The basic philosophical and semantical theory
Richard Routley with Robert K. Meyer
Val Plumwood and Ross T. Brady
1.0 IMPLICATION
THE FUNDAMENTAL LOGICAL NOTION
CHAPTER 1
THE IMPLICATION CONNECTION3 AND THE ENSUING INADEQUACY OF IRRELEVANT LOGICS
SUCH AS CLASSICAL AND MODAL LOGICS
Implication, the main relation studied in this work, is fundamental
in reasoning, particularly in deductive reasoning.
Hence its central
importance in philosophy, logic, and mathematics, where such notions as
entailment and valid argument are central.
The importance of implication
in mathematics, if not entirely obvious, emerges clearly from Russell's
initial definition of pure mathematics (37, p.3):
Pure mathematics is the class of all propositions of
the form "p implies q", where p and q are propositions
containing one or more variables, the same in the two
propositions, and neither p nor q contains any constants
except logical constants.
The definition - whatever its defects, and Russell himself was later to argue
that they are manifold - at least succeeds in revealing the fundamental role
of the implication connection in mathematics (and incidentally, in the sharing
of variables requirement, the importance in mathematics of relevant connections).
Implication is likewise basic in logic, where it (along with its derivatives
such as consequence) is one of the most important relations to be explicated:
... implication seems to be the most important connective
in any logical system (Rasiowa 74, p.167).
It is, moreover, sometimes said to account for the importance of logic:
The chief importance of logic lies in implication ...
(Quine 51, p.xvi).
Lastly, but first in importance in this group, philosophy: since the
essence of philosophy, and especially of dialectics, lies in argumentation
and reasoning, implication is once again a crucial connection. It is fundamental in the formalisation of philosophical discourse and arguments, where
key features of concern are conditionality (represented by 'if ... then ...'),
entailment, and valid argument.
None of the main going systems are at all
adequate for these purposes.
New systematisation is essential, not only for
adequate accounts of implication and valid arguments, but also for derivative
reasons.
A great many important philosophical notions depend, in one way or
another, on satisfactory implication relations, for their semantical analysis,
for a satisfactory explication; for example such diverse notions as aboutness, empiricalness, moral commitment, evaluativeness, semantic information,
degree of conclusiveness of arguments, confirmation and rationality. (Exactly
how improved philosophical explications can be provided, which avoid many
standard difficulties, is explained in detail in BP and, for certain cases,
in UL.)
On the whole there has been far too much effort expended on trying to
accommodate philosophical clarifications to going logical systems, and
especially to prevailing classical-style accounts of implication and valid
argument - rather than on trying to develop logical systems to handle the
evident data and to deal with going philosophical problems.
Thus prevailing
Classical-style logics, and classical logic in particular, have distorted
1
1.1 RIVAL POSITIONS AS TO IMPLICATION
the initially-obvious resolutions of a number of philosophical problems,
and by their distortions generated several gratuitous philosophical
problems (see UL). Classical logic, although once and briefly an
instrument of liberation and clarification in philosophy and mathematics,
has, in becoming entrenched, become rigid, resistant to change and highly
conservative, and so has become an oppressive and stultifying influence.
For instance, classical logic has exerted a very conservative influence
on philosophical problem-solving, especially paradox resolution, and has
hamstrung much enterprise.
Classical logic is, as now enforced, a
reactionary doctrine.
Nowhere is the oppressive effect of classical
logic more evident than in its treatment of implication.
§1. The main vixial positions on isnvliaation. In view of the philosophical
and logical importance of implication and its derivatives, it is hardly
surprising that questions as to the properties of implication have been
a major source of dispute, at least from Greek times, through the Middle
Ages, and into modern times.
In the modern dispute (which, surprisingly,
had a main point of origin at Harvard: see Parry 74) several main schools
can be discerned, many of which have their roots in ancient or scholastic positions.
Briefly the positions are as follows (they will be
characterised in more detail later, when variations and other positions
will also be distinguished):
(1) The PhiIonian position (held by hookers) according to
which implication is adequately (enough) captured by the
material conditional of classical logic.
(2) Metalinguistic positions which claim that implication
is a metalogical relation not to be confused with the
material conditional.
Both these first positions adhere to extensional languages, and depend
ultimately on the mistaken assumption that the extensional way of doing
things is the only clear and unmuddled way of proceeding.
(3) Strict positions (maintained by stricters, i.e. strict
hookers) according to which logical implication at least is
to be explicated through some brand of (C.I. Lewis or
E.J. Lemmon) strict implication.
Though positions under (3) are non-extensional they are closely connected
with those under (2) (which simulate intensionality), the divergence
beginning only with issues as to the iteration of intensional connectives
and their combination with quantifiers.
(4) Positive, intuitionistic and constructive positions
which characteristically base their theory on some
conservative extension of Hilbert's positive logic.
(5) Similarity theories (like those of Stalnaker and
D. Lewis, developing Ramsey's position) according to which
an important class of conditionals, at least, is to be
explicated semantically through similarity relations on the
possible worlds supplied in furnishing semantics for (3).
1.1 RELEVANT, C0NNEX1V1ST AMD C0NCEVTTV1ST
THEORIES
(6) Relevant theories such as those of Ackermann, and
of Anderson and Belnap, which reject the paradoxes of
implication and, most characteristically, the classical
principles of Disjunctive Syllogism, i.e. in symbols to
be explained shortly, A &(~A v B)
B, and Antilogism,
if A & B -* C then A & ~C ->• ~B.
Such relevant positions
will be the main theories investigated and defended in
the text.
(7) Connexivist positions which reject or qualify such
principles of classical logic as Conjunctive Simplification,
A & B -»• B in favour of theses which imply Aristotle's
principle, ~(A
~A) .
(8) Conceptivist theories, developing from Parry's theory
of analytic implication, which qualify Addition, A -*• (A v B) ,
and more generally require that when A -»• B holds the concepts
of B (construed, e.g., in terms of the variables of B in the
case of sentential logic) are included in those of A.
The positions sketched are not exhaustive. There is as well a range
of patch-up positions which try to define a new connective for implication
in terms of the rest of apparatus supplied by (3), there are other positions
which try to characterise conditionals in probabilistic terms, and so on.
None of the first five positions is satisfactory.
For no implication
(or conditional) is going to be adequate for the analysis of natural language
and the logic of discourse, or adequate for most philosophical purposes, which
is not relevant, in which the consequent of a true implication is not somehow
connected with the antecedent, or the antecedent relevant to the consequent.1
A necessary, but not sufficient, formal condition for such relevance at the
sentential level is furnished by Belnap's weak relevance criterion:
WR. That A implies that B is a theorem, in symbols
A -*• B, only if A and
B share a sentential variable, where A and B are well-formed formulae built
up from sentential variables p, q, r, p', ... using just sentential connectives
such as -»-, & ('and'), v ('or'), ~ ('not'), etc.2
It is immediate, then, that logics like strict, positive and similarity
theories which include as theorems paradoxes of implication (such as, in the
first cases, p & ~p
q, q -»• (p -»• p) and q -»-.p v ~p) and so violate WR, are
irrelevant, and hence are inadequate for important purposes.
The emphasis
in the formal work of this volume is on relevant logics: irrelevant logics
on which formal investigations of implication have in the past been excessively
concentrated will be studied only insofar as they appear as extensions or
by-products of relevant systems.
1
Though weak relevance is not a fundamental matter for entailment (see FD
and BP) but a derivative feature of a good sufficiency relation, it
provides an extremely important formal test of adequacy.
2
There is accumulating linguistic evidence that natural language implications
and conditionals are relevant, see e.g. T. van Dijk 74. In fact van Dijk
goes further and advances the interesting, if false, thesis that all
natural language connectives are relevant.
3
1.1
STRATEGIES IN
THE
FACE OF 1MPLJCATJONAL
PARADOX.
There are at least as many sorts of relevant sentential logics as there
are resolutions of C. I. Lewis's "independent proof" of the paradoxical
scheme A & ~A -*• B.
The argument for this scheme - which has always seemed
the hardest obstacle proponents of relevant logics have to surmount - is as
follows:(1)
(2)
(3)
(4)
(5)
(6)
(7)
A & ~A
A
A & ~A
~A
~A -»• ~A v B
A & ~A -*• ~A v B
A & ~A
A & (~A V B)
A & (~A v B) + B
A & ~A -»- B
The distinctive principles used in this argument are:Conjunctive Simplification: A & B -»• A and A & B -*• B, used in lines (1) and (2).
Addition: C -* C v B, used in line (3).
Disjunctive Syllogism: A & (~A v B) -»• B, used in line (6).
Rule Syllogism: A
B, B ->- C
A
C, i.e. where A -»- B and B
C are theorems
(i.e. provable) so is A -»• C, applied in obtaining lines (4) and (7).
Rule Composition: A
B, A -»• C
A
(B & C ) , applied in obtaining line (5).
Alternatively
Rule Factor: A - * B - * C & A - > - . C & B, may be applied to obtain line (5) from
line (3) thereby eliminating lines (1) and (4) and use of Rule Composition.
Strategies abandoning each of the first four (uneliminated) principles have
been investigated to varying extents, and rather evidently (given recent semantical
insights) further strategies could be devised which avoid the paradox but
give up less than tried moves do. The main tried strategies, each of which
can lead to weakly relevant logics, are these:Connexivism, which rejects Conjunctive Simplification.
Since the position
keeps classical connections between &- and v-principles, it is also obliged
to reject Adjunction. However by weakening contraposition principles connexivism could retain Addition. Connexivism, apparently found in Aristotle
and Boethius and rampant among the Stoics, was reintroduced in modern times
by Nelson and developed by Angell and McCall.
That the position has attained
a new popularity can be seen from the number of recent journal articles on
implication espousing connexivist principles (see further 2.).
Conceptivism, which retains all the principles employed except Adjunction.
Obviously since the position retains Conjunctive Simplification it has to
weaken contraposition principles.
Conceptivism, as so far developed,
through the so-called logics of analytic implication, satisfies an (over-)
strong relevance condition, namely:
SR. |-A -*• B only if all sentential variables occurring in B occur in A.
Analytic implication, due to Parry, is supposed to explicate a Kantian
insight.
The logical theory has been elaborated by Dunn, Urquhart, Fine
and others.
Non-transitivism, which rejects only Rule Syllogism, blocks the move from
(5) and (6) to (7).
This desperate strategy, proposed by von Wright, has been
elaborated and defended by Geach. It has also found favour with Prior 67a.
Relevant positions, which reject only Disjunctive Syllogism, and which
retain full contraposition principles.
This strategy, which appears to
have no historical roots, is discernible in Duncan-Jones and was applied by
Hallden. But it owes its clear and formal development to Ackermann and to
Anderson and Belnap, the earliest work dating back to 1956 only.
However
the strategy of defeating the paradox argument by rejecting just Disjunctive
4
7.2 MATERIAL-IMPLICATION, ITS INADEQUACIES
Syllogism was adopted (implicitly) in the minimal logic of Kolmogorov and of
Johannson, which goes back to 1926.
Minimal logic rejects lines (6) and
(7), in effect by throwing out the principle A
B, that a false (or absurd)
statement implies anything, as can be seen upon using Kolmogorov's definition
of negation: ~A =
A ->• A > which converts (7), e.g., to A & (A -»• A ) -*• B.
Sadly "minimal" logic made the mistake of building on Hilbert's positive
logic, and thereby was automatically committed to the paradox that anything
implies a true statement, i.e. B
A for any detachable A, and hence to
weakening contraposition principles in order to save its insights concerning
negative paradoxes.
(The correct course, which minimal logic could historically have taken, would have been to correspondingly weaken Hilbert's positive
logic, as Church went on to do.)
We have reserved the fashionable label 'relevant logics' for logics which
are paradox-free and which reject Disjunctive Syllogism (and its mates, such
as Antilogism) , though these logics are by no means the only ones that satisfy
formal relevance criteria (or for that matter the only ones that satisfy criteria such as those of relevance, necessity, and strength which were supposed
to distinguish the system E of Anderson-Belnap entailment).
The case for
concentrating on relevant logics among the various positions meeting the condition WR - and as to what is wrong with the alternatives - will be picked
up subsequently (in chapter 3).
Here we begin on making out the case for
studying unorthodox and little known logical systems in preference to such
well-marketed alternatives as classical and strict logics - essentially the
well-marketed goods are shoddy and won't perform many of the tasks they are
supposed to.
§2. The inadequacy of material-implication. Since the implication connection
is fundamental, any adequate logic should not fail outright in capturing its
leading properties: classical and modal logics in particular flunk the test,
and accordingly have to be condemned as inadequate.
To begin, the prevailing logic, classical logic (i.e. two valued sentential logic and its quantificational and set-theoretical extensions) offers,
as is well-known, only pathetically inadequate accounts of implication and
of conditionality.
Material-implication or hook, symbolised '=>', has only
a slightly better claim to representing a genuine implication than materialequivalence, '='.
C.I. Lewis, in his long campaign against material-implication as an account of logical implication, i.e. entailment or the converse
of systemic deducibility, has explained in detail why material-implication
gives nothing like an adequate account of logical implication (see the Lewis
references cited). What Lewis did not realise (and would have disputed,
cf. 18 pp.224-6) - something the counterexamples to be presented bring out is that material-implication fails almost as dismally as an account of conditionality.
There are at leaist three statement relations that an adequate theory of
implication and conditionality has to explicate, entailment, law-like implication, and conditionality,1 and material-implication fails for every one of
them.
The most striking manifestations of breakdown are of course the para1
Subsequently we shall find that there are more than three.
The conflation
of entailment and the conditional would involve that piece of Spinozist
rationalism, according to which every truth is a necessary truth, which
Lewis in 14 thought he detected in PM and was concerned to argue against.
5
1.2 COUNTEREXAMPLES TO
MATERIAL
PRINCIPLES
doxes of implication and conditionality.
These not only lead to amazing transgressions of relevance, validating such entailments as "Canberra is located at
the North Pole -*• 2 + 22 = 24" and collapsing theories containing any falsehood, i.e. most theories, to triviality; but they also validate an enormous
range of relevant implications and conditionals which are intuitively invalid,
e.g. the following examples (from Adams 65, p.166 and Stevenson 70, p.28):
John will arrive on the 10 o'clock plane.
Therefore,
if John does not arrive on the 10 o'clock plane, he will
arrive on the 11 o'clock plane (illustrating A entails
that if ~A then B).
John will arrive on the 10 o'clock plane.
Therefore, if John
misses his plane in New York, he will arrive on the 10 o'clock plane.
If he dies tonight he will visit us tomorrow; because he will
visit us tomorrow. (The latter examples both illustrate B entails
that if A then B.)
But it is not simply that material-implication, construed as an implication,
admits of such paradoxes as that a false statement implies any statement at
all - which ruins any direct application of the logic to the investigation
of false hypotheses - but that there is a large class of principles that
hold for material-implication but fail for implication and conditionality.
Consider, for example, the following set of arguments which would be
valid if implication were material-implication, if 'if... then...' were adequately rendered by '
(most of these examples are assembled in Hunter 72).
As Hunter remarks, none of these arguments is valid, for each either has, or
could easily have, true premisses but a false conclusion.
(a) (Switches paradox): If you throw both switch A and switch B then the
motor will start.
Therefore, either if you throw switch A the motor will
start, or if you throw switch B the motor will start.
That is, throwing
one switch suffices: joint sufficiency is eroded to separate sufficiency.
The material principle on which this paradox - which may be illustrated by
any case where joint premisses are both used in the argument - rests is the
following:
A & B 3 C.
Therefore (A => C) v (B => C).
(b) It is not true that if she is over forty she is still young.
if she is still young she is over forty.
Therefore
~(A ^ B). :. (B d A).
(c) If he won't propose to her unless he finds that she's wealthy, then he's
mercenary.
He will find that she's wealthy, but won't propose to her.
So
he's mercenary.
((~A a ~B) ^ C) & A & ~B.
So C.
(d) If this figure is rectangular and equal sided it is a square.
Therefore,
EITHER if this figure is rectangular and not equal sided it is a square, OR
if this figure is not rectangular and is equal sided it is a square.
(A & B) => C.
Therefore [(A & ~B) => C] v [ (~A & B) ^ C].
(e) If John is in Paris, then he is in France.
If he is in Istanbul, then
he is in Turkey.
Therefore, if John is in Paris he is in Turkey, or if he
is in Istanbul he is in France.
(A ^ B) & (C
d
D).
So (A ^ D) v (C ^ B).
6
7.2 FURTHER ELEMENTARY COUNTEREXAMPLES TO CLASSICAL
PRINCIPLES
(f) If Jones comes from Georgia, then he is a southerner.
Therefore, if
Jones rides a bicycle to work he is a southerner, or if he comes from
Georgia he owns a bicycle.
A 3 B.
Therefore (C => B) v (A => D) .
(g) It is not the case that if we follow this road we shall reach the city.
Therefore we shall not reach the city.
~(A = B).
So ~B.
(h) If Goldbach's conjecture is correct, then it is false that if the mayor's
telephone number is an even number, it cannot be represented as the sum of
two primes.
Therefore, if the mayor's telephone number is not an even
number, Goldbach's conjecture is not correct.
A 3 ~(B 3 C).
Therefore ~B => ~A.
(i) It is false that if we can sensibly assert that we are sound asleep then
Malcolm on Dreaming is correct in all his contentions; therefore it follows
that we can sensibly assert that we are sound asleep.
~(A ^ B).
Therefore A.
Consider also (j):- The tax-collector says: 'If you made a mistake in your
income-tax return and owe the government money, you made the mistake deliberately, with the intent to cheat the government.' If 'if' is equivalent to
'3', then I cannot deny this allegation without admitting that I made a mistake and owe the government money.
(g) and (i) may be combined as in the following example of Stevenson (70,
p.28): This is false: if God exists then the prayers of evil men will be
answered.
So we may conclude that God exists, and (as a bonus) that the
prayers will not be answered.
~(A 3 B). .*. A & ~B.
(k) Not both Hunter is a bachelor and Hunter is not married;
Hunter is a bachelor then Hunter is married.
~(A & ~B) .
therefore if
A 3 B.
(1) Either Dr. A or Dr. B will attend the patient.
Dr. B will not attend
the patient.
Therefore, if Dr. A does not attend the patient Dr. B will.
(A v B) & ~B.
~A 3 B.
(m) If John will graduate only if he passes history, then he won't graduate.
Therefore if John passes history he won't graduate.
(A 3 B) 3 ~A.
B 3 ~A.
(n) It isn't true that if he breaks a mirror he will have bad luck.
he doesn't break a mirror he will have bad luck.
~(A 3 B).
So if
~A 3 B.
(o) If Albert's age is greater than twenty and less than twenty-three, then
Albert is either twenty-one or twenty-two.
Therefore if Albert's age is
greater than twenty then Albert is twenty-two, or if his age is less than
twenty-three then he is twenty-one.
A & B 3. c v D.
(A 3 D) v (B 3 D).
(p) He is a former communist. So if he is smiling he has a bomb in his
pocket or else if he has a bomb in his pocket he is smiling.
7
7.2 T H E METHOV OF INTUITIVE
A.
COUNTEREXAMPLES VEFENVEV
(B 3 C) v (C 3 B).
Worse still, the premiss of (p) can be omitted.1
Several of these counterexamples - which destroy any claim materialimplication may have had to credibility either as a conditional or as a general
implication (upon substituting 'implies' for 'if...then') or as an explication
of entailment - also apply, after but minor modification, against other accounts
of implication, e.g. against many-valued implications, against positive and
intuitionistic implication, and against several of the numerous syntactical
adjustments of material-implication that have been rigged up in an effort to
avoid some of the worst features of material-implication.
Although we shall make heavy use of the method of presenting intuitive
counterexamples in establishing the bounds on adequate theories of main topics
we investigate, the method has to be handled with considerable caution and is
far from foolproof (as Simons 65, splendidly, if somewhat unintentionally,
reveals).
Since we are trying to logically capture unformalised notions,2
the method is unavoidable; and intuitive (or preanalytic) countercases form
an important part of the data to be taken account of in designing logical
theories.
Major problems for the use of intuitive counterexamples are caused
by the common occurrence of enthymematic reasoning, where understood or presupposed assumptions or premisses are omitted, and by the common use of arguments which are correct for a restricted class of statements or in restricted
classes of situations and contexts or in a presupposed uniform context.
Our
methodological strategy in the face of these problems is to set our sights on
obtaining first of all accounts of non-enthymematic argumentation which is not
statementally or situationally restricted, but which is universally valid, that
is correct for all (reasoning) situations.3
While this strategy enables us to avoid the problems mentioned, other
problems remain, and the strategy has an important bearing on the use of
examples in trying to establish principles as correct.
For the asymmetry
between falsification and verification - already present since correct principles as universally true are simple universal statements - is accentuated,
because in establishing that a principle holds good it has to be contended on
the basis of intuition that all examples of it hold in an appropriate nonenthymematic and contextually unrestricted way.
But intuition but seldom
provides such general information.
Defences that have been given of the
connexive principle, if A -*• ~B then ~(A -*• B) (one form of the principle
called Boethius), offer a good illustration of failure to meet the methodological requirement that the principle must hold for all statements.
A
common procedure (used, for example by Cooper 68 and Stevenson 70) is to cite
1
The examples have the following sources: (a), (1), (m) Adams 65; (c), (d),
(n), (p) Stevenson 70; (b), (e)-(h), (o) Cooper 68; (i), (j) J. Nelson 66;
(k) Hunter 72. Hunter goes on from the examples to neatly demolish various
defences of the interderivability of If A then B and A ^ B.
2
This clause illustrates nicely a virtue of the split infinitive; for our
original sentence 'we are trying to capture logically unformalised notions'
contained an unfortunate ambiguity.
3
We are left of course with the residual problem of trying to demarcate
reasoning or deductive situations; see UL.
8
7.2 MINOR PROBLEMS WITH SURFACE FORMS OF
ARGUMENTS
one or two instances of the principle and to conclude on the basis of the
acceptability of the instances of that sort that the general principle is
correct, e.g. Boethius is vindicated on the basis of cases like that where
A is 'This is gold' and B is 'This is soluble'.
But the cases considered,
and perhaps envisaged, are ones where both A and B are contingent statements;
so the examples establish no more than what stricters, for instance, would
readily concede, that Boethius is correct where A is possible.
That is,
the examples only vindicate Boethius for a restricted class of statements,
those where A is assumed to be possible; they do not vindicate it generally,
as is required for validity.1
A residual problem with the use of intuitive counterexamples in confining
valid argument arises from omissions, truncations, ambiguities, and contextual
variation in natural language discourse from which the examples are drawn.
The omission of an intended particle such as 'even' or 'still', sanctioned
in English, can have a drastic effect on the validity of an argument, because
meaning is altered: thus the surface form of an argument cannot be relied
upon in a completely unqualified way.2 This is an ancient point, backed up
by examples of invalid syllogisms with ambiguous middle terms and by examples
of the folly of taking surface structure at its face value (e.g. examples
forcing a distinction between the 'is' of predication and that of identity*
such as: The class of Apostles is twelve and twelve is a number so the class
of Apostles is a number); but it is still a serious point of contemporary
relevance.
For instance, some counterexamples that have been presented to
transitivity of the conditional are less than conclusive because they can be
written off as having an ambiguous middle statement.
One of Stevenson's
examples is of this sort, namely the argument: If I pass her I'll be ashamed
of myself.
If she gets down to work I'll pass her.
So if she £ets down
to work I'll be ashamed of myself - the accusation being that 'I pass her'
means 'I pass her under the present circumstances' in the first premiss but
not in the second, something reflected by the shift from 'I' to 'I'll'.
Stalnaker's famous example (in 68) is also of the sort, the argument being:
If J. Edgar Hoover were today a communist then he would be a traitor.
If
J. Edgar Hoover had been born a Russian then he would today be a communist.
Therefore if J. Edgar Hoover had been born a Russian he would be a traitor.
