightly over-stated, he doesn’t know enough, hasn’t time enough and hasn’t the wherewithal for figuring things out that would enable him to meet the performance standards
postulated by standard logics of rationality. Individuals prosecute their cognitive agendas on the cheap. They take short cuts and they assume the concomitant risks. Among
the economising measures of individual cognitive agents are what we have called their
scarce-resource compensation strategies. These include a disposition toward hasty generalisation, toward generic inference and the recognition of natural kinds. Also prominent is an inclination to stay with received opinion, and, in matters of new opinion, to
be satisfied with the say so of others.
It is time to give a case study for our individual (low-resource) reasoning agents, in
which we shall apply two of our low resource principles:
1. low resource individual agents perceive natural kinds around them
2. low resource individual agents are hasty generalisers of generic rules
and obtain immediately the well known and much studied Nute’s defeasible logic
Imagine our agent looking into the world around him and dividing it into certain
natural kind subsets. Let
be the predicate is of kind
and let
be logical
implication (of some known logic) then our reasoning will have rules of the form
þ
3.
4.
þ
þ þ þ
þ
x
þ þ
, where is a constant name
where
are natural kinds.
If we use the Prolog notation we can write (4) as
4*.
and view ( ) as an absolute (non-defeasible) rule.
For example
5. Penguin
6. Bird (Tweetie)
Bird
þ
13.2. CASE STUDY: DEFEASIBLE LOGIC
!#" $ % % %
& ' () ' () 271
Our reasoner also has some positive and negative property predicates
and
as well as a hierarchical understanding of them in the form of
rules
7.
,
or written in Prolog as
7*.
.
Again, (7) is an absolute non-defeasible rule.
Our agent is a hasty generaliser who deploys generic defeasible rules of the form
8.
*
' +)
For example
9. Bird
+ + *, 3 *, *, 3 (/ 3 - Fly
10. penguin
Fly
.
*, - ,*-0 +/ .12 4.5 6 '
+ + + Formally our agent’s logic has predicates
, absolute
rules of the form
, defeasible rules of the form
and facts
of the form
an individual name.
Here is a well known sample database
, are used to name the data items and
rules:
4.5
666 855 777
6 6 9 : 77
Bird
Fly
Penguin
Fly
Penguin
Bird
Bird(Tweetie)
Penguin(Tweetie)
can prove both that Tweetie flies and that Tweetie does not fly. Defeasible logic
allows certain proofs to defeat other roofs, and is thus an example of a genuine nonmonotonic logic. The way in which this system one proof can defeat another proof
is that it is based on a more specific body of facts than the other proof. The proof of
Fly(Tweetie) is based on . The proof of Fly(Tweetie) is based on . Item is
more specific than item ( can prove using absolute rules only). Therefore
overall proves that Tweetie does not fly.
Consider
.
can still prove that Tweetie Flies, as well as that
Tweetie does not Fly. How can we show now that overall,
proves that Tweetie does
not Fly?
We know that Fly is based on the more specific information that Tweetie is a
penguin and this information is more specific because of Penguin
Bird. However,
we have a technical problem of how to bring this out formally in the system?
Let us try making use of the labels. We have
66 9 9 6 9
6:
4,;.!<4.5=?> 6 9 @ 4A;
4A;CBD E F D G F D H
Fly(Tweetie)
6:
4,;
(
6 : 4-5
CHAPTER 13. A GLIMPSE OF FORMALITY
272
I,JCKL M N L O P
QSRUT V W X V Y X V Z [ \SQ RUT V W X\ V ] [ IAJ ^ Q^R_T V W X V Z [ `\ RaT V Z [ ^ QK ^ ^ \^
K
^ Q^ ^ \^
^ \^ ^ Q^
and
Fly (Tweetie)
Let
and
.
The absolute parts of and are
and
.
and
are
equivalent under the absolute part of
and taken as they are, we get
and not
, as we would have liked.
Obviously we have a technical problem of how to formalize the intuition that Penguin is more specific than Bird.
Let us look at the problem slightly differently.
Tweetie Flies because it is a Bird and says that Birds Fly. Tweetie does not Fly
because it is a Penguin and says that Penguins don’t Fly. Never mind how we show
that Tweetie is a Bird and how we show that Tweetie is a Penguin. The fact is that in
, Penguin is more specific than Birds.
So let us view the labels
as sequences of proof steps. We do labelled modus
ponens as follows:
VZ
V]
IAJ
g
where ‘ ’ can be either ‘
Thus we get
QbX \ c d e,f d eUg)h
i c XV V j d h
k I,JCKnl m m L L L
I,I,JCJCKK mm LL M LL N o p N q p
I,JCKL M MM NN oO pp P
’ or ‘
’.
Fly(Tweetie)
Bird(Tweetie)
Fly(Tweetie)
Penguin(Tweetie)
Now we can strip the last label (if it is a defeasible rule) and get the fact last proved
and compare which fact is more specific.
To do this properly we need the machinery of Labelled Deductive Systems. We shall
study this example in detail in a subsequent chapter. (See also our Agenda Relevance,
chapter 13.) Abduction in defeasible logic.
To give the reader an idea of what is involved, note that a database can prove
and
in many ways. This means that there are many labels
such that
and
. The process in labelled deductive systems which decides
whether overall we say
or we say
or neither is called Flattening. This
enables us to define a definite for our case here.
We mentioned that we might not be able to decide whether or
follows. This
happens in our case if the facts are incomparable. Consider
eI)Kt P e e )I Kv e
u
w P I<K e K
e izj h izj
P y izj
We have neither
nor
nor can we conclude
.
I
I<K P e
eh i x jl)y
eh ii zx j jl P
izj
i x ij x
Qr X \s
e Pe
y j
stronger than the other. So we cannot conclude
y izj
13.3. CASE STUDY: CIRCUMSCRIPTION
273
13.3 Case Study: Circumscription
We revisit the Tweetie example. Suppose we factually get the information that Tweetie
does fly. How can we explain it? We have to abduce. We can add Fly(Tweetie) to the
database as a fact and this will defeat Fly(Tweetie) because the latter is based on a
defeasible rule, but this is not explaining. If we think about it intuitively we ‘explain’ by
saying, using our ability to perceive natural kinds, that Tweetie belongs to a subspecies
of Penguins that do indeed Fly. Say Tweetie is an -Penguin.
Thus we add
-Penguin
Penguin
-Penguin
Fly
-Penguin(Tweetie)
{
}} …~ €n‚ | ‚ ƒ‚† ƒ„ ‚| ƒ ‚ ƒ ƒ
} ‡  ||
‚ ƒ„)Š-‚ ƒ
ˆ ‚ ƒn‰
‚ ƒn‰
‚ ƒ„ { ‚ ƒ
‚ ƒ
‹
This reminds us of John McCarthy’s way of writing defeasible rules
not abnormal
we will have
Penguin
not abnormal
Doing circumscription will minimise
is in the data it will contain Tweetie.
Summary of Results
Abnormal
Fly
to be empty. But if Fly(Tweetie)
274
CHAPTER 13. A GLIMPSE OF FORMALITY
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