The argument concerns only conditionality and not of course law-like implication or entailment: for being born in Russia is hardly sufficient for
later being a communist.
The allegation is that for the premisses to be
true 'communist' has to be read differently in each premiss, roughly speaking
as 'Russian communist' and 'American communist'.
The retort might be that
it suffices to take 'communist' as meaning 'communist of some sort'; but
the further allegation would be that either this takes us to a traditional
fallacy of a particularly quantified middle term or else the premisses are
no longer both true.
In any case, the case is stalemated, and the direct
intuitive counterexample lost.
It is more difficult to apply this technique
against examples like the following provided by Adams (65, p.166): If Brown
wins the election, Smith will retire to private life.
If Smith dies before
the election, Brown will win it.
Therefore, if Smith dies before the
election, then he will retire to private life.
For here at least there
appears to be no ambiguous middle term to fall back upon.
What can be said
to have happened however is that the context is changed from the first premiss
to the remaining premiss and the conclusion, and that the antecedent of the
X
Practically all of Cooper's examples of classically invalid arguments which
are intuitively acceptable (68, pp.298-9) can be faulted on this ground.
2
For the most part, however, deviation from surface structure should be
minimised.
9
1.2 A CLASSICAL
RESCUE
ATTEMPT:
THE
DEFINITIONAL RETREAT
latter two conditionals undermines the grounds for accepting the first.1
Adams himself makes a not dissimilar but very general and important point,
that
classical logic cannot be safely used in inferences whose
conclusions are conditionals whose antecedents are incompatible
with the premisses in the sense that if the antecedents became
known, some of the previously asserted premisses would have to
be withdrawn.
Whether such conventionalist stratagems as those designed to defend Conjunctive
Syllogism succeed or not - we maintain they should be avoided as far as
possible in obtaining an applicable theory - they are enough to indicate that
the method of counterexamples is fraught with difficulties.
No simple rescue operation, of the sort just sketched (we do not need to claim
that it succeeded) for transitivity of the conditional, can be successfully
mounted on behalf of material-implication.
No removal of ambiguities, restoration of context, or change of syntax to reveal intended structure, is going
to save material-implication against damaging intuitive counterexamples.
There are, however, other strategies that can, and have, been tried in retaining the entrenched logic's theory of "implication".
What these strategies
have in common is that they all try to weaken in one way or another the
account taken of intuition in delivering counterexamples.
A favourite strategy is what might be dubbed the "definitional retreat", according to which
none of the alleged counterexamples are such when material-implication is
definitionally expanded (e.g. in terms of negation and conjunction or negation
and disjunction).
The move means abandoning the claim that material-implication explicates implication or conditionality - and so concedes the main
point at issue - but this some of its exponents are prepared to do, substituting only such claims as that it is safe, reliable, is all that is required
for the deductive sciences and provides the core of any correct account of
conditionality and entailment (contentions which we try to show are false,
beginning in 1.6).
Other defenders of material-implication are less happy
to abandon the thesis that it provides some sort of explication of the conditional that works for more than isolated cases.
The main alternative strategy has been to claim that intuition is not a
reliable guide, that it is corrigible and sometimes misguided or confused
(points we willingly concede), and that it can be shown, or even conclusively
proved, on the basis of principles that intuition itself supplies, that
material-implication answers exactly to the ordinary conditional.
It is
argued, for example (Simons 65, p.82), that while intuition may supply a set
of principles it cannot ascertain what all the consequences of those principles
are; and among the consequences of intuitive deliverances concerning 'if...
then' are that this connective is none other than the truth-functional conditional.
But, firstly, intuition characteristically rejects cases as well as
accepting cases: indeed intuition tends to work at the level of particular
cases as opposed to that of general principles, so rejection of universal
l
Objections of this general type, that the assumed context or background is
changed from premiss to conclusion, can also be made against Stevenson's
other countercase to Conjunctive Syllogism, to D. Lewis's countercases
(in 73) and to Bennett's countercase (in 74): see further p.46.
10
1.2 INTUITION
VOES NOT SUPPORT THE PARADOXES O F
IMPLICATION
principles is more central than acceptance.
It may happen, as a result,
that intuition occasionally yields contradictory statements: consider, in
particular logical and semantical antinomies.
In the case of implication
and conditionality, however, neither uncorrupted nor refined intuition yields,
or need yield, inconsistent results.
Only if the logically uncorrupted are
gulled into accepting general principles (such as Exportation) which extend
far beyond the reaches of intuition, are the unwary led, or bullied, into
both rejecting and accepting paradoxes of material-implication as logical
features of implication or the conditional.1
The procedure of Simons and others is to represent intuition as accepting
the premisses of independent arguments for the paradoxes (in Simons' case
just variants of the arguments Lewis had considered); and then, naturally,
it is a straightforward matter to demonstrate that ordinary 'if...then' is
really, surprising as the paradoxes and other counterexamples may be, material—
implication.
Unfortunately for Simons the arguments used in his 'fundamental
observation' (65, p.79) - 'that arguments intuitively acceptable taken one by
one collaborate to entail that the conditional which appears in them is truth
functional' - do not rest on generally intuitively acceptable principles,
whatever he may claim.
The main arguments turn on the alleged intuitive
acceptability of a version of Exportation: if A and B entail C then A entails
that if B then C.
But since a paradox of conditionality is an immediate
upshot of this principle using Simplification and Detachment, i.e. it is
immediate that A entails that if B then A, the principle can hardly claim
intuitive acceptability.
A little more generally, identify C with A in the
Exportation,let A be some true statement, and choose B as irrelevant to A, or
in fact as any statement such that "if B then A" is false.
For example,
take A as 'There are still a few tigers in India' and B as 'India is an
island'.
In this way the principle is intuitively falsified.
And that is
not all that is wrong with Simons' "fundamental observation".
For the exportation principle, which is smuggled in as intuitive conditional proof, is
applied to a version of Disjunctive Syllogism, ~(A & ~B) & A -*• B (a principle
flawed in 2.9) to yield the principle that ~(A & ~B) entails that if A then
B, an argument form already intuitively faulted as invalid under example (k)
above.
A less devious attempt - this time to prove that A ^ B is equivalent to
"if A then B", where => is the material-conditional - can be found in C.I.
Lewis (18, p.226).
Lewis's proof is as follows (we set it down verbatim, in
our notation however, omitting only references back to already proven or
assumed premisses, in particular to the crucial postulate that p • (p = 1)
for every proposition p):
A 3
And
B =
B =
B gives "If A - 1, then B • 1", and hence "if A then B".
"If A, then B" gives A = B, for it gives "If A - 1, then
1", and (a) Suppose as a fact A - 1.
Then, by hypothesis
1, and A => B. (b) Suppose that A ^ 1.
Then A • 0, and A => B.
l
The situation with respect to the paradoxes of strict-implication is less
straightforward owing to the widespread (though far from universal)
intuitive acceptance of Disjunctive Syllogism.
The question of the intuitive acceptability of Disjunctive Syllogism, Lewy's charge that the (?)
Intuitive concept of entailment is inconsistent, and the general question
of the role of intuition in settling the (?) logic of entailment, are taken
up in chapters 2, and 4, especially in 2.9 ff.
11
7.2 "PROOFS" THAT I F . . . T H E N IS TANTAMONT TO HOOK
The main trouble with this "proof" may be located in the very first step.
For A => B does not give "If A = 1 then B = 1", i.e. in effect "if A is true
then B is true", unless the unformalised (metalinguistic) 'if-then' is already
assumed to have properties of the material-conditional which are in question.
Lewis has, in effect, assumed the already rejected principle that ~(A & ~B)
gives "if A [is true] then B [is true]", when all he is entitled to from the
truth-table for => is the principle that ~(A & ~B) gives "either A is not true
or B is true" where the 'either-or' is extensional inclusive disjunction.
Thus Lewis has established little more than that A = B is equivalent to "If A
then B" where 'if-then' is understood as the material-implication.
This is
hardly news, and it assumes the point at issue as far as linking materialimplication with 'if-then' is concerned.
A more subtle attempt to establish that A
B and "if A then B" are
genuinely interderivable (i.e. entail each other) rests on the assumption
that conditionality is syntactically enthymematic, that is (in modal form)
(a) a necessary and sufficient condition for the truth of
"if A then B" is (Faris's condition E) that there is a set
S of true propositions such that B is deducible from A together with S.
Thesis (a) is defended in Faris 62 (pp.117-8), and is the basis of his argument that "If A then B" is derivable from A = B.
In fact (a) could also
have served as a basis of a proof that A = B is derivable from "If A then B",
a claim which Faris takes as generally conceded and hardly in need of proof.
For (apparently unknown to Faris) Myhill 53 proved, in effect, that where
(8) "If A then B" is defined as, or taken as logical equivalent
to: for some true propositions p, A and p strictly imply B,
i.e. in symbolism (Pp)(p &. A & p
B),
then A => B and "if A then B" are indeed interderivable.
The proof depends
essentially on certain (but not all) properties of strict implication, in
particular, as Faris brings out in his informal argument (p.117), on the
principle of Disjunctive Syllogism, that B is deducible from A and A
B.
That Disjunctive Syllogism is no correct principle of entailment is a main
negative thesis of this text: but that is not enough to dispose of the interderivability thesis.
For it would be sufficiently embarrassing if A => B
and "if A then B" were strictly equivalent; and more important, the use of
Disjunctive Syllogism can be removed by way of the following relevant characterisation of enthymematic implication:
(y)
A then B is taken as interdeducible with: for some
true proposition p and false proposition q, A and p entail B
or q, i.e. in symbols (Pp,q) p & ~q &. A & p
B v q.
Then, where -*• is the implication of a relevant logic, such as systems E and R,
A s B is interdeducible with ij^ A then B, so characterised (the result is
proved in Meyer 73).
The adequacy of this attempt to equate the conditional with the materialconditional accordingly reduces to the question of the adequacy of (a) and ($)
or, more exactly, of (y).
But we already have a more than sufficient basis
for rejecting (a) - (y) in the shape of examples like (a) to (p) above.
12
7.2 OTHER EXPEDIENTS
IN DEFENCE OF HOOK
Moreover we can see directly that (a) - (y) are false (when 'if-then' is
construed ordinarily); for according to them (read as sufficient conditions),
whenever B is true ±f_ A then B, for any A.
(In (a) take S as {B}; in (y)
let p be B and q be ~B). That is, (a) - (y) at once reinstate the counterintuitive paradoxes of conditionality.
(For the negative paradox let p in
(y) be ~A and q be A; then whenever A is false, if A then B, whatever B.)
This serves to refute Faris, for the only argument he offers for the sufficient
condition in (a) which is crucial to his case is: 'I think that in any case
in which we believed that a set S existed as specified we should be prepared
to assert if_ A then B' (62, p.118).
The paradoxes reveal, however, that for
most of us the claim is false: we are not prepared to assert if A then B
on the strength of the truth of B.
The more general lesson emerging from
such attempts is that enthymematic explications of 'if...then' of the syntactical brand, which impose no restrictions on the class of truths (or falsehoods) that may be tacked on to entailments, are bound to be defective.
Neither 'if...then' nor 'implies' are enthymematic implications of such a
simple syntactical kind.
There remain other more desperate expedients that hookers may resort to
in defence of the material-conditional.
A favourite is this:- Hook is at
least a necessary condition for 'if...then', but you say 'if...then' amounts
to more than =>.
The analysis of conditionality must then take the form:
if A then B is logically equivalent to A
B and J, where J represents some
as yet unspecified condition.
But what can J be? It can't be this and it
can't be that, so it must be nothing. For example, Thomson. 63 argues that J
can't be of the form 'because ...' where a statement of the grounds for the
claim 'if A then B' are given, since this would inject the grounds for a
statement into the meaning or analysis of a statement.
While this is
correct, only a rather blatant commission of a false dichotomy fallacy would
enable the conclusion that J is null to be reached.
Nor is it particularly
difficult to indicate the lines along which J should be filled out (especially
when Hunter 72 has done this for us in rebutting Thomson).
Clause J will
read 'there is some not merely truth-functional connection, of an appropriate
sort, between antecedent A and consequent B'.
In fact, in the light of
metavaluational semantical methods, it is possible to be quite precise about
J: J can be the clause j- jlf A then B, where |- is the provability functor
of the system which formulates the logic of the conditional.
(For the fullest exposition of metavaluational methods, see Meyer 72).
The endemic circularity of such analyses makes it plain that there is no easy escape by way of
the material-conditional from logical field work, from the messy and commonly
inconclusive business of trying to determine the logics (or formal connections)
of conditionality.
In the quest for satisfactory theories of conditionality and implication or perhaps more realistically for systems still in the running for these
esteemed positions - it is salutory to record the negative results obtained
from failed attempts, for these place worthwhile bounds on satisfactory
accounts.
There are, in short, some lessons to learn, as Lewis realised,
from the failure of material-implication, about the character of satisfactory
accounts, about what entailment is not, and likewise what conditionality is
not.
Lewis in fact intended his wider argument to show that no merely
many-valued connective would serve as entailment (see Lewis and Langford 32,
p.237 ff; but the argument is not completely general).
We will try to
record in a more systematic fashion some of the lessons that emerge given a
little reflection; we present them as a set of fallacies that correct
13
1.2 FALLACY
EXCLUSION
ANV FALLACIES
EXCLUVEV
accounts of conditionality and entailment must avoid.
For the avoidance of
fallacies is an overarching requirement of adequacy that should be imposed on
accounts of entailment and conditionality.
We call this general requirement,
the first of many conditions of adequacy we shall impose, the requirement of
fallacy exclusion. Under it, as determinate cases, fall:
1. The truth functional (or matrix-assessibility) fallacy. The fallacy is
that of assuming that whether A
B holds is a function of the truth-value,
truth or falsity (or more generally, in the finitely many-valued case, of the
value) of one or other or both of the components A and B,
But the implications one statement has are never a matter just of its truth-value, and so
knowing whether it is true or false has no bearing on what it implies. Similarly what implies a statement is not a matter solely of the statement's
truth value.
Implication and conditionality are matters rather of the intensionality, or meaning, of statements, not of their extensional values.
2. The finite-valued fallacy. The fallacy is that of supposing that logics of
implication or conditionality can be merely finitely valued logics.
That it
is indeed a fallacy can be shown by arguments of the sort that demonstrate
that no rational logic is finitely many-valued.1
An n-valued logic can only
distinguish n statements; yet entailmentally there are infinitely many nonequivalent statements.
In any sequence of n + 1 variables Pi'-,''Pn+i a t
least two of these miTst be assigned the same value in an n-valued logic. So given that p -*• p, since a correct entailment (and conditional), takes a designated value, and that a disjunction with one designated disjunct takes a designated value - the disjunction (p.. -> p„) v ( p -»• p )v...v (p
p ) v (p
p )v
i z
± j
i n
z
l
...v(p^
Pn_-^) must hold good, for the reason that one component at least
will have the same value as p + p in an n-valued logic.
This chain is
obviously a generalisation to n values of the cycle condition (p^
p^)
v
P^) that material-implication satisfies, and equally fallacious.
It
is not difficult to devise counterexamples to such n cycles.
Consider the
positive integral line (or the natural numbers) and suppose each integral
place is occupied by, or has, a randomly selected colour.
Consider the set
of true elementary statements of colours, e.g. 1 is blue, 2 is black, 3 is
vermillion, etc. By construction of the example, no one of the elementary
statements entails any other; hence no finite disjunctive chain condition
can be correct.
Avoidance of the first two fallacies alone takes the quest for adequate
analyses of entailment and conditionality beyond the confines of many-valued
logic.
We now make a beginning on listing fallacies that narrow the remaining class of logics (and further fallacies will be adduced as we proceed).
3. The truth-copulation fallacy. The fallacy is that of taking A -*• B as
true [valid, a theorem] where both A and B are.
Thus any system of conditionals in which the rule A, B
A -*• B is admissible is fallacious: this
rules out among other things, the system of Stalnaker and Thomason 70, and of
Cooper 68, and the main systems of D. Lewis 73.
Lewis and Langford base
their critique (p.238 ff.) of classical and many-valued accounts of deducibility essentially on the commission of this fallacy; and their case applies
also against accounts of conditionality which incorporate truth-copulation.
1
On this point, which is a further reason for avoiding standard many-valued
logics in any quest for an analysis of implication and of conditionality,
see Routley and Wolf 74.
14
7.2 FALLACIES
OF TRUTH-COPULATION
ANV
MODALITY
They present essentially three points - none of them absolutely inescapable
by a hardened truth-copulator whose logical intuition has become warped in arguing that truth copulation is indeed a fallacy.
Firstly 'two propositions could be independent (one not deducible from the other) only if one of
them be true and the other false.
As a statement about deducibility, this
is quite surely an absurdity' (p.238).
For otherwise, to progress to the
second point, any 'system which... could represent the truth about something
or other, would be deducible in toto from any single proposition of it taken
as a postulate ' (p.2381).
Thirdly, 'the falsity of the assertion "Every
true proposition is deducible from every other" hardly requires proof1 (p.239).
Nonetheless this Moorean appeal can be supported by intuitive counterexamples.
Consider any two (independent) irrelevant statements, e.g. a necessary statement q^ (e.g. 23 = 8)
and a contingent one q^ (e.g. Manaus is situated on
the Rio Negro).
Then none of the following statements are true: "if q^
then q2"» "that q^ implies that q2", "q^ if, and only if, q2".
Moreover
they commit fallacies of other types, namely fallacies of relevance (on
which see ABE2) and of modality.
4. Fallacies of modality. There is really a cluster of traditional fallacies
falling under the heading 'fallacy of modality', and there is dispute about
the extent of the cluster, e.g. in which cases, if any, it is fallacious to
have a contingent statement entailing a necessary one (see FM and ABE). A
commonly-committed modal fallacy, of importance in refuting one form of epistemological scepticism, consists in applications of the principle: if it is
necessary that A implies B then if A then necessarily B.
For example, the
sceptic argues that if one knows that p then p must be the case; but since
contingent statements such as that the cheese is on the table in front of us
never have to be the case (by contraposition and legitimate modal distribution),
one can never really know that the cheese is there.
However the important
fallacy of modality for present discussion is that of necessity-transmissionfailure, which is simply that of violating the necessity transmission principle that if A entails B then DA entails []B, that necessary statements do
not entail non-necessary ones, contingent ones in particular.
This fallacy,
like most of those concerning modality, only applies where a logical implication or entailment occurs.
The statement "that q^ entails that q^" commits
a fallacy of this type, since it would have a necessary statement entailing a
contingent one.
Several systems, both relevant and irrelevant, are ruled out as
systems of entailment because they contain theses which commit this fallacy
of modality.
(It is assumed, what we later argue for, that entailment is
closed under Modus ponens.)
For example, so ruled out are systems which
contain any of the following principles: A -*•. B '•*• A (as B may be necessary
and A merely true); A
(A ->- B)
B i.e. Assertion (since A and B may
both be contingent though A -»• B is necessary); A -*• (B -> C)
B -*•. A
C,
1
Lewis and Langford consider an attempt to escape from this consequence,
but in insufficient detail, considering that they attempt a rather similar
escape from a parallel objection to strict implication (p.252ff.).
2
To have much bite in natural language cases, more than the formal weak
relevance requirement is needed in explaining fallacies of relevance.
15
7.2 AW AVEQUATE THEORY THAT AVOWS
FALLACIES MUST BE INTENS10NAL
i.e. Commutation; A & B
C
A
B-+C, i.e. Exportation.
Given
further principles, subsequently defended as correct, systems containing
the following principles are also ruled out- A & B & (A
B)
~A
~B;
(A ->• B) v (A
~B) , i.e. Stalnaker.
As to the last let A be some necessary
truth, e.g. q^ and B some independent contingent truth, e.g.
Then since ~B is false A cannot imply B, by the fundamental counterexample
principle. But neither can A entail B, without a fallacy of modality, and
this exhausts the disjunctive cases.
Although failure of necessity transmission only shows that the principles
rejected cannot be correct entailment principles, it can be argued by way of
direct counterexamples that none of the principles rejected are correct
(analytic) conditionals either.
The example already given reveals, for example, that Stalnaker cannot be a correct conditional principle, since the
conditional "if q^ then q 2 " is not true, and the conditional "if q^ then ^ 2 "
is not true either, by the counterexample principle.
Adoption of material-implication as an account of entailment [conditionally] would lead to commission of [practically] all the fallacies discussed.
Given the manifold deficiencies of material-implication as an analysis of
logical implication - and there are but few these days who really think that
hook serves to analyse entailment, since it does not even supply a logical
relation - there are two directions in which to proceed:- To put it roughly,
there is an extensional direction, encouraged by the observation that objections are somewhat lessened by equating logical implication with tautological
material implication rather than ungarnished material implication, which takes
shape in a metalinguistic analysis of entailment.
And there is an intensional
direction, which endeavours to analyse entailment systemically by way of intensional or modal notions in addition to purely extensional notions.
The first
direction, that of a metalinguistic analysis, is, so some of its exponents
have tried to argue, compulsory; attempts at systemic analyses all rest on
serious use/mention confusions.
Our next main task will be to defuse this
criticism, and to reveal some of the deficiencies of metalinguistic analysis
of logical implication and deducibility.
V
The need to go beyond classical logic in order to obtain an (even halfway) adequate account of deducibility points up a basic flaw in classical
logic conceived as a general all-purpose logic; for such a logic should be
rich enough to express its own deducibility relation.
Deducibility (i.e.
the converse of entailment) is after all the fundamental logical relation.
Classical logic is inadequate because, as an extensional logic, it cannot
satisfactorily express either overtly or covertly (as in metalinguistic constructions) intensional notions such as deducibility.
And of the intensionality of the entailment and implication connections there can be no doubt:
A -*• B and A = C do not ensure C -*• B and neither do A ->• B and B H D ensure
A -*• D.1 An adequate theory of entailment must be intensional, i.e. nonextensional, and more generally non-value-functional. Thus too no extensional
patch-up of material-implication, by defining a new extensional relation in
terms drawn from classical logic or of a merely extensional cast, can be
adequate.
More generally, no value-functional patch-up can suffice (where
1
= symbolises as usual material equivalence with A = B •**. (A => B) & (B a A).
The various readings of A -*- B, and A «• B (where A -»• B = f (A
B) & (B
A)),
will be a matter taken up in the next section.
7.3 THE mPLJCATlOUAL-COmriONAL
TRANSFORMATION
value-functionality is characterised as in GR) This is the next of the many
conditions of adequacy that we will impose on a correct theory of entailment,
what we call the intensionality requirement. It is evident that, for similar
reasons, an adequate theory of conditionality must also be intensional.
The
metalinguistic proposals do not fail the intensionality requirement because
the metalinguistic procedure represents a backdoor, and somewhat covert way,
of introducing intensionality - allegedly in an extensionally acceptable
form (because it's not too high a grade!).
§3. The trouble with the metalinguistic repair.1 Like most going theories,
especially defective theories, classical logic has a wall of defences against
objections to its theory of logical implication.
The metalinguistic defence
(pioneered by Carnap and popularised by Quine) is to separate implication which is claimed to be a metalinguistic notion combining names of statements from hook, relabelled as material-conditional, which compounds statements,
i.e. declarative sentences on Quine*s account.
Quine's argument turns on
the point that construing hook as any sort of implication involves a usemention confusion (51, pp. 31-3; 60, p.196). Thus Whitehead and Russell are
accused of being careless of the use-mention distinction in writing
(i)
p implies q
interchangeably with
(ii)
if p then q
;
and Lewis is alleged to have followed suit in his account of strict implication.2 Even granting this, the distinction does not rule out the interchangeability of (ii) with
(iii)
that p implies that q
of what we call the implication-conditional (IC) transformation. Both (ii)
and (iii) are of the form
(iv)
... p ... q ...
of statement connectives in Quine's sense, which we may write in symbols
(v)
p -»• q.
In Quine's terms (51, pp. 32-3) 'the relation of implication produces a
derivative mode of composition of ... statements themselves - namely a mode
which consists notationally of compounding the statements by means of
'implies' and two occurrences of 'that'.' Quine however only considers
the case of compounding 'by means of 'implies' and the two pairs of quotation
marks', as would occur if (ii) were equated with
(vi)
1
'p' implies 'q'.
This section was written before the valuable Appendix - Grammatical
Propaedeutic of ABE became available.
The section can profitably, we
suggest, be read in conjunction with the ABE Appendix (about which we
have only a few minor reservations).
2
Lewis and Langford's very suggestive comparison of strict implication with
tautologous material-implication in their defence of strict implication as
an analysis of deducibility (32, p.238 ff.) no doubt encouraged the metalinguistic defence, but at the same time opened the way for Quine's allegation (especially p.242) - which would stand had Lewis and Langford been
using quotation marks in the fashion of the current narrow orthodoxy.
77
1.3 PULLING VOWN THE METALINGUISTIC ACCOUNT Of IMPLICATION
Quine's principal objection to this procedure - of putting quotation or 'that'
into the sentence connective to gain interchangeability with the conditional
(iii) - is that it involves abuse of quotation.
This is true if quotation
is narrowly construed (as forming a constant name even from variables), but
the objection fails entirely for the operator 'that' or if quotation marks are
construed as quotation functions (a not uncommon construal whose liberating
effects are explained in GR).
For the statement variables buried in (iii)
(or in (vi) if the marks are taken as quotation functions) can be treated as
constituents of (iii), just as those in (ii) can.
Thus Quine's argument
rests on a false dichotomy, obtained by conveniently dropping the method of
putting 'that' into the functor.
The method also escapes (in a way Quine
is happy to adopt elsewhere) Quine's objection to introducing 'that..,' as a
substantive clause, namely the obscurity of the items designated; for 'that'
need figure only as an integral part of the functor.
Quine may try to
object that the connective 'that ... implies that
' is (implicitly) metalinguistic, and would have to be so rephrased in any adequate reconstruction
of language.
This we deny (for reasons sketched in GR); and so should
Quine.
For firstly he is prepared to admit intensional predicates containing
'that' into canonical notation (e.g. 60, p.147), and secondly he claims, in
proposing his maxim of shallow analysis, that 'embedded in canonical notation
in the role of logically simple components there may be terms of ordinary
language without limit of verbal complexity' (60, p.160).
The logically
simple statement connective 'that —
implies that
' can be one of these
components.
Thus an implication relation can be introduced into a sentential
object language without any use-mention confusion.1 And this is of course the
way things are in natural languages: the idea that implication is a metalinguistic notion does not stand up to much linguistic investigation.
The implication-conditional transformation integrates, then, implicational
and conditional statements, and, as a corollary, precisely the examples which
show that material-implication is completely inadequate as an account of implication (expressed in form (iii)) reveal that the material-conditional, hook,
is inadequate as an explication of conditional statements in mathematics as
well as elsewhere. In fact the inadequacy of the material-conditional is
obvious from one of the most important tests of adequacy of explication of
conditionals, namely substitutivity tests.
Material-equivalence of statements, i.e. sameness of truth-values, warrants intersubstitutivity in materialconditional statements, but not in the conditionals asserted in any of the
sciences.
The IC transformation also provides a criterion for when an 'if ...
(then) ...' statement is a (genuine) implication.
A declarative sentence
of form (ii) is an implication if it is interchangeable (preserving values,
and sense) with one of form: that p implies that q.
The requirement rules
out as implications such statements as 'if you want nuts there are some on the
the sideboard', since there are familiar circumstances where this is true but
'that you want nuts implies that there are some on the sideboard' is false.
-j
Quine's proclaimed policy of admitting none but truth-functional connectives
(51, p.33) can be seen to be quite arbitrary as soon as the further desirable
step of admitting a that-operator (or quotation functions) into the object
language is taken.
For then logical Implication as defined in the metatheory can be mapped into the object language.
Quine's policy is extraordinarily narrow, and apparently based on little but extensional prejudice.
18
1.3 ENTAILMENT, VEVUCIBILITY AMP LOGICALLY NECESSARY IMPLICATION INTERRELATE!?
The criterion also excludes non-conditional uses of 'if (discussed in 1.5),
e.g. 'I rather like your paper, if I may say so', and a variety of counterfactual and non-contraposable occurrences of 'if', e.g. 'If he had telephoned
he would have got me', 'If he's here I didn't see him'.
Central among implication relations, and crucial for logical reasoning
is the relation of entailment.
Entailment here has the meaning assigned to
it by Moore, namely the converse of deducibility.1
That is
(vii)
that p entails that q iff that q is deducible from that p.
Is there any way of picking out what sort of implication entailment is?
A familiar answer which we shall argue for is that entailment is logically
necessary (or analytic) implication, where furthermore the implication is a
sufficiency relation and logical necessity satisfies the postulates of a
normal Lewis modal system (specifically those of system S5).
Thus in
particular
(viii) that p entails that q iff it is logically necessary that
that p implies that q.
The argument for part of our claim is straightforward.
It is generally
conceded that entailment is a sort of implication and that where it holds it
holds necessarily.
What is more likely to be disputed is that where it
fails it fails necessarily, and more generally that logical necessity meets
not just S4 postulates but those of S5 as well, in particular that
(ix)
~Q]p
CHlp,
i.e. that it is not logically necessary that p implies that it is logically
necessary that it is not logically necessary that p.
We have defended the
correctness of S5 as an analysis of logical necessity before (Routley 69),
and later (in chapter 10) we take the argument further.
Later too we shall
expand the case for characterising entailment as in (viii), i.e. for defining
A =» B as D(A •> B) . 2
Given the standard connection forged between deducibility and valid
argument, namely
(x)
that q is deducible from that p iff there is a logically valid
argument from p to q,
it follows
(xi)
that p entails that q iff there is a logically valid argument
from p to q.
With the introduction of these connections the metalinguist will return to
the attack.
He will argue that such notions as validity and deducibility
are metalinguistic, not systemic, and what is more they are relative to
1
There are also, of course, several analytically-linked analyses of entailment that we will pick up subsequently, e.g. those of valid argument and of
the availability of a valid derivation, of logical sufficiency and of inclusion of logical content.
2
The arrow symbol -*• is the way we represent the determinable, 'implies' and
'if-then'; but when we need to distinguish implicational determinates such
as entailment, implication and the conditional, we use the double arrow,5*,
for entailment and the single arrow for implication.
Similarly where we
want to specifically show that the relation is material-implication or
strict-implication we use the symbols => and -i respectively.
19
7.3 EXPLICATION VOES NOT HAVE TO BE METALINGUISTIC
particular logical systems.
Moreover the resources of classical logic in
providing an adequate implication relation are by no means exhausted by
material implication.
Indeed implication can be defined metalinguistically
along the lines of Tarski's classical definition of logical consequence.
For
example, for classical quantification logic Q, A logically implies B wrt Q
iff B is a logical consequence wrt Q of the class consisting just of A.
There are several different objections to unscramble here.
To begin with,
there are stronger and weaker versions of the thesis that logical implication,
like validity, has to be defined metalinguistically relative to a given system.
The weaker claim, which we concede, is that there are implication-type relations which can be defined metalinguistically relative to given systems.
These metalinguistic relations (which characteristically correspond to firstdegree strict implication) have however decided peculiarities, especially once
matters of quantification and iteration are considered (as explained in
Routley 74) - peculiarities which rule them out as worthwhile explications of
the pre-analytic implication relations of philosophy and mathematics.
Moreover these system-relative relations do not exclude either absolute implication
relations (any more than language-relative notions of truth rule out an absolute notion) or systemic explications of implicational relations and corresponding conditionals.
The stronger claims say that the explication has to be metalinguistic. One
basis for this claim, that a systemic explication of implication involves a usemention confusion we have already met, essentially through the IC transformation.
Another ground for the claim appeals to limitative theorems which are alleged
to show that semantical notions, of which validity and so deducibility are
specimens, are not fully explicable systemically.
It would take us too far
afield to take issue with this dogma here.
It is enough to point out that
the limitations are quite remote and will certainly not be felt at sentential
or even quantificational levels; and accordingly do not preclude complete
systemic investigations of deducibility at these levels.
Indeed the most
that this objection shows is that a systemic investigation of deducibility is
bound to be incomplete at the edges, something that would be unsurprising on
quite independent grounds (such as failure of explications to take account of
quirks of context and so on).
Furthermore if the objection were right metalinguistic explications would in due course suffer precisely the same fate.
In fact a purely metalinguistic explication of implication is bound to be
more inadequate than a systemic explication since it provides only part of
the requisite story.
Roughly speaking it accomplishes only a somewhat
anomalous first degree explication without iterated occurrences of implications,1
and everything it does can be systemically reflected - in a system of quantified
strict implication in the case of the classical account of logical implication
according to Q.
But it is quite evident, for example, that implications can
be iterated and that variables in implication statements can be quantified,
that the fact that p ->• q and q -*• r can, and does, imply that q -»• r, that for
every x f(x) -*• g(x) does imply that for some x,fx -*• gx, and so on.
These
quite ordinary truths the metalinguistic account cannot accommodate - without
1
Iteration of a sort can be achieved by the artifice of ascending to a
sufficiently high level of language.
It is then transparent, however,
how far removed the metalinguistic account is from the familiar notions we
are trying to isolate and explicate, and that it is a fantasy to suppose
that the metalinguistic construction can serve as any sort of replacement
for the originals.
20
1.4 THE THESIS THAT ENTAILMENT IS STRICT-1MPLI CATION
enlarging the usually admitted metalinguistic apparatus (e.g. by the introduction of unstructured quotation functions) in a way that facilitates an
elementary systemic mapping of metalinguistic relations such as implication
anyway.1
In short, the proposal that investigations of implication should be, or
have to be, metalinguistic is bound to lead to only quite inadequate explications of implication.
The reasons advanced so far are not the only reasons
for dissatisfaction with the metalinguistic accounts we have been offered.
Firstly, standard accounts only characterise logical implication.
As in the
case of strict implication we are left with n£ analyses of those very common
brand of implications, non-logical implications, better than the quite inadequate material connection or less than satisfactory enthymematic accounts
(criticised below).
Secondly, standard accounts correspond to a part of the
theory of strict implication (in a way that Carnap 56, p.l73ff., makes clear)2,
and -accordingly suffer from the main defects of strict implication as a theory
of deducibility.
§4. What is wrong with strict-implication. Strict-implication, or fishhook,
symbolised
and defined: A
B
& ~B) or equivalently as D(A ^ B),
is admittedly much superior to material-implication as an analysis of entailment.
For example, Lewis strict-implicational systems avoid the modal fallacies that material-implication would admit, such as that statements which are
logically necessary can entail merely contingent truths.
Nor does strict
implication have to be given up for the petty grammatical reasons Quine
elicits - 'p
q' can be read unproblematically 'that p strictly-implies that
q'.
But, like its mate material-implication of which it is the necessitation,
strict-implication tolerates not merely a large class of paradoxical principles
but also very many other principles to which there are intuitive counterexamples.
And the peculiar properties of it are neither important
logical discoveries nor absurdities; they are merely the
inevitable consequences of a novel denotation for an old
and familiar word, long used in common parlance in a
different meaning. (Lewis 30, p.34, but referring to
material-implication, not strict-implication.)
Despite all this the thesis that entailment is strict-implication boasts
many advocates.
Before we begin to detail the case against strict-implication
as the analysis (it is substantial and central parts of the argument extend
far into chapter 2), let us hear from the friends of strict implication, and
from the stricters themselves.
For these people think that they have an
1
We should really like to propose, what is highly disputable these days,
that the metalanguage should be reflectable in a sufficiently rich
(object) language (which includes its own semantics).
This is one step
in the direction of abolishing the metalinguistic hierarchical structure
altogether, a plan we begin to outline in chapter 15.
As to how we live
with the semantical paradoxes and avoid Tarski's strictures, see 1.8,
5.3 and 6.5.
2
This is not an essential feature of metalinguistic accounts as Routley 74
shows.
In fact, a variety of first-degree theories of implication have
metalinguistic analogues; for instance every semantical account in
chapter 2 can be represented metalinguistically.
8
1.4 NEUTRALITY OF THE VEFINITION OF ENTAILMENT THROUGH INCONSISTENCY
easy time, that they can remove rival analyses with a few of the following
worn-out charges:A first charge is that the strict definition of entailment is the only
one that approaches real clarity (suggested in the case Hughes and Cresswell
68 make for strict-implication, p.336).
This is simply false: for if the
strict definition is clear then the Philonian definition of material-implication
is at least as clear (and many would say decidedly clearer since no modal notions
are involved).
But of course the Philonian definition is patently inadequate
and does not even make entailment a logical matter.
So the stricter may contend that his definition is the only definition which is not patently inadequate
which has real clarity.
To start with this begs one of the questions at issue,
as to whether the strict analysis is or not patently inadequate.
We think it
is patently inadequate, for reasons we will detail.
But letting this pass,
and conceding, as we believe, that some modal and intensional notions have
requisite clarity, it is still the case that practically every rival analysis
can produce a definition of entailment that is at least as satisfactory as that
of strict implication.
Some connexivists have in fact used the very definition
stricters propose, only they offer a non-extensional account of conjunction
(see Routley and Montgomery 78).
More important, Nelson, another connexivist,
long ago revived the following definition of entailment that Lewis had once
used:1
(xii) that p entails that q iff that p is inconsistent with that ~q,
or, in symbols and notation that we will subsequently prefer: A -»• B
~(A o ~B),
where o is our transcription of the two-place consistency connective first introduced by Lewis.
This definition can be agreed on by most parties, in particular
by strict, relevant, connexive and conceptive positions; and it is just as clear
as the strict reduction - which other perceptive parties reject - of consistency
to a one-place modal notion.
From this perspective the strict analysis is
distinguished by its insistence that A « B be analysed as 0(A & B), that the
consistency of A with B is nothing but the conjoint possibility of A and B.
But the strict analysis of consistency is mistaken in the same way as the strict
analysis of entailment; it would make the impossibility of one component A,
for example, suffice for its inconsistency with any other component B, quite
neglecting the connection there should be linking B with A (cf. 5.2).
Syntactical definitions of entailment, whether of the relevant or strict
kind, though important, are in the end of limited merit.
For they amount to
trading one notion in for another that either raises practically as many problems, e.g. consistency, or if it doesn't, as in the material case, is wrong.
For where the properties of entailment are in doubt so are those of consistency
or possibility, and while the properties of consistency or possibility remain
in dispute, e.g. the iterative properties of possibility, so correspondingly do
those of entailment.
In fact the strict analysis of entailment no more leads
to a unique theory of entailment even when conjunction and negation are extensionally battened down than does the relevant analysis in terms of an irreducible
consistency relation.
There are as many strict analyses of entailment as there
are systems of strict-implication, that is, infinitely many.
It is surprising,
1
But the account, like so much in the implicational area, is much older, and
may be found stated in Sextus Empiricus:
... those who introduce the notion of a connexion say that
a conditional is sound when the contradictory of its consequent is incompatible with its antecedent.
(in Kneale and Kneale 62, p.129)
22
1.4 THE REQUIREMENT ON ENTAILMENT OF "MEANING CONNECTION"
in view of the supposedly luminous nature of strict implication, how reluctant
most stricters are to let it be known which system of strict implication represents entailment, and why that particular one.1
Accounts which reject the strict analysis of entailment characteristically
do so, it is alleged, in a second group of charges,
on the ground that a further condition of q's being deducible
from p is that there should be some connection of 'content'
or 'meaning' between p and q.
It is, however, extremely
difficult, if not impossible, to state this additional requirement in precise terms; and to insist on it seems to introduce
into an otherwise clear and workable account of deducibility a
gratuitously vague element which will make it impossible to
determine whether a given formal system is a correct logic of
entailment or not (Hughes and Cresswell 68, pp. 336-337).
Similar charges are levelled in many other places (e.g. in Bennett 54, p.460 ff.
and, in a rather better substantiated way, in Bennett 69, p.214 ff.).
But
the fact is that accounts of entailment as clear and workable as strict analyses - relevant analyses in terms of inconsistency - already yield formally
precise accounts of content linkage, which there is no difficulty in stating,
e.g. the sharing of variables requirement WR.
Nor is there any difficulty
in defining an appropriate exact notion of content - it is defined in the
classical way - or the appropriate connection of content - it is that of inclusion.2
That is, A entails B iff the content of B is included in the content of A, in symbols c(B) £ c(A) (for details see UL).
This is just one of
several, interconnected, analyses which meets Bennett's requirements on an
appropriate notion of "meaning-connection", namely
The failure in more than 60 years of research to resolve Lewis's early
puzzle as to whether S2 or S3 or some other strict system really captured
entailment, or to advance the issue of which system of strict implication,
if any, represents entailment at the statemental level, is only one of the
surprises in the history of strict-implication.
Another is the fact that
strict-implication has rarely been applied in theory construction, even in
such initial enterprises as the formulation of set theory and number theory.
There is also another stricter inclusion-of-meaning relation - that tied to
sense as distinct from informational content - which diverges from entailment.
The stricter relation, important in assessing the paradox of analysis
and in helping to clear up some similar paradoxes of Lewy 58, yields an implication relation which does not satisfy intersubstitutivity of coentailments.
The "paradoxes" and the implication relation are studied in chapter
14; the notion of sense and the various meanings of "meaning" concerned are
examined in Routley 77.
Lewy's "paradoxes of entailment" (76, p.117 ff.)»
which depend primarily on illegitimate suppression steps, are dealt with in
2.9 ff.
Note well that it does not matter for those who say that entailment
requires a connection of meaning that there are accounts of meaning inclusion
which entailment relations violate: it is enough that there is some satisfactory account of connection of meaning which goes with entailment (and
there is, as UL shows for content inclusion).
This point is enough to
rebutt Pollock's case (66, pp. 186-191) against requiring that an adequate
account of entailment meet a connection of meaning condition.
23
1.4 RELEVANT VALUES OF BENNETT'S FUNCTOR <p
that (a) where there is an entailment there is a meaningconnection, and that (b) for some Q there is no meaningconnection between (P & ~P) and Q (69, p.214).
For the content of q is not included in that of p & ~p where p is a distinct
variable from q (a result that holds for a wide range of relevant sentential
logics).
Another, weaker, connection which provides the appropriate linkage
is given by the sharing of variables feature of relevant logics itself.
Content inclusion and variable sharing represent just two values of <p
which will serve in strategies Bennett allows for arguing against the hardest
Lewis paradox argument:
One strategy for arguing against the paradox is to defend a
principle of the form "If A -*• B, then <j»(A, B)" and then to
argue for the falsity of <K(P & ~P)» Q) for some Q (69, p.216).
There are other values of (J) that will also serve, among them essentially all
proposals that Bennett dismisses (69, pp. 216-218), namely:(3) <t>(A, B) is A
B, i.e. 'if A then B', where the connection is that of
a non-logical relevant conditional, e.g. a subjunctive conditional.
(4) <p(A, B) reads 'A gives a (logically) sufficient reason for B' or 'A is
(logically) sufficient for B'.
(5) <J>(A, B) iff
appropriate weakness;
A -*• B, where L is a genuinely relevant logic of
or equivalently (for many such systems),
(5') <J>(A, B) iff there is a route from A to B licensed by natural deduction
rules for L; or equivalently again, but less formally, in advance of explication of 'use of premisses',
(5") <J>(A, B) iff there is a derivation (according to system L) of B from,
or using, A.
(5)-(5") take up the traditional characterisation of entailment in terms of
which A -»• B iff B can be derived, or inferred, from A by valid rules.
The
issue is then: what constitute valid procedures? Values (5) - (5") are
spelt out in detail in ABE for favoured exportative relevant systems, but
similar syntactical methods extend to a variety of other relevant systems.
And there are other syntactical explications which fulfil an analogous role
and so likewise provide values of (j), e.g. the tableaux and Gentzen methods
studied in subsequent chapters.
More satisfying perhaps, there are semantical
explications, the fundamental one, in terms of which sufficiency as deployed
in (4) can be explained, being as follows:(6) <|>(A, B) iff for every (L-)situation or world a, when A holds in a then
B holds in a; or, put differently, A is logically sufficient for B iff without exception A situationally guarantees B, or, differently again, iff A on
its own without any other assumptions (which would be peeled off by suitable
falsifying situations) guarantees B.
Such semantical analyses, especially
(6) itself, are the main object of subsequent investigations (from chapter 2
on).
For all the main relevant systems studied Bennett's requirements are
met, entailment ensures that the <}> relation obtains, but <|>(P & ~P» Q) is falsified for (weakly) irrelevant Q.
The main charge, on which stricters always fall back, is that the standard
relevant logics involve giving up the intuitively valid principle of Disjunc-
24
1.4 THE BURDEN OF PROOF AS REGARDS V1SJUNCTJVE SYLLOGISM
tive Syllogism.
This is prima facie extremely implausible ....[Disjunctive
Syllogism is] not to be rejected except on the basis of
strenuous arguments (Bennett 69, p.205).
Disjunctive Syllogism is not intuitively valid; it is easily falsified by
inconsistent situations (as we explain subsequently in this chapter and show
in detail in 2.9).1
Briefly, A & (~A v B) is not logically sufficient for B
because in inconsistent logical situations both A and ~A may hold though B
does not.
Such logical situations are readily envisaged and described (see
1.7 and 1.8).
But the antecedent of a valid entailment has to guarantee the
consequent over all logical situations.
A & (~A v B) does not so guarantee
B.
Hence Disjunctive Syllogism is not valid, as this intuitive argument
shows.
As to whether the argument sketched is strenuous enough for Bennett,
we make no conjectures; but we do disapprove mildly of the transparent way
in which Bennett, here and elsewhere in his defence of strict implication,
has shifted the onus of argument.
It seems to us that intuitively correct
principles should have sound intuitive supporting arguments: that it is not
that classical, or strict, implicational principles should be accepted until
they can be knocked down, but that they should not gain accreditation without
intuitive supporting considerations.
Methodologically a principle of weakness, not the all too familiar principle of strength, should be adopted in
assessing the correctness of logical principles (as reason indicates2, and
we try to explain in chapter 3).
Similar points tell against other attempts to reinforce the status quo that entailment is some brand of strict implication or its metalinguistic
reformulation - by shifting the onus of argument to the opponents of strict
implication.
Although methodologically unsound procedures abound among
defences of strict implication, Pollock 66 surely deserves some sort of prize
for his effort.
Pollock's overall argument takes the following lines:Firstly he tries, on the basis of Lewis's arguments, to shift the burden of proof:
i) All the inferences in the derivations of the hard Lewis paradox are
just too intuitive and 'obvious to be denied unless some very good reason
can be given for their denial' (66, p.183).
By i) intuition is supposed to show
ii) 'there is a very strong prima facie case for thinking that the paradoxes
of ... implication are logical truths' (p.183).
iii) No philosophers have produced a reason as required by i), i.e. have
undermined the "intuitive" case of ii).
Therefore, Pollock concludes,
iv) 'the paradoxes are in fact true principles of logic, surprising though
they may be' (p.180; also p.195 and p.196); indeed, this is 'the only
justifiable conclusion'!
1
We also show in Chapter 2 that Disjunctive Syllogism involves illegitimate
suppression. At the same time we deal with Lewy's failed attempt to
accurately locate suppression in Disjunctive Syllogism and show where Lewy's
stimulating but thoroughly unsatisfactory discussion (in 76) goes astray.
2
And Lewis observed:
see footnote 1 p.36.
25
1.4 POLLOCK'S DEFENCE OF STRICT-IMPLICATION AS ENTAILMENT
Similar, invalid, arguments would have entrenched the Aristotelian theory of
motion (before Galileo) and the Newtonian theory (before Einstein), and more
generally are of great assistance to the establishment.
But the fact that
no investigators have so far upset a prima facie case hardly suffices for the
conclusion drawn, unless the investigations have been suitably exhaustive.
Pollock adduces no evidence that they have, merely claiming, without any
requisite argument, what appears false, that he has examined the three most
plausible arguments philosophers have located.
Not only is the overall argument invalid, the premisses are also defective.
If intuition has too commonly supported the argument for the paradoxes, it
has also supported the feeling that there is something badly wrong with the
paradoxes.
In short, intuition commonly generates a conflict about what
holds good for implication, and because it does, it does not produce such a
very strong prima facie case for thinking that the paradoxes of implication
are logical truths. (Compare other paradoxes: does the Liar paradox create
a strong prima facie case for thinking that some statements are both true and
false? That is not what is usually said.)
Since intuition too commonly
yields inconsistent findings about the logical truths of implication, on a
strict account (consistencizing and applying Antilogism) one perhaps ought
to hold also that the Lewis argument provides a very strong prima facie case
for thinking that at least one of the premisses in each "intuitively convincing" Lewis argument is incorrect.
It is a piece of history that this was
the position taken by Nelson 30, Duncan-Jones 34, and other investigators, all
of whom found the paradoxes more intuitively outrageous than certain of the
premisses of the Lewis arguments.
Pollock tries to circumvent this problem by denying that there is - or
can be - anything intuitively wrong with the paradoxes.
First he appeals to
his own case, that he has 'no intuitive reaction to the paradoxes one way or
the other*.
How much is this case worth?
The fact that students brainwashed
by orthodox logic courses frequently have no reaction even to the paradoxes of
material-implication hardly assists Pollock's case: many students could almost as easily be made flat-earthers.
More to the point, for students of
relevant logic the principles of Disjunctive Syllogism and Antilogism are not
intuitively acceptable, for advocates of connexive logic other Lewis principles
are unacceptable, yet Pollock does not relinquish the first premisses of his
argument.
Pollock is favouring strict intuitions over others: on what basis?
Pollock contends, against intuitive rejections of the paradoxes, that 'a philosophical position cannot be refuted by an intuition that is not generally
shared' (p.183-4).
If so, it would seem that a philosophical position could
not (perhaps even less) be so established, thereby undercutting Pollock's
overall procedure which begins from intuitions which are "not generally shared".
Pollock is sunk unless he can find some clear asymmetry between acceptance and
rejection intuitions.
He has two suggestions (p.183).
One is that those who
reject the paradoxical implications have defective intuitions, resulting from
confusing implication with some other relation such as a meaning-inclusion one.
The move fails to establish the requisite asymmetry because it can equally well
be applied against those who accept the Lewis premisses; for example Anderson
and Belnap have repeatedly argued that intuitive acceptance of Disjunctive
Syllogism rests on a conflation of extensional and intensional disjunction
(see especially 62).
The second suggestion seems to be - and will have to be
if it is to stand a chance of success - that we cannot have intuitions as to
invalidity, but only of validity.
The assumption is that complex validity
can be determined by putting together elementary steps, but that we can never
26
1.4 THE METHOV OF OVERKILL, APPLJEV TO THE PARADOXES
be intuitively certain that by a suitably complex argument that some inference
step will not be derivable.
Setting partly aside the syntactical-semantical
confusion, it is simply false that creatures never have correct intuitions
of invalidity.
Furthermore validity and invalidity are semantical considerations, often determined, not step by step, but by model-theoretic considerations; and an intuitive countermodel to an implication is enough for intuitive invalidity.
There need be no intuition, or argument, about an unlimited
number of steps or a tree path leading to the result.
Pollock's picture is
far too simple to handle evident intuitive data.
The second suggestion,
like the first, fails.
It has become fashionable to follow up emphatic assertion - without any
of the necessary supporting argument - of the intuitive validity of all the
principles used in the Lewis paradox arguments with statements as to the
innocence and sheer desirability of the paradoxes - what might be called the
method of overkill.
Thus, for example Hughes and Cresswell:
... we are inclined to go further: the "paradoxes" seem
to us on reflection not to be tiresome (though harmless)
eccentricities which we have to put up with in order to
have the disjunctive syllogism, transitivity of entailment and the rest, but sound principles in their own right:
a logic of entailment ought, for example, to contain some
principle which reflects our inclination to say to someone
who has asserted something self-contradictory, 'If one were
to accept that, one could prove anything at all' - and the
princ iple that (p.~p) entails q expresses this in just the
way that a formal system might be expected to (68, pp.338-9).1
In rather similar vein Bennett contends that the hardest Lewis paradox
is not 'something unpalatable which must be choked down because there is no
convincing way of faulting the Lewis argument which supports it', but that
'the fact that [Lewis's] analysis generates the paradox is part of the case the overwhelming case - for the analysis' (69, p.198).
But how this paradox
is part of the positive case is never clearly revealed.
The most we are
2
explicitly offered depends on a figurative connection of contradictions
which conflict with 'the system of move licenses'.
The 'positive merit in
Lewis' analysis that it implies that each impossible proposition entails every
proposition' is supposed to emerge in the following way according to Bennett:
'what the paradox says ... is that if you start with something conflicting
with the system of move-licenses there is nothing you cannot arrive at' (69,
p.218).
When this is cashed out it can be seen to depend upon having already,
quite circularly, adopted a strict system of inference rules.
It has been unkindly remarked that Hughes and Cresswell were so little
interested in alternatives to strict implication that they did not even
trouble to get right the axioms of the one alternative to strict implication
that they do specifically mention in the text of their appendix on entailment and strict implication, the system E.
Axiom El for entailment should
be ((p -*• p) -*• q) ->• q, not the special case presented (68, p.299), namely
((p -»• p)
p) pWe will take up shortly a different respect in which paradoxes of the form
that anything at all entails a necessary truth have been claimed, both by
Lewis and by followers like Bennett and more reluctantly Lewy 76, to be
essential to deductive reasoning, namely through the suppression of
necessary truths they admit. Compare also Popper's use of paradoxes
in characterising derivability.
n
1.4 THE PARADOXES ARE NOT INNOCENT OR HARMLESS
The same arguments which show that Disjunctive Syllogism is invalid also
show that the paradox A & ~A
B is not sound.
And it is widely recognised
as unsound and as counterintuitive.
Correspondingly the inclination to say
that someone who has accepted or is committed to some particular contradiction
is bound to, or might as well, accept anything at all, is by no means widespread - for good reasons: There are many statements that dialecticians and
others who accept certain contradictions will refuse to accept.
More generally,
we shall argue that a logic for entailment ought not, as a condition of
adequacy, to contain paradoxes such as A & ~A
B because it thereby precludes
the investigation of non-trivial inconsistent theories (1.8).
This is part
of our case for claiming that the paradoxes are not simply tiresome eccentricities (or Don't Cares), but are, to understate the matter, positively harmful.
The larger case (a beginning on which may be found in UL, §4) is that the
paradoxes have done untold damage, especially in philosophy, where time after
time they have subverted straightforward resolutions of philosophical problems.
Thus it is not just false, it is decidedly misleading to make the common claim
that the principle A & ~A
B 'is at worst an innocent one:
it could never
lead us astray in practice by taking us from a true premiss to a false conclusion' (Hughes and Cresswell 68, p.338).
The paradox is not innocent
because of its effects in applications; and it can lead us seriously astray
as we shall see: just as we can go wrong with material-implication (1.6) so
we can go wrong, seriously wrong,with strict-implication.
The semantical arguments which counter Disjunctive Syllogism and the
negative paradox A & ~A -*• B also invalidate the positive paradox B
A v ~A
and the main disputed assumption in Lewis's "independent" argument for this
paradox, namely the expansion principle B -*-. (B & A) v (B & ~A).
For suppose
B holds in an incomplete situation where neither A nor ~A holds.
Then A v ~A
fails to hold, and likewise (B & A) v (B & ~A) fails to hold; so the antecedents of the implications are not sufficient for their respective consequents;
and the implications are falsified.
(The details of such intuitive countermodels to the positive paradox principle and to Expansion will be elaborated
and clarified in chapter 2.)
Stricters commonly defend Expansion on the
grounds that A and ~A exhaust the cases in every situation, so that there are
1
Lewis and Langford (32, p.252) make an analogous defence:
the logical division of any proposition B with respect to any
exhaustive set of other alternatives, A v ~A, does not alter the
logical force of the assertion B.
But on our account the division does alter the logical content.
following in Lewis's footsteps, asks:
Bennett,
When we take a premiss and split it up into jointly exhaustive
subpossibilities, are we arguing invalidly or "suppressing" something which needs to be explicitly stated? (69, p.230).
The short answer is: Yes, the expansion argument is invalid; the further
necessary premiss A v ~A has been suppressed, and without it the subcases
are not exhaustive.
But the larger issue which emerges, the inadmissibility
of necessary premiss suppression, remains to be sorted out.
Stricters are, all the same, a little uneasy about proofs using Expansion.
The proofs have the air of clumsy legerdemain, not because they are
sophistical, but because they are 'so unnecessary' (Lewis and Langford
32 p
252)
' p*
(footnote continued on next page)
28
1.4 FAULTING, ANV CORRECTING, EXPANSION PRINCIPLES
no incomplete deductive situations.
But all the evidence points to the
fact that the evaluation of entailment requires that all situations be taken
into account, including incomplete and inconsistent situations, where deduction
and reasoning do not cut out.
Semantical analyses of systems S2 and S3
reveal, through the situations admitted where entailments fail to hold (the
so-called 'non-normal' or 'queer' worlds), that even the framework of strict
implication allows to a limited extent for such situations.
But of course
strict implication does not go nearly far enough in this direction; for
although it recognises situations where Identity, A -*• A, does not hold (in
correctly rejecting the paradox B ->. A -»• A and such principles as A
A
A),
it fails to observe that there are likewise situations where other laws of
thought fail to hold, where in particular Excluded Middle, A v ~A, breaks
down.
Yet surely if A
A can fail in some situations so can A v ~A.
We
shall find that it is easy to locate situations where Excluded Middle does
fail, incomplete ones - and these serve to falsify Expansion.
Unlike the negative paradox where the various relevant approaches are
divided, several approaches join forces against Lewis's positive paradox
argument in rejecting Expansion; relevant logics, conceptivism and connexivism all reject Expansion.
Indeed the positive paradox argument - though
according to Bennett (69, p.229) simply the contrapositive of the negative
argument (it is not) - has not enjoyed anything like the reputation or plausibility of the negative argument.
Lewis's independent argument is as follows:B -*•. B & A v. B & ~A
(B & A) v (B & ~A) ->-. B & (A v ~A)
B & (A v ~A)
A v ~A
B -»-. A v ~A
Expansion
Converse Distribution
by Simplification
applying Rule Syllogism
repeatedly.
The commonplace objection to this argument is that Expansion is false and
should be corrected to
B & (A v ~A)
B & A v. B & ~A
Corrected Expansion,
i.e. to a form of Distribution.
This is a correction upon which conceptivism
and connexivism agree with the relevant position; and it reduces the paradox
conclusion to the principle B & (A v ~A)
A v ~A, to nothing but a form of
Simplification.
Bennett, in his apologia for strict-implication, has endeavoured to come down heavily upon the commonplace objection.
He charges (in
69, p.229) that resulting positions deny the widely accepted principle
A & C + B, DC-OA + B
Necessary Premiss
Suppression.
This is not true of connexivism.
But it is true of relevant logics, which
view necessity suppression as a widespread fallacy, as an extremely damaging
principle which leads directly to paradoxes.
For in virtue of Simplification,
•B -* A -»• B at once results.
So likewise the Lewis paradox and much more
result, shortcircuiting the "independent" argument.
Bennett contends, how(footnote 1 continued from previous page)
And Bennett, though he first suggests (p.230), without any evidence at all,
that Expansion is part of 'the common concept of entailment', then concedes
that it may look odd.
We suggest that Expansion is not part of the common
concept - to the limited extent that there is such a common concept.
Our
evidence is circumstantial: part is the consensus among a variety of nonclassical positions that Expansion is invalid, and part is the fact Lewis's
argument for the positive paradox has never carried the conviction that the
argument for the negative paradox has, and that the step in the argument
students regularly baulk at is the first expansion step.
29
1.4 DEFEATING THE LEWIS CARROLL ARGUMENT FOR SUPPRESSION OF NECESSARV TRUTHS
ever, that anyone who rejects Necessity Suppression totally is caught in an
infinite regress, as Lewis Carroll found (and Lewy 76 makes the same, invalid,
point)•
For if Suppression is rejected every necessary assumption used in
an argument must be stated and conjoined to the given premisses; but to this
procedure there is no end.
If this picture were right there could be no deduction in relevant logics since these prohibit necessary premiss suppression:
but as there are deductions, the picture cannot be right.
It is very far from
right.
The myth that deduction not merely permits, but requires for its very
operation, the deletion of (conjoined) premisses when they are logically necessary, is embedded deep in the modal Weltanschauung; it is worth some detail to
try to dispell it.
Let us begin with the exportation-importation confusion that is one of the
muddles underlying the assumption that conjoined necessary antecedents must be
sometimes 'deletable without loss of validity' (Bennett 69, p.230).
According
to Bennett,
As Lewis Carroll once showed, we cannot automatically declare
Q -*• S to be false just because the derivation of S from Q rests
upon, presupposes, or requires a necessary truth which is not a
conjunct in Q; for this would commit us to denying every entailment statement.
For example, the derivation of (2) [i.e.
(Q & £) V (Q & ~P)] from A [i.e. Q & (P v ~p)] rests upon
B: A D (2) ;
and the derivation of (2) from A and B rests in turn upon
C:
(A & B) 3 (2)
and so on backwards and outwards.
The argument confuses 'resting upon' a premiss in the sense of being implied
by a premiss, with its imported form, being conjoined with a premiss.
Consider
the example.
The derivation of (2) from A applies Modus Ponens, not Material
Detachment, and rests upon
B' : A
(2) .
The derivation of (2) from B' and A "rests", for what it is worth, on
C* : B' -»•. A + (2)
,
i.e. on a case of Identity: A -*• (2)
A -»• (2), not on its imported form
A & (A
(2))
(2).
Thus with C*, i.e. B' -*• B', the regress effectively
terminates.
All steps 'backwards and outwards' are cases of Identity,
C' -»• C', C' -»• C' -»-. C' -* C', etc.
There is thus no vicious regress, and no
basis for conceding that Necessary Premiss Suppression is "sometimes valid".
The Carroll argument does not show, what Bennett claims (69, p.230), that
someone who denies that (1) -*• (2) [i.e. Expansion, using the
commonplace objection] must say that necessary premisses are
sometimes but not always deletable without loss of validity.
But non-classical logicians will in any case admit that antecedents, whether
necessary or not, are sometimes deletable without loss of validity, notably
where the resulting implication is independently valid; e.g. B is deletable
in A & B -*• A because A
A is valid.
This admission naturally does nothing
to reinstate Expansion.
30
1.4 ADMISSIBLE SUPPRESSION, ACCORDING TO HOOKERS AND TO STRICTERS
Relevant logicians agree with stricters that Rule Exportation is inadmissible, that true (or even proven true1) antecedent conjuncts cannot be
omitted from entailments without invalidity.
Exactly the same holds for
necessarily true components; but this the stricter denies:
in stating a strict implication one cannot omit a merely
true premise which is one of a set of premises which together give the conclusion; but one can omit a necessarily
true premise.
The omission of a premise which is a priori
or logically undeniable does not affect the validity of a
deduction; but the omission of a required premise which is
true but not a priori leaves the deduction incomplete and,
as it stands, invalid (Lewis and Langford 32, p. 165).2
Such an omission would, prima facie, leave the deduction incomplete, and so
invalid, irrespective of the modal status of the omitted antecedent, and
irrespective of whether it was known to be necessary or not.
There is
nothing special about necessarily true premisses that permits their omission.
The stricter thinks there is, but much of his case relies on circularly
appealing back to what is at issue, the positive paradox and the assumption
that all statements entail ones that are necessarily true.
Indeed the latter,
e.g. DC -fi> D
C, together with Disjunctive Syllogism, yields (in the framework
of basic relevant logic) Necessary Premiss Suppression, as follows:
A & C - > B - * ~ B - > ~(A H C)
A & C
B, DC -F ~B
C, ~B -> ~(A & C)
~B ->. C & ~(A & C)
-o ~B -> ~A
A -> B
Rule Contraposition
Paradox principle
Rule Composition
Disjunctive Syllogism, Rule
Syllogism
Rule Contraposition.
The hooker can similarly argue, using the positive material paradox C -*»D -»• C,
that true premisses can be suppressed; and the stricter who mistakenly
accepts Necessitation, C-^QC, everywhere can -likewise argue in this way to
the (exportation) rule
A & C - > B , C-fA^-B
The rule is defective, and is defeated by any case where A and C jointly
entail B but do not do so separately and C happens to be true (e.g. by the
switches paradox presented above).
So the stricter is right as against the hooker; but he is only part-way
right, for what he says against the hooker's omission of merely true premisses
can be duplicated for the omission of necessarily true premisses.
That is,
counterexamples and countermodels invalidating the Necessity Suppression
Rule are readily devised. Inversely, what the stricter urges against those
who reject the suppression of necessarily true premisses can be recast as
castigation of those who reject the suppression of true premisses, whereupon
the inadequacy of the case becomes evident, even to some stricters. Consider,
for instance, Bennett's assertion (69, p.233) so adapted:
Because in S3 and subsystems the Necessitation Rule: A
DA is inadmissible.
Entailments, though true, are not according to the systems necessarily true.
Thus the stricter is committed to one of Quine's two dogmas of empiricism,
that the necessary/contingent distinction can be made good.
The relevant
position need not depend on this distinction at all.
31
1.4 HOW STRICT SYSTEMS ERASE A FUNDAMENTAL DISTINCTION
Someone who thinks that ... validity may be lost by the deletion
of - - - true premisses must presumably be muddling validity with
educational or heuristic satisfactoriness.
That is certainly not a claim Bennett would agree to, though it differs from
his remarkable assertion only by deletion of the word 'necessarily' (where
the three dashes occur).
Furthermore it points out the way in which it would
be argued, against Bennett, that in the necessary case there is no muddle:
it would take almost precisely the lines along which Bennett should argue that
in the truth case there is no muddle.
To put it briefly, there is no muddle
because validity requires a universal guarantee over all deductive situations:
the deletion of a true, or necessarily true, premiss can easily undermine this
guarantee (as the semantical analyses of chapters 2 and 4 will make very clear).
What Bennett really wants to insist upon is not this warm-up point but
that the suppression of necessary truths (and the corresponding negative
suppression of logically impossible statements) which the paradoxes facilitate,
is an inevitable feature of deductive reasoning, and ipso facto of applications
of entailment within philosophy.
It is on this sort of ground that the paradoxes are often claimed to be essential, and desirable, outcomes of a correct
analysis of entailment (and this very likely covers what Bennett intended, in
saying that the paradoxes are part of the case for Lewis's analysis, 69, p.198).
The critical issue here, and also underlying Lewis's independent argument for
the positive paradoxes, comes down to whether a distinction can be made between
two roles for necessary entailment statements, an inference facilitating role
and a conjoined premiss role (cf. Bennett 69, p.230 ff).
Despite Bennett's
doubts, such a distinction of roles can be drawn very easily, in both syntactical and semantical fashion.
Syntactically the distinction is that between
A -*-. B
C and A & B -»• C with A an entailment (i.e. of the form D -*• E) ; and
what permits inference is Modus Ponens, not Necessary Premiss Suppression.
The requisite distinction is obliterated by all strict systems that have ever
been seriously considered as offering analyses of entailment; for even in
system S2° A -»•. B •*• C and A & B -*• C are interderivable where A is necessitated.
Nonetheless the distinction between exported and imported forms is syntactically
quite obvious and is clear from elsewhere, indeed not just from relevant systems
but from weaker modal systems such as SO.5 which still maintains Lewis's definition of strict implication.
Semantically, the distinction is that between
situations conforming to an entailment (or principle of valid inference) and
an entailment's holding in situations (see FD and UL). Once again strict systems
erase this fundamental distinction in the case of necessary truths (so it is
not altogether surprising that Bennett fails to see it); for necessary truths
are supposed to hold everywhere.1
We can usefully adapt the distinctions between premiss roles in dealing
with attempts that have been made to reinstate the material-conditional as the
conditional and strict-implication as entailment.
Consider, for example,
the material case.
Simons argues for intuitive recognition of that version of
Exportation he calls 'conditional proof', namely: if A and B entail C then A
entails that if B then C, by way of examples like the following:1
In S2 this result is only achieved by not asserting D(A
A).
That is,
A -»• A can fail in some situations because the theory does not require that
it is a necessary truth - and really cannot so require.
32
1.4 HOW WHEW EXPORTATION VIELVS A CORRECT RESULT IT IS OTIOSE
From the theory of universal gravitation (G) together
with the initial condition that x is unsupported near
the earth (U) it follows deductively that x moves towards
the centre of the earth (C).
This is to say
(29) G, U therefore C
is a valid argument.
But "if U then C", the singular
consequence of the lawlike generalisation is thereby
established as a consequence.
But if so, it seems to
be established because
(30) G, therefore if U then C
is taken to be valid if (29) is valid (Simons 65, p.83).
The mistake in this argument can be pinpointed in the word 'thereby1; for
the point at issue is thereby assumed.
Less obliquely, (30) is not established from (29), except incorrectly; but (30) may be established along with
(29).
The argument, which illustrates a method of avoiding "conditional
proof" of wide applicability, namely an Instantiation method, is as follows:Firstly, G entails that, for every y, if y is unsupported near the earth then
y moves towards the centre of the earth. Then (30) follows by Instantiation
and Syllogism.
Simon's more general argument for conditional proof in the
sciences, from which he argues to the result that the conditional of the
sciences is truth-functional, may be dealt with in the same way. The correct
argument is not: T (Theory), A (antecedent) .*. C (consequent), so by "conditional proof": T
if A then C; but T so (lawlike) generalisation: always
if Ax then Cx, and so T
if A then C.
The incorrect exportation step is
otiose.
The Simons example also illustrates in a very elementary way more
general features of the way in which relevant theories, such as relevant
arithmetic and set theory, can operate without suppression of necessary truths;
namely it is very often possible to prove relevantly what classically or
modally is proved by suppression means or using paradoxes.
For example, in
standard relevant logics and many relevant theories it can be shown that
Rule y of Material Detachment is admissible, that whenever both A and ~A v B
are provable then B is also relevantly provable.
Modally, of course, B
results immediately by detachment from Disjunctive Syllogism, but this illegitimate step does not furnish a relevant proof of B.
A relevant proof can however always be obtained by different, and legitimate means.
The situation is
familiar from mathematics and the theory of constructive proofs: often a
theorem proved by indirect means can be proved (in a less controversial way)
constructively.
For instance, the whole of classical Peano arithmetic can
be reworked intuitionistically.
Relevant logics require not that valid arguments be constructive1 but
that for an argument to be valid it conform to relevant standards.
An argument which proceeds by suppression of necessary premisses will be admissible
only if it can in principle be reconstructed relevantly or replaced by a
relevant argument.
A great many classical arguments are so reconstructible
or replaceable (cf. investigations of relevant arithmetic).
This is more
1
Constructive relevant logics where proofs conform both to relevant and to
constructive standards can, of course, be designed: sentential parts of
such logics are studied in chapter 6.
33
1.4 BENNETT'S CASE FOR THE POSITIVE LEWIS PARADOX IS UNCONVWCINGLV
CIRCULAR
than enough to refute the view that necessary premiss suppression is an inevitable feature of deductive reasoning.
The strategy of replacing the Necessary
Premiss Suppression,
A & C
B, DC-0A->B
by the case of Modus Ponens,
C ->•. A
B, C - P A + B
,
is just a very simple example of one way in which replacements can sometimes
be effected.
Bennett has, he believes, a case for the positive paradox 'which rests
neither on Lewis's analysis nor on the second Lewis argument* (69, p.232).
It
is based on the feature of derivation and proof that we may at any stage in a
proof introduce a previously proved statement, or, in the parallel informal
case, that 'a necessary proposition is entitled to crop up anywhere in an argument without being explicitly related to anything that has gone before' (p.233).
But contrary to what Bennett assumes, this is not a 'stronger truth' than 'that
a necessary proposition is entailed by every proposition': it does not imply
the paradox at all without further assumptions, such as that the procedure can
be extended to hypothetical proof and a deduction theorem applied.
Derivation
and arguments of the line-by-line type introducing established statements are
relevantly acceptable and do not lead to paradox (as chapter 4 will make plain).
It is only if classically designed inference rules are erroneously reconstrued
as representing valid arguments - they do not - that anything like Lewis's
positive paradox is produced: the derived rule
B-» A v ~A
,
of relevant logic E, for example, does not however represent valid argument in
any sense.
There are derivation rules, especially derived rules, which
correspond to valid argument and derivation rules that do not.
The logic of
entailment distinguishes the cases, but there are other ways of doing so, e.g.
natural deduction explications of proof from hypotheses, and explications of
formal deducibility as distinct from derivation.
Now Bennett does realise
that it can be objected that the introducibility of a necessary proposition in
an argument without loss of validity is not the same as the deducibility of a
necessary proposition from any premisses (69, p.234); but he does nothing
effective about meeting the objection.
Instead, to erase the introducibility/
deducibility distinction, he appeals back to the strict analysis he is supposed
to be defending: accordingly the "case" for the positive paradox reduces to an
unconvincing circular one.
The case for the strict paradoxes is, despite
initial protestations, the strict theory itself.
For Bennett simply accepts
the Lewis thesis that
arguments purporting to prove necessary propositions cannot be sorted
out into valid and invalid simply on the basis of whether each nonpremise line is entailed by earlier lines (p.234).
And this is simply to accept from the outset, what was to be defended, that
necessary propositions are entailed by every proposition.
There is no argument, but instead another piece of tournament philosophy: no argument, but
another challenge.
Bennett invites those who do not accept the amazing Lewis
thesis to defend a rival position.
But each different theory of entailment
can furnish such a sorting of the arguments in question into valid and invalid,
at least for sufficient range of cases, e.g. by use of canonical systems the
challenge can readily be met (later chapters show how in detail).
34
7.4 THE MISTAKEN SUBSTITUTION PRINCIPLES STRICT-IMPLICATION SUPPLIES
Just as the framework of the material-conditional, classical logic, is
inadequate to define a decent conditional, so, to turn now to the attack,
the framework of strict-implication systems is inadequate to entailment.
The
basic trouble with strict-implication can be located in the substitution conditions it admits. (The key to the logical behaviour of a connective lies in
its substitutivity conditions.)
The equivalence which guarantees intersubstitutivity within strict implication sentence frames is strict equivalence,
defined as strict coimplication thus:
A 6-4 B =
(A -» B) & (B
A) .
For systems which include Feys' system S2° or Lewis's system S2 - the only
strict systems ever seriously considered as offering an analysis of entailment - coimplication can be redefined:
A M B = D f d(A = B).
The upshot is that provable material equivalents are everywhere intersubstitutible in these systems, so that a great many undesirable implicational
theses are immediately derivable.
In particular, no discrimination in
deductive power can be made between theorems, since any two have exactly the
same set of consequences - something that is obviously false.
Similarly no
discrimination can be made between logically false statements: they all have
the same set of consequences, namely everything. These disastrous results are
magnified in theories based on entailment, such as theories of logical content
and semantic information (as FD and UL explain in detail), e.g. all necessary
truths have the same logical content, namely none, and convey the same information, again none.
The worst manifestations, in purely entailmental terms, appear in the
paradoxes - that a contradiction entails every statement and that any statement whatsoever entails each necessary truth.
For any contradiction is
strictly equivalent to B & ~B, and B & ~B entails B for arbitrary B;
hence,
by substitutivity, any contradiction entails B.
Similarly using substitutivity,
B entails, on the strict account, every necessary truth.
These results can
be traced back of course, to the fact that strict implication, as an analysis
of entailment, amounts to necessary material implication.
As in the case
of material implication, however, it is not just that the analysis admits
these paradoxes.
There are several other principles admitted which (though analytic of
course for strict implication) fail to hold when -3 is construed as entailment.
The stronger the strict implicational system the more of these unwarranted and unwanted principles there are as theses.
Because of these
significant differences it is important that stricters tell us which system
of strict implication it is that really captures entailment.
But this most
stricters are extraordinarily reluctant to do, even when they are informationally equipped to make a choice1 - perhaps for good, if rarely divulged, reasons.
1
This applies in part even to Lewis, who wrote in 1932, at a stage when he
was still informationally unequipped to distinguish S2 and S3:
Prevailing good use in logical inference - the practice in
mathematical deductions, for example - is not sufficiently
precise ... to determine clearly which of these five systems
expresses the acceptable principles of deduction. (Lewis and
Langford 32, pp. 501-2).
(footnote continued on next page)
35
1.4 IMPLAUSIBLE "ENTAILMENT" PRINCIPLES OF SB
The few who do choose often plump for S51 which, despite its merits as an analysis
of modal notions such as logical necessity, is hopeless as an analysis of entailment, containing some singularly undesirable principles.
For example,
Peirce*s "law" ((A ^ B) o A) => A, rightly rejected by all logics (such as
intuitionistic logic) built on positive logic as an unsatisfactory implicational principle, reappears in an unconvincing modalised form in system S5
(but not in S4 or weaker systems) as
((A -f B) H A)
A, where A is strict, i.e. of the form C -3 D.
In fact S5 is distinguished as a strict implicational system from its rival
S4 by the modalised Peirce "law" or one of its complications such as:(((A-?B)-JC)-JB)-i. A -J B
(((A -J B) -3 C)
B)
B-iD-J. A W D
(((A-3 B)
C)
B) -5. B -i D -?. E -i. A—? D
(((AH
( ( (A
B ) - I C ) - J D ) -4. B
D -3. A -3 D
B)
C)T3 D )
B - » D -3T. E
. A-* D
Although none of these principles is evident2 or indeed at all plausible as an
(footnote 1 continued from previous page)
Apart from the limitation to the five systems S1-S5, this is a claim with
which we have a good deal of sympathy (cf.chapter 3). Lewis's indecision, such
as it was, was not to last.
Already in Lewis and Langford (32, p.178) it is
said:
It is this System 2 which we desire to indicate as the System of
Strict Implication;
and by 1960 Lewis wanted to insist:
I wish the system S2 ... to be regarded as the definitive form of
Strict Implication (Preface to Dover Edition of Lewis 18).
l
The choice is often made on such insufficient grounds as strength, mathematical
elegance and simplicity: cf. Lewis's astute observation (32. p.502):
Those interested in the merely mathematical properties of such systems
of symbolic logic tend to prefer the more comprehensive and less 'strict'
systems, such as S5 and Material Implication.
The interests of logical study would probably be best" served by an exactly opposite tendency.
Occasionally however the choice is based, more satisfactorily, on the semantical structure of S5 and its logical theory of modalities.
This points up
the unfortunate historical conflation of entailment with modality, and how
a very likely account of logical modality can lead to a most unlikely account
of logical implication.
2
AS Prior (62, p.49) remarks of Peirce, p -»• q -*• p -»• p, itself:
Its truth is not evident at a glance, but we might look at it
this way: it means
If p does not imply q without being true, then it i:s true.
That is, it means
Either p implies q without being true, or it is true.
If p is true the second of these alternatives holds.
And if p is false, then it does imply q (for COq = 1,
whatever q may be) and of course does so without being
true, so in this case the first alternative holds.
(footnote continued on next page)
36
1.4 LEWIS SYSTEM S2 AMD OTHER MODAL AFFIXING SYSTEMS
like the system S3 that Lewis once favoured, has to be made.
It is easily
demonstrated (using matrices or a semantical analysis of non-normal modal
systems, e.g. Kripke 65) that S3 has no theses of the form
A -5 B.
The focussing by Lewis of the choice of a correct (or as he would have
said, acceptable) logic of deducibility on the systems S3 and S2 undoubtedly
represented a vast improvement over most of what had gone before him, especially in the direction of the formalisation of the logic of entailment.
Since
S2 is, according to its author, 'to be regarded as the definitive form of
Strict Implication' (Lewis 60) and since it provides the locus of much subsequent discussion of entailment, we exhibit the system in explicitly entailmental form, not in the usual modal form.
Connectives -*• (symbolising Lewis
entailment, i.e. Strict Implication), &, and ~ are taken as primitive, and
further extensional and modal connectives are defined thus: A v B
~(~A & ~B); A a B - d £ ~A v B; A = B « D f (A = B) & (B => A) ; A
B » D f (A
B)
& (B -*• A); DA
~A
A; 0A » D f ~0~A. (A detailed discussion of the morphological structure of sentential entailmental systems may be found in 4.1.)
The postulates of S2 in Lewis entailmental form are these:- A •* A (Identity)
(A -*• B) & (B -*• C)
A -»• C (Conjunctive Syllogism)
A&B+A
A&B+B
(Conjunctive Simplification)
(A -*• B) & (A
C)
A -»» B & C (Composition)
A & (B v c )
(A & B) v C
(Distribution: the scheme is not independent in S2)
~~A -»• A (Double Negation) A •*• ~B •*•. B -»• ~A (Contraposition)
(A
~A)
~A (Reductio) A&B-+C->-. A&~C->-~B (Antilogism)
A, A -*• B
B
(Modus Fonens) A, B
A & B (Adjunction)
A -*• B, C
D-»B
C •*•. A
D
(Affixing, or Becker rules).
Other strict implicational systems of entailmental interest are these
(the classification anticipates that we subsequently adopt in 3 for entailmental systems):Other Affixing Systems: These result by varying the axiom schemes for S2;
e.g. Feys' system S2° results by deleting Reductio, or what perhaps surprisingly
(and wrongly in the larger relevant context) is axiomatically equivalent:
A & (A
B) -»• A, (Conjunctive) Assertion.
Another example is the relevant
logic DL (of DCL, formulated without sentential constants) which results from
S2 by deleting Antilogism; and yet another example is the relevant logic DK
(of UL) which results from S2° by deleting Antilogism, i.e. from S2 by deleting
both Antilogism and Reductio. The formulation of S2 given thus brings out
sharply the key separation point between strict and relevant accounts of
entailment, a point to which we will repeatedly allude (especially 2.6 ff),
namely the principle of Antilogism and its outcome, Disjunctive Syllogism. It
was this one major obstacle that kept Lewis forever out of the relevant paradise.
(But really the unquestioned loyalty to Disjunctive Syllogism and its
mates, such as Rule Antilogism, is part of the larger classical-modal psychosis
which we subsequently analyse.)
Hitherto the main concentration in relevant
logic studies has been on analogues of the exportative affixing system S3,
obtained by addition to S2 of one form of Exported Syllogism, i.e. A -*• B
C -*• A
C -*• B (Prefixing) , or A
B
B -»• C
A -»• C (Suffixing), or of one of
its modal equivalents, e.g. A-*B-*-. DA-*- []B (Necessity Distribution), or
A -»• B -»-. OA -*• 0B (Possibility Distribution).
Plainly both the Affixing rule
and Conjunctive Syllogism are otiose in such exportative affixing systems.
Other Exportative Affixing Systems.
These include S30 - S2° + Exported
Syllogism; S4 which adds to S3 any of the following schemes: B
A -*• A,
•(A -»• A), DA -*• CDA, OCA
<A; 0(A & ~A) -»•. A & ~A; and relevant system T
38
1.4 A CLASSIFICATION OF MODAL LOGICS OF ENTAILMENTAL INTEREST
which adds to DK both forms of Exported Syllogism and A -*• (A -*• B) -»-. A -»• B
(Absorption).
(On formulations of modal systems, see Feys 65; for formulations of exportative affixing systems, see ABE p.340 ff.) The very close
connection of Anderson and Belnap system E with Lewis system S3 can be
brought out by a slight reformulation of S3.
Take the formulation of S2
given, and replace Conjunctive Syllogism by its exported form, Suffixing,
i.e. A
B -»•. B -»• C
A •* C, and Affixing by N-necessitation, i.e.
A-fcA -*• B •*• B; so results S3.
Now E is obtained simply by deleting
Antilogism; i.e. there are formulations of E and S3 which are separated
just by the principle of Antilogism.1
In the quest for a correct entailment system or group of such systems,
it is important to look inside S2 and DK as well as beyond.
So result two
further classes of systems, both non-affixing systems:Replacement Systems. These systems result from affixing systems by replacing
the Affixing rule by a Replacement rule, i.e. a rule of intersubstitutivity
of provable coimplicants:
A **• B, C(A)
C(B)
(Replacement or Substitutivity),
where C(A) is some wff containing zero or more occurrences of A as a well
formed part and C(B) results by replacing zero or more of these occurrences
of A by B.
The affixing systems introduced are all replacement systems in
that Replacement is a derived rule.
The new class of systems of interest
comprises replacement systems which are not affixing systems.
Best known
of these in the modal arena are the systems SI and Sl° which result from S2
and S2° respectively simply by replacing Affixing in the initial formulations
given by Replacement.
Relevant analogues of those systems result in a
similar way from DL and DK.
Non-replacement Systems; These systems dispense with replacement rules, and
so characteristically lack--affixing rules also.
Examples are provided, in
the modal case, by systems in the vicinity of SO.5 and, in the relevant case,
by Arruda-da-Costa P systems.
The issues raised by these various different classes of systems, in
particular the correctness or satisfactoriness of Replacement, of Affixing,
and of Exported Syllogism, are taken up in chapter 3.
The classes of systeas
differ not only syntactically, but the difference is reflected in a deeper
way semantically: the various semantical analyses are the main concerns of
chapter 4 and later chapters. The criticisms already made and now to be made
of modal affixing systems, and especially of preferred Lewis systems S3 and
S2, extend, for the most part, to non-affixing systems.
Although systems S3 and S2 avoid some of the worst of the paradoxes of
normal strict implicational systems, e.g. Ackermann fallacies,2 they hardly
offer a panacea.
We set out criticism of these systems serriatim.
1
The analogues of a system obtained in this way are far from unique.
For
example, corresponding to S2 is not only DL but a system ES2 obtained
from reformulating S2 with N-necessitation as an added rule by deleting
Antilogism. ES2 retains the restricted Commutation principles
of E while abandoning Exported syllogism principles.
2
Ackermann fallacies are inevitable in normal modal systems.
The argument
which proves this relies on the standard definition of normal for these
systems, and takes the following semantical lines:Suppose where A
B
is a theorem, that p
A -»• B is not valid.
Then, presupposing details
(footnote continued on next page)
39
1.4 RELEVANT CRITICISM OF SVSTEMS S2 AND S3
1) The systems still contain a great many undesirable principles as theses,
the worst of which are of course the much advertised paradoxes of strict implication which occur in S2 in the forms: OB
A
B and
A -3 B (in SI
there are equally damaging T-forms of these paradoxes).
Since every truthfunctional tautology is provably necessary and every truth-functional contradiction demonstrably impossible in S2 (and in SI) the familiar "independently
provable" paradoxes already discussed emerge, and also typical irrelevant
entailmental principles such as C ~t D
A ~f A and C
D
A
B where A -3 B
is any theorem.
But the systems also contain several principles independent
of the paradoxes which, if not objectionable, are least very questionable.
(We will question them, and reject them in chapter 3 where we present our
case for concentrating on different entailment logics from the usually favoured
systems E and NR.)
For example, S3 and stronger systems share with relevant
system E such theses as A -*• (B
C) -*• D -»-. B
C
A
D (Restricted Commutation) and (A -*• B) -»-. (A •*• B)
C
C (Restricted Assertion) . In S2 these
counter-intuitive principles appear in rule form, and in both cases they lead
to such defective theses as ((A
A) -»• B) -»• B (Suppression) and, more generally,
(D
B) -»• B where D is a thesis, i.e. to the Anderson and Belnap rule of
necessitation A
NA where NA = ^ (A -*• A) -*• A. In these entailmental respects
53, like its relevant mate E, is too strong: but in modal respects S3 is too
weak.
2) The elimination of the worst of the paradoxes of implication by non-normal
modal logics is far from costless.
For it is achieved only with the simultaneous elimination of apparent implicational truths. In particular, S2 and
S3 lack the expected principles 0(A
A) and D(A & B-^A); more generally
they lack D(A -3 B) for each tautology A ^ B.
The addition of any one of
these principles takes S2 into modal system T (von Wright's M) and S3 back into
54, and thus leads to Ackermann fallacies.
Yet surely the entailment A -*• A
is necessarily true, and a satisfactory modal system should assert its necessity. 1 So emerges a serious dilemma for the stricter trying to choose the modal
system that really captures deducibility, a dilemma which is a direct outcome
of the erroneous strict conflation of entailment and modality (a conflation
that is retained in systems like E).
The rejection of incorrect entailment
principles, e.g. the avoidance of Ackermann fallacies, limits the choice of
acceptable modal systems for entailment to non-normal systems, whereas the
retention of correct modal principles for logical necessity, such as D(A -> A)
and DA -> QUA, forces the choice back to normal modal systems.
Relevant logics
escape this dilemma, most simply by observing a proper separation of entailment
and logical modalities, which modal logics confuse.
(footnote
2
continued from previous page)
of the semantics to be developed in chapters 3 and 6, for some normal world
a, I(p,a) = 1 and I (A •*• B) = 1, whence for some normal world b, for which Rab,
I(A,b) = 1 ^ I(B,b).
But as A -»• B is a theorem and so valid, for every normal world if I(A,b) = 1 then I(B,b) » 1, which is impossible.
Thus p -*•.
A -»- B is valid, and so a theorem.
1
In this respect E appears to represent a substantial improvement upon S3,
since D(A -*• A) (with • equated with N) is a thesis of E.
However, as
E c S3, N(A
A) is also a thesis of S3.
One of the troubles with S3 is,
to put the point under discussion differently, that S3 admits of two nonequivalent accounts of necessity, • and N.
In a satisfactory modal theory
these accounts would coincide; however from the relevant viewpoint to be
advanced neither account is satisfactory (see further, chapter 10).
On the normalising effect of [D(A" => B), where A => B is a tautology, in
SI, S2 and S3, see Feys 65, pp.99 and 127.
40
1.4 FURTHER FALLACIES TO AVOW
IN THEORIES Of ENTAILMENT ANP
COmTlONALlTV
3) The modal weakness of S2 and S3 can be exposed in another, equally
damaging, way - through their Hallden incompleteness.
A sentential system
is Hallden reasonable iff, whenever A v B is a theorem with A and B wff which
share no sentential parameter, either A is a theorem or B is a theorem.
Hallden showed (in 51) that both S2 and S3 (and indeed a range of modal systems
from SI through S3) are unreasonable in this sense, and argued that this revealed a serious semantic incompleteness in these systems, namely that there
are no normal interpretations with respect to which the systems are complete.
But strengthening S2 and S3 to remove this incompleteness leads back yet again,
if the modalities are construed as genuine logical modalities, to normal
modal system S4 (alternatively, under conventional construals of modalities,
removal leads to S6 and S7).
Yet again this problem is avoided by relevant
analogues of the modal systems, which are (as we show in chapter 5) Hallden
reasonable.
In sum, there are severe problems with the non-normal strict implicational
systems favoured as analyses of entailment, problems which are in each case
overcome by analogous relevant systems. The most notable problems with strict
implication, the validation of a great many undesirable principles, are an
inevitable outcome of the fact the strict implication is a modal account of
implication, which tries to account for two-place entailment connections in
terms of a one-place modal connection in tandem with truth functions. Such
a modal reduction cannot, of course, succeed with entailment; nor can it
succeed for conditionals.
Determination of adequate theories of entailment - and, so it will turn
out, of conditionality - requires, to summarise then, avoidance of the
following further fallacies:5 . The modal dependence fallacy. The fallacy is that of supposing"that
whether A -*• B holds true sometimes depends just on the modal value (e.g.
necessity, impossibility) of one of the components A and B - independently of
any connection between A and B.
Neither the implications a statement has,
nor what implies a statement, are ever functions of the statement's modal
values, of its impossibility, contingency or necessity.
The Stalnaker and
Lewis theories of conditionals commit the modal dependence fallacy as well
as the truth copulation fallacy. A special case of the modal dependence
fallacy is
6. The necessity suppression fallacy. The fallacy is that of taking A -> B
as true where B is necessary.
7. The modal reduction fallacy. The fallacy is that of defining A -»• B as
i(f>(A,B) where i is a (one-place) modal connective and 4>(A,B) is a truthfunction of A and B.
More generally, neither entailment nor conditionality are modal. Any
adequate account will meet this strong intensionality requirement; and this
is the basic reason why the investigation of implication generally has to
be in the area of the intensional beyond the modal, why - to state a central
thesis soon to be advanced - it has to be ultramodal.
§ 5. Conditionals: the theory sought in contrast to extensional and modal
attempts. The orthodox attempts at gaining an analysis of entailment - the
metalinguist ic shift to avoid overt systemic introduction of modal and intensional notions, and the strict implication account in terms of modality - leave
one with no satisfactory account of non-logical implication. Thus, given that
our case against material-implication stands up, the problems of a natural implication, and of conditionals, are not satisfactorily resolved at all within the
usual metalinguistic and strict implicational frameworks. Yet a great many
41
7.5 'IF' STATEMENTS, AMD WHV SOME NEW CLASSIFICATION OF THEM IS ESSENTIAL
implicational statements are not analytic, for instance the lawlike statements
of science and everyday conditionals, and many conditionals are, as we have
noticed, not implications: but such statements are commonly deployed in
reasoning, e.g. in the empirical sciences, and in legal and ordinary arguments,
and they have a logic, distinct from material-implication, which it should be
the business of a full theory of implication to capture.
In an effort to fill these lacunae there have recently appeared on the
market various theories of conditionals, which are supposed to supplement the
modal view and thereby overcome its acknowledged weaknesses on the conditionals
front.
Logical space is made for these theories, not by abandonment of the
thesis that entailment is necessary implication but by the distinction of nonnecessary implication from the conditionality relation.
The conditional
connections these theories are intended to capture are supposed to be neither
sufficiency relations nor extensional connections such as material-implication
(cf. Stevenson 70, pp.28-29).
So much is not in dispute: There certainly is
a class of non-sufficiency conditionals, in common usage in natural languages,
whose intensional analysis is an important logical matter.
Although some
have hoped that these conditionals could be defined enthymematically in terms
of a satisfactory sufficiency relation, there is nothing to stop, and reason
to encourage, their independent logical investigation.
For, among other
things, if enthymematic reducibility is to be established as distinct from just
claimed, an independent account of -the class to be reduced needs to be given.
There is not only a strong case for independent logical investigation of
(non-sufficiency) conditionals - and conditionals are after all central in much
everyday reasoning - but there is also a good case for jettisoning many former
divisions and exclusions built into or presupposed by theories of conditionals.
For example, there are semantical reasons for not trying to make the artificial
and never satisfactorily determined or vindicated division of English conditionals into "counterfactuals" and "noncounterfactuals", where a "counterfactual" usage is not simply (a counterfactual) one where the antecedent is
false, but 'one in which its antecedent is uttered in the knowledge, belief,
or temporary supposition that it is false' (Cooper 68, p.291).
Part of the
argument is that there is no evidence that 'if' differs in semantical meaning
when contextual conditions are so varied, and indeed evidence that it does
not - just as there is evidence that the correctness of an implication cannot
be determined from the truth- or modal-value of its consequence.
Likewise
there is a prima facie case for not separating out counterlogical or counterphysical conditionals, i.e. those with logically or physically impossible
antecedents.
Nonetheless it seems that any non-vacuous logical theory of
conditionals is bound to exclude some natural language occurrences of if ;
that is, it seems that there is no non-vacuous way of encompassing all occurrences of 'if' within a single uniform analysis.
The argument, such as it is,
is that any all-encompassing theory would be vacuous because any proposed
thesis it contained could be defeated by a counterexample of sorts.
Remember
that in English 'if' can do duty for such connectives as 'whether', 'even if*
and 'as if', as well as for 'if ... then'.
The claim is that the intersection
of the logical properties of these diverse connectives is null.
Consider, to
illustrate the inconclusive case-by-case method of confirming the claim, a
few of the more likely or desired principles of an i^ of argumentative value.
Firstly, consider detachment, i.e. Modus Ponens.
This appears to fail in
cases where 'if' can be replaced by 'whether', e.g. from "you see if you can
turn the wheel" and "you can turn the wheel" we cannot validly infer "you see".
Next, consider Identity, and let 'if' do duty for 'even if* in the intended
42
1.5 GENUINE CONDITIONALS SURVIVE THEN-TRANSFORMATION
sense where this makes a contrast, as in 'he is intelligent if crazy*.
Then "he is intelligent (even) if he is intelligent" is not just weird but
not true. Differently "I am wondering if 1 am wondering" is not always
true. In the same way, by taking examples where 'whether* or 'even if' can
be substituted for 'if', countercases to such basic principles as Conjunctive
Simplification can be devised.
No principles, it would seem, survive this
sort of procedure.
The procedure does indicate, however, the first kind of condition that
should be used to restrict the class of genuine conditionals, namely that
uses of 'if' where 'if' does duty for 'whether', 'even if' or 'as if' should
be excluded (at least from the core theory).
There is a simple linguistic
test which shakes out many of the undesired cases, namely reordering and then
insertion.
The test, which we call then-transformation, is this: given
'A if B' reorder to 'if B, A' and insert then to obtain 'if B, then A';
finally ask whether the result makes sense or has the same sense as the
original.
Examples: 'if you can turn the wheel then you see' does not have
the same sense (as qualified salva veritate tests will show) as 'you see if
you can turn the wheel', and 'he is sound if unimaginative' does not have the
same sense as 'if he is unimaginative then he is sound'.
A necessary condition on a genuine conditional is that it should survive
then-transformation. Genuine conditionals do appear to have a logic, even if
the pure conditional logic itself is not of vast logical interest, consisting,
as it seems to at bottom of Modus Ponens and Identity (such a logic is reviewed
in 2.1 and subsequently in chapter 5: the minimal B conditional logic is in
fact the pure implicational fragment of system B of chapter 4, and all its
theorems are instances of Identity).
But, rather like the parallel logic of
deducibility, the interest and difficulty of a general conditional logic
quickens when further connectives, such as those of the orthodox set {&, v,
are introduced.
Many principles that appear correct for an implicational logic fail for
a general conditional logic - something to be expected since the I-C transformation (of 1.3) fails for many genuine conditionals.
For example Contraposition which appears correct at least in some forms for implication (e.g.
in the form: when A ->- B then ~B -»• ~A) fails for conditionals taken generally.
Counterexamples to Contraposition abound: while "if you are tired then we
will sit down" may be true "if we won't sit down then you are not tired" is
commonly false; "if it rains then I'll take my umbrella" rarely ensures "if
1 don't take my umbrella then it doesn't rain"; and so on.
The failure of
Contraposition (especially for "temporal" if-thens) is indicative of the fact
that conditionals do not in general meet sufficiency requirements: that the
antecedent A of a true (or acceptable) conditional A •*• B is commonly not
logically or naturally sufficient for the consequent B.
When A is not sufficient for B, further background assumptions are involved in getting B from A,
so that when ~B holds it may not be A that is negated but some of the further
assumptions that lie in the background.1 But not all negation principles fail:
1
Syntactical enthymematic representation of the conditional A 3 B in the
relevant framework as A & t -»• B, where t represents some background class
of truths, shows up a technical reason for Contraposition failure.
~B D ~A would require ~B & t -»• ~A, that is it would depend on the rejected
Antilogism. But the syntactical representation is of only limited value,
since it sanctions irrelevance; it also happens to validate Augmentation.
43
7.5 THE PROBLEM OF THE WATERSHED PRINCIPLE OF AUGMENTATION
Double Negation, for example, appears to hold in both directions. Contraposition is not the only implication principle that is faulted when ordinary conditionals which do not pretend to offer sufficient conditions are countenanced.
With conditionals, as distinct perhaps from implications and entailments, very
many commonly accepted logical principles are up for reconsideration, and
rejection.
A critical issue in determining the general logic of conditionals is
whether the rule Affixing holds; for the rule is a watershed principle in
semantical analyses of implicational-conditional logics (as will become evident
in subsequent chapters). An important test case for Affixing is provided by
the principle
A -*• C -*•. A & B -*• C
((Antecedent) Augmentation)
in combination with A & B -*• A; for were Affixing correct, Augmentation would
result from Conjunctive Simplification. But according to recent logical wisdom
Augmentation fails for the conditional relation.
The case against Augmentation
is two-fold; it tends to be based on intuitive counterexamples, these being
backed up by various, incompatible, theoretical explanations drawn from the
different theories that have emerged. A familiar counterexample (Stalnaker
68, p.106; Stevenson 70, p.27) runs as follows: there are cases where it is
true that if you strike that match it will light without it being true that if
you wet that match and strike it it will light.1
Another example is this:
"If I walked on the ice, it would remain firm" could be true, while "if I
walked on the ice and I wore 60 lb. boots, it would remain firm" is not
(cf. Bennett 74, p.384).
There are ways of explaining away examples like
these; but there are, so Lewis, Bennett and others contend, too many examples
for the dismissals to continue to look convincing. As almost always, the arguments for this kind of claim are not decisive. Bennett's argument, for example,
makes the assumption that moves to explain away apparent counterexamples to
Augmentation must treat conditionals as elliptical (or enthymematic), to be
expanded, in determining the logic of conditionals, by further clauses conjoined
to the antecedents.
And it is true that there are then well-known difficulties
about ever completing such clauses satisfactorily, since the background conditions to be incorporated are open-textured in several ways which defy complete
linguistic description.
But the strategems aimed at explaining away the
counterexamples do not have to be of this sort; they can legitimately refuse
to take up background or contextual conditions syntactically in antecedents.2
It is enough, presumably, to meet the "counterexamples" to contend that
the conditions which falsify the conclusion of a deduction of A & B
C from
A
C (to take an entailmental form of Augmentation), that is typically which
guarantee A & B but falsify C, undermine the basis for, or are incompatible
with, or background conditions for, A -*• C.
Thus in Bennett's example, my
wearing 60 lb. boots undermines (or in Adams' revealing terms, is incompatible
with) the background or contextual conditions under which the conditional "If
I walked on the ice, it would remain firm" is held true.
This strategem, which
we call the A-stratagem (after Adams 65), is of very general application,
defeating, in its fashion not merely all counterexamples to Augmentation but
also those to Conjunctive Syllogism and to other conditional principles that
initial intuitive considerations might seem to reject (cf. p.10 above).
1
It is enough that there be appropriate cases to rebut an entailment or conditional.
The fact that the Greenlite matches that Routley often has to use
at Plumwood Mountain will light after being dipped in water does not count
against familiar cases where non-waterproof matches are used. The fact does
help expose however one of the background assumptions - that that match is
non-waterproof - thereby emphasizing the non-sufficiency character of the
connection.
(footnote 2 on next page)
44
7.5 THE SEPARATION OF RULES FROM THESES IN THE LOGIC OF CONDITIONALS
On the other hand, there are, rather obviously, various difficulties
with this sort of stratagem. For one, it is no longer possible to assess
on their own untreated conditionals, conditionals that have been subject
to no analysis or investigation1: to know whether a conditional holds one
needs to know, so to speak, what is supposed to follow from it.
And this
makes conditionals virtually unassessible, and a great many ordinarily acceptable conditionals false - because otherwise, were they true, we could, by
defeating background conditions, counter Augmentation.
Given that Augmentation is to be rejected for untreated conditionals,
and also for important subclasses of conditionals such as counterfactuals,
given indeed that Augmentation is a "fallacy" (as D. Lewis puts it), an important question for formalisation arises: What forms of the principle are
fallacious? The same difficult question arises also for other rejected principles, Contraposition and Syllogism.
Granted that both conditional and entailmfental forms of Augmentation fail for conditionally, i.e. neither Antecedent
Augmentation as formulated above nor its entailmental reformulation,
A -* C
A & B
C (where
distinguishes entailment), are correct, what of
the rule form: A
C —t>A & B -*• C?
The rule is evidently separable from
theses forms, and does not succumb to the counterexamples given to theses
forms.
It is noteworthy, for instance, that, despite Stalnaker's emphasis
on the breakdown of Augmentation, the formal system Stalnaker arrives at admits the rule form.
Similarly, though Hunter and Graves 73 consider Augmentation to be a principle which differentiates conditionality from entailment,
they adopt principles (including Conjunctive Syllogism) which lead to the
rule form of Augmentation.2
Thus an easy answer to the question about the
rule form is that we can leave the matter open for the present: what fallacies
there are, if any, are avoided by avoiding the theses forms.
The distinction of rules and theses is important not merely with respect
to the axiomatisation of the logic of conditionals, but also as regards the
common claim that the failure of Transitivity (in conjunctive form) follows
from the failure of Augmentation (Stalnaker 68, p.106; Lewis 73, p.32ff;
Bennett 74, p.385).
Unless entailment is erroneously identified with strict
implication, this common claim is false.
The assumption is that Simplification, A & B ->• A, being logically necessary, can be suppressed; so given
Conjunctive Syllogism in the form (A & B
A) & (A
C)
A & B
C, Augmentation results (and similarly where the main connective is strengthened to an
entailment).
But such suppression of conjoined necessary premisses is inadmissible, and fallacious.
All that the failure of Augmentation establishes
against Transitivity is the failure of Suffixing (in thesis and rule forms),
not the failure of Conjunctive Syllogism.
Only the failure of the rule form
(footnote 2 from previous page)
Bennett has, it may be alleged, made the similar error - in assuming that
background conditions can be taken up as antecedent conjuncts - to that he
made in his "case" for the general suppression of necessary truths.
1
Thus heavy use of the A-strategem would violate our earlier methodological
strategy.
2
The collapse of the original Hunter-Graves system does not detract from
this point.
For the point is correct for subsystems of the HunterGraves one.
45
7.5 THE FURTHER CASE AGAINST CONJUNCTIVE SYLLOGISM, AND ITS DISSOLUTION
of Augmentation would undermine Conjunctive Syllogism, as it would undermine
even Rule Syllogism, A
B, B
C
A -»• C. Whether Rule Syllogism holds or
not makes a substantial difference to the semantics of a logic (as will become
evident in chapter 13); so the determination of exactly what is supposed to
be rejected in conditional and counterfactual logics is not a merely idle matter.
From this perspective the way the common rejection of transitivity is stated
is rather sloppy, not to say misleading; thus, for instance, Stalnaker writes:
'... the conditional corner is a non-transitive connective.
That is, from
A > B and B > C, one cannot infer A > C* (68. p.106).
But in the usual way
in which inference rules are stated (e.g. in Church 56) one is entitled to
infer, in Stalnaker*s system C2, A > C from A > B and B > C (for when A > B
and B > C are valid so is A > C).
The claim intended is that A > B and B > C
do not entail A > C; that there is, in this sense, no valid inference from the
premisses to conclusion.
The remaining case, directed against Conjunctive Syllogism is again twofold; arguments based on direct counterexamples, and more theoretical arguments from the analyses of conditionals proposed.
We have already glanced at
the sorts of direct counterexamples offered, and observed that there are conventionalist strategems by which they can be disposed of; and one stratagem
will do, the A-strategem.
Consider how this disposes of a counterexample to
Syllogism from Bennett:
If there were snow on the valley floor, I would be skiing along it;
and if there were an avalanche just here, there would be snow on
the valley floor; but it is false that if there were an avalanche
I would be skiing on the valley floor —
(74, p.385).
According to the stratagem, the antecedent of the conclusion, that there be an
avalanche, is incompatible with the truth of the initial premiss, because if
there were snow from an avalanche I (or Bennett) would not be skiing along it.
Insufficiency implications, such as conditionals are, are contextually-bound:
they are true at best under present conditions, and when the conditions are
changed, as by an avalanche, they fail.
The apparent counterexample emerges
only, so the A-strategem complains, by changing the contextual conditions as
we go from premisses to conclusion.
As we have observed, however, in the case
of Augmentation, the A-strategem, may do too much; and there is, moreover, a
good methodological case for a theory of (relatively) untreated conditionals.
Furthermore if a general theory of untreated conditionals can be devised, we
can then check the adequacy of reductions and likewise of strategems rigorously,
and we can also investigate the particular logics of conditionals which satisfy
given classes of conditions, e.g. contraposable conditionals, transitive conditionals, augmentable conditionals, to begin on a classification by logical
principles of conditionals (cf. Rennie's procedure in the case of adverbial
modifiers, in 74).1
The idea is that the logic of augmentable, contraposable
conditionals, for example, will emerge as a special case of a more general
conditional logic, in something the same way that the' theory of commutative
groupoids comes out as a special case of the theory of groupoids.
Similarly
the logic of counterfactual conditionals, as a case of the logic of lawlike
conditionals where the antecedent is false, will be a subcase of the logic of
l
The classification of conditionals by way of logical principles they satisfy
is only one, among many, of the classifications that might be tried. Other
classifications, most of them less than fully satisfactory, already occur in
the literature, e.g. Goodman's classification in 55.
46
7.5 FAILINGS OF RECENT (IRRELEVANT) THEORIES OF CONDITIONALS
transitive augmentable contraposable conditionals, assuming (as we shall argue
in chapter 8) that these properties hold for lawlike connections.
The acceptance of Conjunctive Simplification separates main modal theories
of conditionality from connexivism, a position with which, because of the
rejection of Augmentation, they otherwise have a good deal in common.
For
the deletion and defeating features of the added antecedent in Augmentation is
exactly what lies behind the connexivist critique of Augmentation, a critique
which extends automatically to apply against Simplification (see chapter 2).
It is not too surprising, then, that modern theories of conditionality are
split by Conjunctive Simplification into two factions, a connexivist group
(which includes Cooper 68, Stevenson 70, Downing, e.g. 61) and a non-connexivist group (which includes not only resemblance theorists such as Woods 67,
Stalnaker 68 and Lewis 73 but many others, e.g. Hunter and Graves 73, Xqvist
71, Gabbay 73).
Many of the recently proposed theories of untreated conditionals, whether
connexivist or not, have the alleged virtue that they do not retain the faulty
principles of Augmentation, Contraposition and Transitivity.
But most of
these theories are nonetheless defective, in that they incorporate principles
which do not hold for conditionality, in particular relevance-violating principles.
Thus they succumb, exactly like the theories they reject, to intuitive
counterexamples.
For example, the most publicised systems (American ones of
course), those of Stalnaker and Thomason 70 and of D. Lewis 73 are badly
irrelevant.
Both the Stalnaker-Thomason system and Lewis's official logic of
counterfactuals have as theses (and in rule form) the paradoxes: (~A -*• A)
(B A), i.e. anything implies a conditionally necessary statement, and
A & B
A -»• B, thus any truth conditionally implies any other truth.1
In
short, the theories provide no requisite connection between true antecedents
and true consequents in a conditional or between any arbitrary antecedent and
any necessary consequent: in such cases the accounts are every bit as fallacious as material-implication and strict implication, committing truth-copulation and modal-copulation fallacies.
These systems also contain several
other bizarre or undesirable principles - arising in part from the fact that
they were obtained principally by working back from a possible world modelling
to an axiomatic basis.
For example, Stalnaker's system (investigated formally
by Stalnaker and Thomason) includes the principle (A
B) v (A -»• ~B) already
criticised, the paradoxes (A -*• ~A)
A -*• B (a principle which also rules out
proper treatment of counterlogical conditionals) and A = (B =. A •*• B), and the
dubious principle ~(A -»• ~A)
A -*• B =. ~(A -* ~B) .
In rejecting these theories we also reject the particular semantical and
philosophical analyses in terms of which the acceptances and rejections of
logical principles are justified.
The worst faults of the semantics, the
modal framework, we shall criticise, but the question of what can be salvaged
from the extraordinary theories of neighbouring worlds and similar worlds
which are written into the intended construal of the semantics, we postpone
until we elaborate our own (ultramodal) theory of conditionals (at which stage
resemblance theories are severely criticised).
1
~A -»• A
B -»- A holds in all three Lewis systems, and A,
in two systems.
47
A ->• B
7.5 THE INADEQUACY OF MODAL THEORIES
Other recently proposed theories of conditionals, especially those that
are properly equipped with semantical modellings, are equally defective, both
in terms of the principles they contain and as regards the underlying philosophical analysis.
For example, according to Aqvist's 71 system a principle
of connectivity: (A -*• B) v (B -*• A) , where B is of the form C •> D, holds,
while according to Gabbay's 73 system, if A s B is a theorem, so is A •*• B,
whereupon the full force of the paradoxes is immediate.
One trouble with all these theories of conditionals is that they take as
entailment, as the underlying logical implication, strict-implication - something we have already seen to be inadequate.
But the basic trouble with these
theories of conditionals - of which the adoption of strict-implication as
logical implication is symptomatic - is that, like strict-implication they
operate essentially within the possible worlds framework of modal logics, and
so are subject to the inevitable deficiencies of such approaches, namely modal
substitutivity conditions and ensuing paradoxes and sundry other unpleasantness.
As customary, we say that an n-place connective $ is modal in the ith
place iff whenever A B B is true (or A = B is a theorem) $(... A ...) iff
$(... B ...).
A functor is modal iff it is modal in each place, and a sentential language is modal iff all its connectives are modal.1 Possible world
modellings are only adequate for modal languages.
Modal notions have been raised to a position of excessive prominence in
recent philosophy, partly due to the mistaken idea that the entailment connection, propositional identity, and conditionality are modal matters, and partly
owing to technical limitations, that known semantical analyses - possible world
semantics, and their metalinguistic correlates involving state descriptions succeeded at best for modal notions.
This elevation of modal notions has
done an enormous amount of philosophical damage, not merely in the case of
entailment and conditionality, but with a range of other notions of major
philosophical importance and interest such as evidence, confirmation, belief,
knowledge, perception and obligation (see UL).
It has even fostered the
grossly mistaken, but widespread, idea that all intensional notions are modal
or, if not, merely sentential (admitting replacement of no logical equivalents
of any sort but only of identical sentences).
The fact is that a great many
notions fundamental in philosophy (including all those mentioned above and
many more) are of more than modal strength: they are ultramodal intensional
notions, which cannot be forced into the modal straitjacket without grave distortion and the generation of many gratuitous philosophical problems (as UL
and BP explain).
Most important here, none of the main notions we are seeking
to explicate, namely entailment or logically necessary implication, law-like
implication, and conditionality, nor any of the associated notions, such as
propositional identity, are modal.
All are ultramodal. Take implication.
It is not a modal matter; for the substitutivity conditions required are
stronger than modal, i.e. formally the truth (or provability) of A H B guarantees neither A -»• C iff B
C nor C -*• A iff C -*• B. The ground we have already
traversed shows this, and more.
More generally, there can be no reinstatement, by patch-work repairs and qualifications from strict-implication and
material-implication and similar purely modal and extensional notions such as
the modern probability relation, of strict-implication or material-implication
as satisfactory accounts of entailment and conditionality, and similarly there
can be no constructions of satisfactory accounts from these elements.
For
such accounts always satisfy modal substitutivity conditions.
This disposes
of many proposed analyses (e.g. those of Adams 65, Stevenson 70, Geach 74).
1
These technical senses take up one (the main) traditional sense of 'modal'.
48
1.5 CHARACTER OF THE SOUGHT THEORY OF CONDITIONALS
So results a further condition of adequacy on theories of entailment
and conditionality, the strong intensionality, or ultramodal, requirement,
that adequate theories must be ultramodal.
Just as strict-implication is
insufficiently intensional to provide a satisfactory account of entailment,
so, as we have now seen, modal accounts of conditionality, such as modal
similarity theories, are likewise insufficiently intensional. Actually, as
compared with implication and conditionality, which are fundamental notions
in argumentation and reasoning and so central in philosophy and also in such
more practical matters as the law, modal notions are of relatively slight
importance.
Through modal muddlement, however, which has taken entailment
and conditionality as modal notions, the importance of modal notions and
modal logics has been much exaggerated - though not of course to the extent
that the importance of classical logic has.
Modal befuddlement is not the only factor at work in modal-style analyses
of entailment and conditionality.
There is also the older, associated,
assumption to be found in C. I. Lewis and in Burks' analysis of causal implication 51, but reappearing in Stalnaker and Thomason 70 and D. Lewis 73, that
conditionality can, like implication, be correctly defined in terms of a oneplace connective (characteristically but not necessarily, modal), such as
physical necessity or conditionally-defined necessity, in combination with
truth functional connectives.
Meyer has put the kibosh on this assumption
(in 74b; adapted in ABE under the misleading title 'Relevance is not reducible
to modality', since neither relevance nor modality are to the point).
There
is no need to appeal to the outcome of modal substitutivity conditions to
show that genuinely two-place connectives such as entailment and conditional
relations cannot be defined in terms of some one-place connective, modal or~
not, applied to a combination of truth functional connectives.
That is,
A ->• B cannot be defined as $$(A,B), where $ is a one-place connective satisfying a fairly minimal set of conditions and \|»(A,B) is some truth-functional
combination of A and B.1
To begin, then, from the assumption that conditionality can be captured by simply adding a one-place connective to the logic of
truth-functions, is to make a false start.
Some of the newer systems of
conditionals have implicitly recognised this limitation, that conditionality
will have to employ genuinely two-place connectives.
However they have persisted with the assumption that conditionals are nonetheless modal. This is
false, as relevance considerations alone reveal in the case of natural language conditionals.
Assembling the conclusions reached sets the shape of the general theory
of conditionality sought.
The theory will contain an irreducibly two-place
conditionality connective which is ultramodal.
The theory will be relevant,
and ideally will be coupled with a relevant theory of entailment - certainly
not with a strict one.
The theory will, in its most general form (of 13),
fail such principles as Contraposition, Augmentation and Transitivity, but
validate such principles as Simplification.
The theory should be underpinned
by a semantical analysis preferably of a suitable philosophical cast.
There
are several hitches to this ambitious undertaking, to some of which we now
turn, the main intellectual obstacle being the mistaken idea that any satisfactory theory must include a classical sublogic.
1
For the conditions required, see ABE, p.466, p.471.
Note that system E
can be replaced by any subsystem of RM3 which includes the basic logic of
chapter 13; and accordingly that the argument applies both to the desired
system D of deducibility, whatever its precise final form, and to that
sought for conditionality.
49
7.6 DISTINGUISHING FORMS OF THE CLASSICAL DISEASE
§6. The classical hang-up; how one can go wrong with material implication,
and why no classically-based logic can be adequate. The trouble with all the
attempts, so far criticised, to produce satisfactory theories of conditionals
and logical implication is that they are far too classical in orientation features which emerge, firstly, in the way they all proceed by adding to
classical logic and aim to incorporate classical logic as an integral part,1
and, secondly, in the semantics where only classical-like possible worlds are
admitted.
For no such classical-type account is going to provide a tight
enough connection of antecedents and consequents in conditionals or therefore
to preserve relevance.
Strict-implication and irrelevant conditionals, are
both classical in a worlds way, in that each world used in their semantical
analysis conforms to classical canons. It is from this classicalness that
their main inadequacies stem.
Classicalness is a persistent and widespread disease (which affects
even relevant logicians who like the logical ease and "smoothness" of classical semantical corditions which involve no new or less familiar functions).
It has resulted in a vast overemphasis of modal notions, and underlies
attempts to provide modal analyses of a great many - sometimes, in extreme
cases, all - intensional notions, though it is evident that many of these
notions are bound to resist modal, or extended-classical, analyses. But in
contrast with implication - which is the basic argument relation* and hence
fundamental in philosophy - modality is much less important; it has assumed
an exaggerated importance largely through defective accounts of implication
and conditionals, which reduce them to modal notions and reduce coimplicational
substitutivity conditions to modal interchange conditions.
A much weaker claim is sometimes advanced in favour of classical logic
or modal logic, than that they are adequate.
It is claimed that even if
material-implication [strict-implication] is defective as an implication
[entailment] and perhaps needs some augmentation, classical logic is nonetheless correct and classical logic has to be included in any adequate logic of
implication.
Moreover, since one can not go wrong using classical logic,
it provides a minimal working base, which there is a good case for using in
controversial areas.
It is, so to speak, a core logic with which one can
comfortably rest.
This is all too familiar bunk, which we will try to
expose.
The core-correctness claim may be weakened still further to: one can't
go wrong with material-implication so long as
simply figures as the main
connective in inferences - which is all it is ever needed for.
Even this
latter assertion is of pretty doubtful correctness unless only very limited
purposes are envisaged - certainly not formalisation of much of discourse as
it figures in commerce and in newspapers and occurs on television, nor formalisation of the intensionally less rich mathematical discourse, where
nonetheless second degree arguments and inferences with iterated implications occur.
We will examine these stronger and weaker forms of the claim that one
can't go wrong with hook together. First of all, we do not of course dispute
the truism that classical logic is correct in classical contexts, i.e.
1
Modal logic is incorrect for the same reason, because it incorporates
classical logic as an integral part of its structure, whereas a correct
modal logic would be built on a relevant base. Similarly for modal
conditional logic.
50
7.6 WAV'S OF GOING WONG
WITH CLASSICAL LOGIC
in those contexts where it is correct.
But these contexts provide only a
quite proper subclass of the contexts where some logic is needed, and where
classical logic is supposed and intended to apply.
There are then logical
contexts which are not classical, and within these classical logic is incorrect (see also UL).
Moreover while we agree that the & - v — theorems of classical logic though not doxastically compulsory - are true (for the extensionally-refined
connectives of natural language that they consider), and accordingly should
be theorems of any adequate logic of implication, it does not follow that
classical sentential logic is correct, or should be included in every theory,
or can never go wrong.
Classical logic is only correct if it is coccoonised,
if it does not assert too much more than the & - v — tautologies.
It goes
wrong all too quickly under needed and intended applications (as the counterexamples of 1.2 reveal).
One can go wrong with classical logic in two main sorts of ways.
One
can go wrong interpretationally, by interpreting => as amounting to some sort
of implication rather than as simply short-hand for an extensional combination
(e.g., not-or)1, and one can go wrong in a deeper way by making the consistency and completeness assumptions of classical logic and their worlds.
We
deal with these ways of going wrong in turn.2
A common view - summarised in the slogan 'You can't go wrong with
material-implication' - is that because => is truth-preserving it is safe to
use for valid argument.
The common view is well-represented in Lemmon (65,
p.60):
While admitting that this discrepancy [between ^ and if-then]
exists, we may continue safely to adopt 'A => B' as a rendering
of 'if A then B' serviceable for reasoning purposes, since,
as will emerge ... our rules at least have the property that
they will never lead from true assumptions to a false conclusion.
Indeed how can any truth-falsity counterexample (i.e., where the antecedent
is true and conclusion false) to any of these laws be possible, when these
laws are laws of material-implication,
which is, as everyone knows, truthpreserving? The truth-falsity counterexamples are possible firstly because
A ^ B is at best a necessary, and is not a sufficient, condition for A -*• B.
The truth-preserving properties of material-implication mean that one cannot
find truth-falsity counterexamples - of an extensionally-admissible kind (i.e.
one world examples) - to A -* B when A ^ B holds; but if A ^ B is only a
necessary condition for A -*• B, one can find a truth-falsity counterexample
to the main connective in A => (B •* C), when A = (B => C) holds.
If B
C
is only a necessary, and not a sufficient, condition for B -*• C, B
C may
be false when B a C is true.
Hence one has only to find one of these cases
to obtain a higher-degree truth-falsity counterexample to the main connective
in A 3 (B -*• C) when A is true and A ^ (B ^ C) holds. (Exportation and Commutation may be countered in exactly this fashion: cf. 3.7 where counterexamples to Commutation are presented.)
1
The counterexamples of earlier sections all rely on intended interpretations
of connectives, on the interpretation of extensional connectives in terms
of natural language surrogates, and of
and -3 as 'if-then* and 'entails'
respectively.
2
In discussion of the first way we borrow heavily from V. Routley 6 7.
37
1.6 WHV HOOK IS MOT SAFE TO USE IN REPRESENTING CORRECT ARGUMENT
The second reason why counterexamples are possible - though not counterexamples of a one-world truth-falsity variety - is because, though A => B is
true, A
B is rendered false by a situation, distinct from the actual one which
determines truth, where A holds but B fails to hold.
Modal counterexamples
to A -*• B, where A and B are both contingently true, are of this sort; a
possible world is envisaged where A => B fails to hold.
And the specific
countercases to Disjunctive Syllogism (given in 2.9) are of this kind; though
A & (~A v B) a B holds in the actual situation, deductive situations can be
designed where A & (~A v B) holds but B does not.
In the light of this, the common view, that because
is truth-preserving
it is safe to use for valid argument, or as a surrogate for correct conditionals, must be defective.
First, the truth-preservation property of => does not
show that it is safe unless it is assumed quite circularly that the only way
an implication can be wrong is by having its premiss true and conclusion false
in the actual situation, the real world, T.
Second, o is not truth-preserving
in the right sense to guarantee safety for valid argument.
For it is not
truth-preserving in the sense that if we substitute 'there is a valid argument
from ... to ...' or
is deducible from ...' or even 'only if' for all
occurrences of '=>' in the laws of =>, what we obtain will continue to be true
where the original => law was true.
Similarly, once such a substitution is
made, => can lead from true premisses to a false conclusion, as counterexamples
show.
Admittedly all the false conclusions are themselves about valid argument (or conditionals), but failure of this sort can not be discounted given
the uses of deducibility in assessing responsibility.
Nor is it an adequate
defence against these counterexamples to claim that the supposedly false conclusions are always about valid argument and that since truth-preservation, as
represented by =>, is all that is required for deducibility, they must not be
false at all, but true, even if surprising.
Such a defence is quite circular,
for we can only decide whether a particular connective, say
ij^, as claimed,
adequate for valid argument, by substituting 'there is a valid argument from
... to — ' for '=>' and seeing whether the results continue to hold; hence
we could not without circularity decide these results on the basis of what
holds for the connective itself. (A similar objection applies to the claim
that the material-implication paradoxes are not paradoxical for deducibility
because they follow from the truth-table for =>.)
A connective is only safe to use for valid argument if upon substitution
of 'there is a valid argument from ... to ...' for all occurrences of the
connective in its laws, the result continues to be true, i.e. if every law of
the connective is also a correct law of valid argument.
It must licence no
methods which are not also methods of valid argument; but many of the principles discussed, which are => laws, do license methods which are unacceptable
as methods of proof or valid argument.1 And thereby one is licensed to go wrong.
1
These points also destroy van Fraassen's curious claim that classical logic
is correct for mathematical English (71, p.4).
Insofar as mathematics relies
on valid argument, its proper formalisation is not in terms of classical
logic.
While it is true that much of classical mathematics can be reformulated using classical logic, such a formalisation is hardly unique, and alternative formalisations using strict systems or even relevant systems can undoubtedly be devised.
In fact classical mathematics uses only comparatively
weak logical inference principles, few highly nested implications and nothing
like the power of classical logic or stronger Lewis systems.
Finally what
'the correct logic for a language' is remains a somewhat obscure matter; but
what seems clear is that a language may have a variety of logics associated
with it (see Routley 75, for further discussions of this issue).
52
1.6 THE PEEPER TROUBLE WITH CLASSICAL THEORIES: MATERIAL DETACHMENT
Even if the interpretation of ^ as any sort of implication or validargument indicator or conditional is given away entirely and = construed as
no more than an extensional abbreviation (e.g. for not-or) classical logic
and classically-based logics remain in real trouble.
For there is a deeper
way in which classical logic is wrong, and in which exponents can, and do,
go wrong with classical logic.
Even though all extensional tautologies are
true, one can still go seriously wrong through assuming the unqualified
correctness of the principle of Jfeterial Detachment:
Y- Where A and A => B are theses, so is B,
or, in application to theory c, through assuming its theory analogue:
Yc. Where A and A => B hold in c, so does B.
Classically, of course, yc is assumed not only where c is the class of all
truths T but also for every theory (i.e. y is assumed to hold not just for
the actual world but for mathematical and scientific theories).
This is
part of the (mistaken) assumption that classical logic is universally applicable, to every (scientific) theory, and that it is fundamental in reasoning.
It has too often escaped attention that principle y is a fundamental
component of classical logic and of all its patch-ups. Classical sentential
logic, as ordinarily presented and understood, consists not just of and-ornot schemes, or some batch of these taken as axioms, but also of the rule
of material detachment.
The deeper trouble with classical systems lies in
the unrestricted assumption of this rule - which would be correct were => indeed a sufficiency connective - in the assumption, that is, that y holds for
any and every application of the logic, for every theory.
Consider what happens when classical logic is applied as the underlying
logic to a theory which is simply inconsistent but not trivial, in that not
every proposition holds in it.
Such theories are not just important, they
are common: philosophical theories, logical systems, dialectical positions,
and belief systems furnish a wealth of examples (detailed cases are set out
in DCL and in Routley2 75).
But the presence of material detachment excludes
them, and trivialises every simply inconsistent theory, thus:- Since a
simply inconsistent theory contains B and ~B for some sentence B, it also
contains ~B v C, i.e. B
C, for arbitrary sentence C.
Hence by Material
Detachment it contains C, and so it contains every sentence.1
That is, it
is trivial. Exactly the same problem also arises from use of any implication,
such as strict implication, which satisfies the principle of Disjunctive
Syllogism,
DS . A & (~A v B)
B.
For this principle yields, by Modus Ponens closure, y.
Or, more directly,
since ~B v C and B belong to the simply consistent theory, so, by DS and
implication closure, does C.
The limited applicability of the rule of Material Detachment emerges
from the conditions necessary and sufficient for its correctness.
We have
already observed that simple consistency is a necessary condition for its
correctness.
For sufficiency, let c be a theory which is prime, i.e. whenever A v B e c either A e c or B e c, and simply (i.e. negation) consistent,
i.e. for no A do both A and ~A belong to c.
Then y holds at c, i.e. yc is
correct.
For suppose A e c and ~A v B e c.
As A e c, ~A / c, by negation
consistency.
But, as c is prime, either ~A e c or B e c; so B e c.2
1
It is assumed that the theory meets certain quite minimal normality conditions - which will happen in main cases of interest.
(footnote 2 on next page)
53
1.6 TYPES Of OBJECTIONS TO MATERIAL VETACHMENT
The first class of objections to Material Detachment, and derivatively to
its thesis analogue Disjunctive Syllogism, centres, then, on the point that
they obliterate a class of theories of considerable philosophical and logical
interest.
A second class of objections arises from this, namely that many
other important theories may, for all we know, belong to this class (cf.
Lukasiewicz 70).
In particular, it may well be that some important mathematical theories
fit into the category of simply inconsistent but non-trivial theories.
Naive
set theory is a prime candidate.
It is certainly simply inconsistent, but it
is only obvious that it is rendered trivial by paradoxes like Russell's when
it is classically formalised.
We now know
that, on the contrary, naive set
theory is non-trivial when given a suitable relevant formalisation - a far more
appropriate formalisation of the intuitive theory than conventional classical
ones (as EMJB explains; see also 4.2).1
(footnote 2 from previous page)
The sufficiency conditions also indicate the line of a main strategy for
proving that y is admissible for a given logic L, namely by showing that each
L-theory c which keeps out D can be replaced by a prime simply consistent
L-theory c' which keeps out D.
The feat is accomplished, where it can be,
by first expanding c, by Lindenbaum methods, to a prime L-theory c + . But c + will
commonly be negation inconsistent, so the technique is to select a subtheory
c' of c + with the desired properties.
This can be done, in some cases, by
splitting c + , or, what accomplishes the same, by metavaluation methods (as in
Meyer 72).
*Russell observed long ago (06) that any way of avoiding the logical paradoxes,
as classically formulated, involves changing the logic (of set theory), and
accordingly he went on to consider three important ways of amending the logic
of sets.
But what he did not consider was the obvious move of changing the
underlying classical logic.
Yet this was the change of logic that really
should have been considered.
The option opened up, of retaining all of highly appealing intuitive
theories by adjusting the underlying, usually classical, logic presupposed
for formalisation, raises some awkward questions for the dedicated pragmatist;
for example as to whether to trade off little used sentential principles for
the powerful unqualified abstraction thesis in a formalisation of set theory.
But the classical hang-up most pragmatists have has blocked investigation of
this attractive alternative.
The abandonment of the unwarranted classical assumption that every simply
inconsistent theory is trivial also raises other severe problems for pragmatism.
For the pragmatist conception of logic which is supposed - in opposition to absolutist views on logic - to make the choice of underlying logic a
pragmatic matter, in fact in the presentations of its leading exponents,
assumes that classical consistency conditions must be met - otherwise revision
in the light of recalcitrant experience could not play its assumed role.
In
short, the pragmatist view presupposes part of the absolutist position it is
supposed to be rejecting.
It is facile to try to meet this objection as
Haack (74, p.36) and others do by trying to rule out simply inconsistent logics.
Dialectical logics are formally viable, and are coming to be part of the logical
scene: they cannot simply be ruled out of court as not 'logics'.
Nor can
they be dismissed on the grounds that they fail to discriminate valid arguments since an inconsistency entails everything.
For such paradoxes of
deducibility any worthwhile dialectical logic would repudiate.
54
1.6 GROUNDS FOR SCEPTICISM ABOUT MARKETED SET THEORIES
Set theory of adequate power may well be inconsistent.
(No doubt consistent theories may be obtained by sufficient mutilation - simple type
theory without an infinity axiom provides a familiar example - but so far
these theories are far too underpowered to haul the classical mathematical
train.1)
Most important, any classical set theory that approximates closely
enough to naive set theory to undertake its work is likely to be simply inconsistent. Marketed set theories don't approximate too well at all, and there
is a chronic need for better set theories than the inferior goods we have so
far been offered in the way of formalisations.2
The fact that Quine's
theories appear in some respects better than most others is indicative of
the situation.
Marketed set theories don't however have to approximate
very well for inconsistency to threaten. Indeed it seems to us that some
marketed set theories - Quine's system ML is the most conspicuous example are quite likely inconsistent. So scepticism about set theory and set theoretical foundations for mathematics has some warrant.
A trivial foundation
for mathematics - which is all that classically formulated inconsistent set
theory would provide - is hardly of much merit, and one does not have to go
through the laborious circuit of set theory to provide it.
The conventional classical reply is that if a theory such as set theory
is simply inconsistent it might as well be trivial; simple inconsistency is
hardly any better than triviality.
This supposes that a theory which is
simply inconsistent but not trivial is of no logical interest, an assumption
we have already seen to be false. Abstract set theory, with abstract sets
characterised in terms of the abstraction axiom, contains, as Cantor saw,
inconsistent sets, and accordingly it provides a good prima facie3 candidate
for a simply inconsistent theory; but that's no good reason for concluding
that the theory is trivial.
Thus universal use of classical logic in set
theory is without justification.
The situation with regard to various other parts of mathematics is
scarcely better than the set theoretic situation (cf. DLS).U
Other historical mathematical theories have also proved to be simply inconsistent, most
conspicuously the theory of infinitesimals and the theories of calculus and
analysis that this theory supported.
It was not evident - it is still not
obvious - that these theories were thereby trivialised - and under a dialectical formalisation the theories may well turn out to be viable, as historically they were thought to be.
1
With only a small increase in capacity, however, they are powerful enough
to haul the main semantical train of subsequent chapters.
2
The classical story of what mathematicians do when they use intuitive set
theory, of what they used to do before the paradoxes were discovered and
continued doing afterwards, is largely mythology - there is the parable of
the consistent subtheory, of the restricted domain of discourse where sets
are well-behaved, and so on.
3
The case so provided is only prime facie because there can of course be consistent theories of inconsistent objects.
Other considerations lead, however, in the case of set theory to the conclusion that the theory is simply
inconsistent, especially the intuitive arguments for the logical paradoxes.
u
Furthermore, through classical set theoretic formalisation, much of the rest
of modern mathematics has been contaminated by set theory.
But really
classical set-theoretic formalisation is too rough a device to capture subtler parts of mathematical argumentation and inference, and even at quite
(footnote continued on next page)
55
1.6 LIVING WITH CLASSICAL LOGIC IS LIVING DANGEROUSLY AND IRRATIONALLY
Worse still, the consistency of a sufficiently rich classically formulated
mathematical theory can never be classically assured, and so conventional use
of Y can never be justified.
As a result of the classical limitative theorems,
and especially Godel's consistency theorem, classically based theories are
forced into a completely untenable position.
For let MT be any sufficiently
rich classical theory, i.e. any classical theory in which every recursive
function is representable.
Most mathematical theories satisfy this requirement by virtue of including arithmetic.
Then, by Godel's theorem, the consistency of MT cannot be proved using just methods of MT; any attempt to prove
the consistency of MT in a non-question-begging way will lead to an endless
regress through larger and larger, and increasingly suspect, theories.
In
short, a non-circular proof of the simple consistency of MT is classically impossible.
But vindication of the use of y in MT requires a non-circular proof
of the simple consistency of MT.
Thus use of y can never be classically vindicated in the case of any sufficiently rich mathematical theory. Accordingly,
too, use of classical logic in formulating such theories cannot be justified.
There is indeed far too much complacency about formalisations, especially
classical formalisations, of set theory and other substantial regions of the
foundations of mathematics.
The fact is that they are in dreadful philosophical and logical shape.
A lot of glib and supposedly reassuring tripe is trotted
out about how everything is alright - like the economic experts on an obviously
ailing economy.
But really the crisis in logic is far from over.
It is
simply that, like hyperinflation in some economies, people have come to live
with it. The fact that one can live with crises, can live dangerously with
classical logic (as van Fraassen is quite explicitly prepared to do: 71, p.5),
is hardly a good reason for continuing to do so when the risks can be reduced as they can be giving up material detachment everywhere except where consistency
can be established, that is in most places.
There is no need to live in the
shadow of possible disaster in the fashion of classical set theorists, and it
is mostly irrational to do so.
Nor can there be any doubt that allowing for the possibility that a set
theory is a simply inconsistent non-trivial theory substantially reduces risks.1
Naturally where a version of Godel's theorem applies, risk cannot be entirely
eliminated; but firstly, Godel's result has not been established for theories
which are not classically based (see UL), and, secondly, there is good prospect
that for such theories a non-Godelian strategy can succeed (for example, dialectical theories automatically escape the main impact of limitative theorems,
since they admit the paradoxes, whereas the limitative theorems, and the scepticism they inject, are consequences of getting burnt by the paradoxes after
their main fire has allegedly been extinguished).
(footnote u continued from previous page)
elementary levels creates distortion. (Consider, for example, the settheoretic representation of groups and semi-groups discussed in van Fraassen
71.)
Too many logicians are prepared to tolerate this level of distortion,
to accept isomorphisms for identities, etc.
Probability considerations support this claim.
Just as, to use a
geometrical analogy, it is probable that isolated singularities remain
isolated and do not connect with every point, so there is a considerable
probability that isolated inconsistencies do not spread everywhere.
56
1.6 THE NEED FOR REASSESSMENT OF THE PLACE OF CLASSICAL LOGIC
In view of all this, and especially the severe limitations on the
reliable applicability of the core classical rule of Material Detachment, and
(as we shall see) the defensibility of dialectical theories, we believe that a
reassessment of the place of classical logic in theory formulation is wanted.
Classical logic was the first formalised logic of much scope on the scene and
it remains by far the best developed.
Practically all the extant formalisations of mathematical theories have been carried out on a classical basis not because classical logic was necessarily or obviously the best medium for
all - or any - of these formalisations, but in part because of historical
accident and vagaries and of conservatism of the educational process and in
part because it alone was there.
Nowadays classical logic, which has many
powerful adherents, has consolidated its position and become entrenched,1 and
much mathematical theory is being rewritten in terms of it.
A comparison
with the rewriting of history in terms that suit a new dominant ideology is
not inappropriate.
For it appears that the informal reasoning used in classical mathematics at least until the mid-century rarely2 or never deployed the
excessive power of, or, for that matter, the paradoxical features of, material
implication, and it also appears that most of the reasoning can be reset on
alternative logical bases including relevant ones.
It is our view that the importance of classical logic - which captured
the market because first into it and because of its childish simplicity - has
been grossly exaggerated.
Given time and relevant effort it is our hope that
history will in due course bear out our claim on this matter.
What importance
classical logic does have is in providing a simple model, and testing ground,
for arguments that can then be re-engineered to extend to more adequate
alternative systems.3
1
Thus rival logics are derisively dismissed as 'deviant' in (much of) U.S.A.
and its intellectual satellites: elsewhere they are 'non-classical'.
2
If there were many such cases, which we doubt, they could be catered for in
a relevant formalisation by additional mathematical postulates - the classical postulational bases having in any case a measure of indeterminacy.
3
We have already, impetuously, proposed relevant logics as candidates for
the job from which we are arguing classical logic should be dismissed (see,
e.g. UL).
As a matter of history, relevant logics were not introduced
merely for recreational or academic advancement, or demotion, purposes, but
were intended as serious alternatives to classical logic and to such improvements on classical logic as modal logics.
We certainly believe that relevant logics should, because of their unusual scope and range of other virtues,
be very seriously considered as alternative logics.
So it pained us to
encounter Haack 74 - a whole book on alternative logics in which relevant
logics get no mention.
Of course recognition of relevant logics would have destroyed the
organisational framework and several of the arguments of Haack's book.
To
begin with, relevant logics do not fit into S. Haack's oversimple classification (in 74, p.4ff) of alternative logics.
For although all theorems of
classical (sentential) logic are theorems of the main relevant logics
studied (those including system G below), these relevant logics do not in
general simply extend classical logic, but lack (except as an admissible
extra in many cases) the characteristic classical rule of Material Detachment.
Moreover non-trivial inconsistent extensions of the best relevant
logics - the dialectical systems - no longer allow material detachment as an
admissible addition. Thus relevant logics - with -*• considered, like the strict
implication and necessity of modal logics, as a further connective beyond the
standard set {&, v, ~} of classical logic - though they typically include
classical logic as admissible, are not essential extensions of that logic.
Doesn't that make relevant logics quasi-deviant logics on Haack's classifi(footnote continued on next page)
1.6 THE RATIONAL LOGICAL POLICY IS TO REDUCE RELIANCE ON CLASSICAL LOGIC
To sum up the main drift thus far:- Not even the residual claim made for
classical logic - that classical logic will have to be part of any adequate
logical system - is correct.
On the contrary the residual claim is drastically mistaken.
Admittedly the tautologies of classical logic are analytically true (though, as noticed, not compulsory, in that one doesn't have to
believe them).
Material Detachment is however an integral part of the classical scene, and this rule is formally unsatisfactory and becomes positively
undesirable when investigating the consequences of philosophically and mathematically important theories which may be inconsistent.
Then one can go
seriously wrong using classical logic.
One can, in short, go wrong with
classical logic if one applies it in the wrong situations.
And if one applies
it to discourse concerning a range of worlds that are not classical such as
inconsistent and paradoxical situations, then one is quite certain to go wrong:
and so of course for any arbitrary situation there's a chance of going wrong.
Furthermore for most theories of interest the requisite conditions for the
application of classical logic cannot be classically established.
Standard
logic is not, ther, a device that one can apply with real confidence anywhere
much beyond decidable theories.
Only the completely unfounded, and classically unsupportable, presumption of correctness props up the universal application of classical logic, and a rapid - but impermissible - retreat towards
pure logic is often attempted by exponents in the face of criticism.
Rational
applications of logical theories should include logical caution, which means
dropping classical logic as a universal, or even extensively used, tool especially in investigations of consistency.
For we do not check situations
carefully to make sure they are not bugged by contradictions somewhere before
applying logical reasoning - yet that is what the safe application of classical
logic would appear to require.
Moreover, even if, as is classically assumed,
the world is consistent it is not generally safe to apply classical logic in
investigating theories (which may diverge from the world).
The rational course
would seem to be to reduce risks and adopt a relevant logic (a thesis developed
in detail in DCL).
§ 7. Dialectical logic, and the repudiation of the dogma that the world is
consistent. So far we have been arguing against classically-based logics primarily on the grounds that there are non-trivial inconsistent theories which just
cannot be discounted, and that, moreover, a great many of our other intuitive
theories may, for all we know, be of this sort (and that classical theory prevents us from ever knowing otherwise).
(footnote
continued from previous page)
cation? No, for the theorems of the common vocabulary with connectives &,
v, ~ do not differ.
But the matter of the choice of connectives of alternative logics and the
comparison made with the connectives of classical logic is important, and,
unfortunately for Haack's classification, the status of a logic is not invariant under choice of its formulation.
For example, Lewis's systems are
classed by Haack as extended logics.
But should we consider Lewis's systems
of strict-implication, as formulated in a quite standard way with connective
set {&, v,
=>} with = as a strict implication, as an alternative to classical logic (the way Lewis and others have thought of these logics), then Lewis
systems are deviant logics on the Haack classification - not extended logics
as Haack has it.
Similarly relevant and intuitionist logics so considered
are deviant logics, and so likewise are the very different connexive logics
which contain decidedly non-classical theses and which should be separately
classified.
Accordingly the Haack classification is neither suitably stable
under systemic formulation nor sufficiently revealing, and it is neither
exhaustive or appropriately exclusive.
58
1.7 PARACONSISTENT LOGICS, ANV THE DIALECTICAL CRITICISM OF CLASSICAL LOGIC
The dialectical criticism of classical logic is much harsher.
For
according to it, there are true theories that are inconsistent, and thus the
world T (considered as everything that is case, as the class of truths) is
simply inconsistent.
If so, classical logic cannot be reliably applied,
even in its home (extensional) territory, to true theories: it is incorrect
in a quite drastic way.
The new dialectical criticism of classical logic and repudiation of the
consistency of T is based on the development of paraconsistent logics.
The
new logical case reinforces the older intuitive case, in a quite remarkable
way (as we shall see in 6.5), and furnishes many of its main theses.
A
paraconsistent logic is, at bottom, a logic which admits arbitrary contradictions without thereby being trivialised, i.e. a necessary condition for a
paraconsistent logic is that it does not contain (as a derived principle)
the spread rule: A, ~A -frB.1
Let us take this necessary condition to
characterise weakly paraconsistent logics, so leaving it open what additional
conditions have to be met by paraconsistent logics proper2 - just as we have
characterised weakly relevant logics in terms of satisfaction of Belnap's
weak relevance requirement and so far left open what additional conditions
must be met by relevant logics (but this question is closed in chapter 3,
where relevant logics are characterised as logics which are weakly relevant
and conservatively extend distributive lattice logic). Since most positive
logics would qualify as paraconsistent under the account we also require that
the logics concerned should be sentential and contain as well as •*• a full
stock of extensional connectives, typically &, v and
1
A less sweeping requirement that merits some investigation is that under
which a logic admits some contradictory pair A and ~A of further theses
without being trivialised.
2
Da Costa in a number of publications, particularly 74, has imposed further
conditions on paraconsistent logics, and similarly Jaskowski has required
that his discussive logics - which are paraconsistent logics - should meet
further conditions.
But some of these conditions are not formally tractable (and are too strong insofar as they are), e.g. that a paraconsistent
logic should have an intuitive sense or that is should be sufficiently rich
to formalise the bulk of useful sentential reasoning (Jaskowski 69,
D'Ottaviano and da Costa 70), and others are undesirable, e.g. the condition
that a paraconsistent logic should not contain the thesis ~(A & ~A).
Some
important dialectical logics have ~(A & ~A) as a thesis though for some
B, B & ~B is also a thesis, e.g. the system DL of 6.5.
Though the modern logical theory of formal inconsistent systems originated
in Poland with Jaskowski - its roots go back to Lukasiewicz 70 - extensive
development of theory has occurred in Brasil, and is to be found in the work
of da Costa, Arruda and coinvestigators.
The theory of formal inconsistent systems, or as they are now called, paraconsistent systems (the term
'paraconsistent* being coined by Quesada), has become very much a Latin
American institution, and distinguishes Latin American logic in much the
way that the identity theory of mind used to distinguish Australian
philosophy.
Important work on paraconsistent systems has also been done
by Asenjo, Apostel, Priest, Routley and others (see DLS).
59
1.1 THE CLASSICAL FAITH (IN CH) CONTRASTED WITH DIALECTICAL HARDHEADEDNESS
It is evident that the classes of logics distinguished, weakly relevant
and weakly paraconsistent logics, properly overlap.
The relevant affixing
systems (of chapter 3) generally fall into the overlap, but most of the systems
studied in depth by da Costa are irrelevant (the P systems are an exception),
while the original rigorous implication systems of Ackermann 56 are relevant
but not paraconsistent since they contain the rule y.1
The usual irrelevant
logics which all contain Disjunctive Syllogism are neither relevant nor paraconsistent.
A dialectical logic is a paraconsistent logic that realises the potential
that a paraconsistent logic allows for, that is, a dialectical logic is simply
inconsistent as well as non-trivial, i.e. it contains contradictory theses.2
Thus dialectical logics are a subclass of paraconsistent logics, and relevant
dialectical logics - those of main interest to us - of relevant paraconsistent
logics.
Let c be a situation that conforms to a dialectical logic.
The semantics
of relevant logic will reveal that such non-classical situations are easily
supplied.
Situation c upsets an application of y, namely yc.
So much we
have already seen.
The additional dialectical thesis is that T is such a
situation, and that so also are certain subsituations, such as c, of T.
Hence,
by the negation of the consistency hypothesis, yT is upset.
There are three positions that can be taken with respect to the consistency
hypothesis, CH, that the world T is consistent, and so three positions with
respect to yT - positions which correspond to the three standard positions with
respect to the existence of God.
There are, firstly, theists or believers
who accept CH, mostly as an article of faith.
Such is the classical position,
such also was Ackermann's position, and, so it appears, that of Anderson and
Belnap.
Secondly, there is the agnostic position, argued for as the rational
position in DCL.
Finally, there are atheists or infidels: such are thoroughgoing dialecticians (and the same dialectical position is argued for in UL and
in DLS).3
l
It is assumed in characterising paraconsistent logics that the extensions
considered are closed under the rules of the logic: otherwise even classical
logic could be paraconsistent.
2
0ur use of the term 'dialectical' has encountered a fair measure of criticism,
partly because 'dialectical' and 'dialectic' have another important (related)
sense, and partly because the links of dialectical logic with the enterprises
of Hegel and the dialecticians have not been made sufficiently clear.
But, firstly, our use of 'dialectic' falls squarely under the second part
of the dictionary listings for the terms (see OED):
dialectic, n (often in pi.) Art of investigating the truth of
opinions, testing of truth by discussion, logical disputation;
(Mod. Philos; not in pi.) criticism dealing with metaphysical
contradictions 6 their solutions.
For we are certainly concerned with the application of dialectical logic in
resolving metaphysical contradictions (see below): this is the central reason
for investigating them. (Jaskowski's discussive logics and paraconsistent
logics are intended to tie in also with dialectics in the Greek meaning, i.e.
as under the first part of the dictionary listing.) Secondly, Hegel and his
followers do not have a monopoly on the use of the term 'dialectic' even if
(footnote 2 and 3 continued on next page)
60
1.7 WHAT IS AT STAKE IS THE QUESTION OF THE CONSISTENCY OF THE WORLD
For present purposes we do not need to settle the issues between consistency agnostics and consistency atheists, even if we could; it is enough to
knock out the classical, theistic, position.
What we shall do is to set
out the seemingly powerful, and pretty impregnable, case that the agnostic
has (adapting the argument from DCL) and in the course of presentation,
indicate how it can be biased in favour of the dialectical position.
We have seen that what is at stake is nothing less than the question of
the consistency of the world T, that if the world is simply consistent (and
also, as against the intuitionists, complete) then the classical position is
correct, whereas if the world is inconsistent then the dialectical position
is correct.1
Whichever is the case, however, the relevance position does
not go wrong.
This provides a major reason for claiming that the relevance
position is more rational than the other positions, should it turn out that
the matter of the consistency of the world cannot be definitively settled.
But the question of the consistency of the world cannot be conclusively
settled in a non-question-begging way - so at least the agnostic argues hence the keeping-options-open strategy of (non-classical non-dialectical)
relevant logic is the rational one from a decision-making viewpoint.
The
rival positions of course contend that the matter can be settled, in their
way - not perhaps definitively, but in the way that other high level theoretical hypotheses are settled.
Here the dialectician has a point, as we shall
see.
What is at stake is not the absolute consistency or non-triviality of
the world, or the logic or theory whose truth it reflects.
All the positions
can agree that the world is absolutely consistent - which is as well for them,
since the absolute consistency of the world can be empirically verified.
The
statement, q , "R. Routley is cutting firewood at Plumwood Mountain on the
(footnotes
2
and
3
continued from previous page)
they were responsible for the modern usage;
and though it would be very
interesting to apply logical and semantical methods to the explication of
Hegel's philosophy, we are under no obligation to do so.
We do believe,
however, that the dialectical logics we have so far investigated have a
direct and important application in clarifying the logic of Soviet dialectical philosophy (see DCL).
3
0ne of the fringe benefits of going dialectic is that one can no longer
be condemned quite so easily for the occasional inconsistency in one's
work.
But of course we can make the classical plea that our position
has changed - improved we should like to say - over time.
x
The dialectician can admit with the intuitionist that the world is
incomplete, but the intuitionist cannot admit with the dialectician that
the world is inconsistent; for intuitionistically, since A -*• B, A -+A •*•. A -*• B,
whence —iA
A -*• B and A & —|A -*• B, i.e. intuitionistic inconsistency
implies collapse into triviality.
Of course a minimalist who rightly
rejects A -»• B can go some way with the dialectician. In general, minimalism
is a relevantly rather more congenial position than intuitionism: if only
however "minimal" logic had been based on a relevant positive logic
instead of on Hilbert's positive logic (the so-called "absolute calculus").
61
1.7 THE CHARACTER OF THE CONSISTENCY HYPOTHESIS
afternoon of August 1, 1976", for example, can be empirically falsified by
observing Routley in Brasil at that date. Hence q Q is false and q^ does not
belong T.
Therefore, T is absolutely consistent.
It is often supposed that
given the absolute consistency of the world, the (simple) consistency of the
world is also proved, and the classical position thereby established.
However the "proof" of simple consistency must depend on the special rule
n.
A, ~A -o B
which transforms simple inconsistency into absolute.
But this paraconsistencyexcluding rule is tantamount to rule y, even in weak sublogics of classical
logic (e.g. in DML of 2.8), and is semantically equivalent to the assumption
that T is simply consistent (given absolute consistency or non-degeneracy of
the world).
Hence the question is begged by rule n> since precisely what is
at issue is rule y and the consistency of T.
The question of the consistency of the world is not, it seems, empirically
decidable. Nor i~ it a high level scientific hypothesis.
It is a metaphysical
thesis, but a perfectly significant one.
Since it involves a universal claim to the effect, as canonical semantical modellings will show, that for every
statement A exactly one of A and ~A belongs to T, i.e. holds true - it would
appear that the dialectician is in a much stronger position to establish his
claim than the classicist, since universal claims can be falsified in principle
by a single counterexample.
To establish his thesis however the classicist
has to establish a claim that extends over all true theories, whether scientific, mathematical, evaluative, or whatever.
Should a contradiction lurk
deep in some still unpenetrated mathematical or other theory then the classicist is undone.
Only a priori "evidence" that such a contradiction or inconsistency cannot occur would seem to exclude the possibility: thus the classical position if true is not empirically so in any simple way, no endless search
is contemplated, nor would its results be relied upon.
The dialectician, on
the other hand, has the chance of falsifying the consistency hypothesis by a
solitary counterexample. But, as any dialectician soon finds out, a simple
falsification of the consistency hypothesis which would satisfy the opposition
is not so easily achieved, as each alleged contradiction can be avoided,
though often none too convincingly when the wider theoretical framework is
remembered, by theoretical shifts (coupled with conventionalist strategems).
In this way too it quickly becomes apparent that the consistency hypothesis is
a very highly theoretical claim.
Characteristically dialecticians have appealed to paradoxes to establish
the inconsistency thesis.
Hegel was, it seems, convinced both by the Kantian
antinomies and by Zeno's paradoxes of motion; Soviet philosophers are more
impressed by Zeno's paradoxes and invest heavily in the contradictions they
observe in motion (which generalise to development).
None of these cases are
however decisive, and even if the classicists do not have really convincing
solutions, say to some of Zeno's paradoxes, they do have alternative classical
resolutions which so far get by.
Much more convincing examples may be drawn
from the logical and semantical paradoxes (this is the basis of the argument
in UL against CH), but dialecticians have not generally made much appeal to
such examples. Our case however is based, in the first place, squarely upon
the logical and semantical antinomies and their like (self-referential arguments, including those of the limitative theorems).
The arguments for the
Kantian antinomies are conspicuously fallacious, and Zeno's arguments generally
fail to convince. The antinomies we depend on have a different character:
the arguments are convincing and are not conspicuously fallacious, if falla-
62
7.7 THE ARGUMENT TO INCONSISTENCY FROM PARADOXES
cious at all.
The treatment of these antinomies within the framework of
classical logic leaves an enormous amount to be desired, philosophically,
linguistically, and mathematically.
The artificial hierarchies of languages
or types that classical semanticists appeared forced into, to avoid the
catastrophic effect of semantical antinomies in combination with classical
logic, are frankly unbelievable.1
One basic problem is that classical
logic has to exclude deductive reasoning within and surrounding the antinomies, though it undoubtedly occurs (cf. the Prior-Mackie exchange, especially
the initial paper, Prior 61).
What has seemed especially puzzling is that
paradoxical situations seem to be perfectly possible, in the sense that they
could quite easily occur, and perhaps sometimes do occur.
And this would
make T inconsistent - which is one reason why attempts to dispose of the
antinomies have exercised philosophers and logicians so extensively.
Consider, for example, the policeman-prisoner situation, where the prisoner
states only that everything the policeman says is true while the policeman,
whose statements are otherwise unquestionably true asserts that whatever the
prisoner asserts is false.
Nothing stops such a situation from occurring indeed it could be arranged in a real-life courtroom situation.2
But no
catastrophical breakdown would occur: legal reasoning could go on as before.
And in fact such a paradoxical situation could go unnoticed. Paradoxes just
do not spread and affect everything else: paradoxical situations do not
have the logical features that classical analyses ascribe to them.
The case for dialectical logic and the inconsistency of the world is
not confined to that based upon "paradoxes" of one sort or another, of logic
and semantics, of mathematics and physics.
There are inconsistent principles
in many legal codes which are valid (even if they are not said to hold true),
and within the shadows of which legal reasoning takes place.
To put it
bluntly and without the detailed argument really called for: the prevailing
law is sometimes inconsistent, yet deductive reasoning, incorporating the
principles of the law as postulates, continues both within and outside the
courts without being trivialised.
Legal logic would accordingly appear to
be dialectical logic.
The argument we have outlined (more detail appears in UL and DLS) does
not pretend to be entirely conclusive against the opposition, though the
case seems to us very damaging. The classicist will insist of any such
theory which contains a contradiction that it is not true, that it is (if
perchance absolutely consistent) a non-standard theory distinct from the
actual one which must be consistent.
Thus counterexamples lead to a
stalemate: the consistency hypothesis is not straightforwardly falsifiable or, the relevance position tries to insist, falsifiable at all.
A useful example of the theory-saving methods of the classicist in the
face of apparent inconsistency in nature is provided by Rescher 73, who,
confronted by the incompatible actual states assumed by the Everett-Wheeler
1
The manifold deficiencies of the hierarchial approaches are well enough
known: they are brought together, e.g. in Routley 66. The hierarchical
approaches are not of course completely unavoidable classically; for
every consistent, classically-based set theory will, in principle, yield
an analogous protothetic; but these will typically be even less satisfactory
than the set theory from which they derive.
2
How it is described is another matter, of course.
63
1.7 REASONS FOR REJECTING CLASSICAL THEORY-SAVING STRATAGEMS
theory of wave packet reduction in quantum physics, casts around for a difference
of respect between the incompatible states in order to save the law of non-contradiction, and "locates", i.e. postulates, such a suitable distinguishing respect
in a further time dimension.1 Rescher's difference-of-respect procedure (a
method that goes back to the Socratic dialogues) thus offers further confirmation for the popular claim that a logical or mathematical theory2 can always be
saved - at varying costs - by making sufficient changes or complications in
scientific theories, since logical principles rarely confront empirical data
in isolation and generally only do so rather indirectly in combination with
other theoretical assumptions (as pragmatists have long pointed out).
But
though a non-empirical principle, such as the consistency hypothesis, never
directly encounters the hard empirical data and can always be saved in one way
or another, with greater or less cost, by changes elsewhere, the costs may be
too high, and it may be better to give up the principle.
A convincingly microphysical theory based on dialectical logic might provide such a reason.
For
success with theories based on classical logic has always been at the finite
macro-level (as Br^uwer would have said): there is, in principle at any rate,
no reason why classical logic should not go the way of Euclidean geometry satisfactory locally in regions of consistency, but globally defective.
It
should now be obvious that such a change would be perfectly possible; and
persuasive evidence supporting the call for a conceptual revolution in logic,
which queries the consistency hypothesis, is fast accumulating.
If the consistency hypothesis is not straightforwardly falsifiable, it is
far less readily verifiable.
There is no easy or obvious way of surveying
or sampling the class of true theories, particularly those yet or never to be
discovered.
Moreover by virtue of a by-product of the logical paradoxes, the
situation appears to look still blacker for the classical position.
Consistency is a logical matter, and if it can be established it will presumably be
by logical means.
Suppose, with a view to proving consistency, that the
theory of T is formalised as far as it can be classically, and suppose that the
truths of T included provide the principles of Peano arithmetic: then, according to Godel, the consistency of T is not incontrovertibly provable classically
except by means at least as demanding as the resources of T (nor can they
exceed the resources of T without appealing to falsehoods?).
As we have observed (in §6), the same problem arises each time a reformalisation is attempted
1
To what extent the Everett-Wheeler theory can be satisfactorily constructed
on the basis of a dialectical logic is an interesting question that takes us
beyond the scope of the present venture.
The same goes for questions as to
whether such discredited theories as the classical theory of infinitesimals
can be "satisfactorily" reformulated in the framework of dialectical logic,
and the connected question of the extent to which the admission of inconsistencies at the infinitesimal level would help to resolve Zeno's paradoxes.
For each of these theories, each of which leads to at least isolated inconsistencies at micro-levels, would demand a substantial amount of technical
development before a worthwhile discussion could get off the ground.
2
An analogous, equally unsatisfactory, method in legal theory is to insist
that there is an implicit priority ranking on laws, which, should a contradiction become manifest, the courts will explicitly determine.
But what
such post-hoc determinations really indicate is that before the consistencizing step, legal reasoning had been operating with inconsistent premisses.
64
1.7 THE MODAL STATUS OF THE CONSISTENCY HYPOTHESIS
aimed at incorporating outstanding truths syntactically and thus approximating
syntactically to T.
In short, the classicist can never conclusively establish
his position; for, any formal proof of his hypothesis will use means at
least as powerful and accordingly as questionable as his claim. Although we
can be sure that the world is locally consistent for some regions we work
in, like locally Euclidean (more exactly, there are consistent subtheories,
like pure quantification theory, that we commonly use),1 globally we can
have no such confidence, thanks to the fine dose of scepticism Godel's classical results inject (cf. 1.6).
Therefore the question of the consistency of the world cannot be conclusively resolved classically in favour of the classical position, which was
one of the theses to be argued.
This in turn suggests the uncharitable proposition that belief in the consistency of the world is a mere act of faith,
a part, so to speak, of the classical religion.
While the belief may be an
act pf faith for many people, it doesn't have to be quite so obviously one;
for, the consistency hypothesis may be represented as part of a correct scientific theory, and overarching principles of sufficiently comprehensive scientific theories are in somewhat the same kind of epistemological plight as
the consistency hypothesis.
There can be good evidence for such theories
even when they cannot be conclusively verified.
The uncharitable proposition
then rests on a false dichotomy between conclusive resolution or else a mere
act of faith.
Even so the comparison of CH with an overarching scientific
principle does not, as already observed, stand up to too much examination:
rather the hypothesis functions classically as an unfalsifiable metaphysical
hypothesis.
This takes us straight into one remaining puzzle, namely the modal status
of the consistency hypothesis: is it a contingent statement, or a purely
logical one, or something else again?
For example, what seems to us more
likely, synthetic non-empirical?
It certainly does not appear to be empirical in the way that the deeper assumptions of physics such as invariance
principles are.
The alternative, on the simplistic classification that positivism so obligingly imposes, seems even less satisfactory, namely that the
hypothesis is logical, and so if true analytic and if false logically false.
The alternative does however have the virtue of grouping the consistency
hypothesis with a range of other principles which raise similar epistemological problems, in particular the axiom of choice and axioms of infinity.
The
consensus that seems to have been reached regarding these principles is that
we reason as far as possible without them, note carefully where they are used,
try to establish whether or not they are essential where they are used, mark
out those cases where they are essential, and so on.
And the consensus view
reflects a rational course of action, namely to minimize questionable logical
assumptions.
The same method would incline one to the relevance position to eschew the consistency hypothesis and to work, where consistency cannot be
established, i.e. in most classical places, with a relevant logic or, at
worst (as in 3.3), with some restriction of CH as a further explicit assumption - since this seems the rational course of action.
But there are further
factors that are not to be forgotten, in particular that, in contrast to the
axiom of choice, there is a substantial dialectical case against CH.
In fact
1
The official Soviet view is not dissimiliar; e.g., classical logic does
apply in the case of simple, relatively stable objects and relations: see
the discussion of Kedrov's position in Kline 53 p.84, and compare and contrast
the intuitionist position on the areas where classical logic is correct and
may be safely applied.
65
7.7 COROLLARIES
OF REJECTION
OF THE CLASSICAL
FAITH
the comparison of CH with the axiom of choice is in some ways misleading, and
a better comparison is that (emerging from Rescher 73) with the principle of
the uniformity of nature or that, hinted at above, with the assumption of the
existence of God.
Admittedly there are differences between these cases but
the similarities are striking.
In each case the assumption underpins the
application of an extensive theory, classical logic, theories of induction
and religions respectively; in each case the assumption is not, in its standard senses at least, an analytic one1, and would fail in its intended role
if it were; in each case, however, it is extremely doubtful that the assumption is an empirical one - the assumptions being impervious to falsifying
empirical evidence - yet in each case these are considerations, metaphysical
arguments for instance, which incline many people to conclude that they are
false; and yet in each case the assumptions can be maintained, despite all
counter-considerations, and they can be bolstered or supported by a variety
of expedients, by faith, by metaphysical arguments (invariably of a questionbegging cast), by the purchase of Kantian glasses (the wearing of which makes
the assumptions part of one's perceptual scheme), or by other familiar political
means, e.g. exhortation, persuasion, propaganda, repression, force, and so on.
But we don't have the classical faith, we are not persuaded by the questionbegging classical arguments for CH, we don't wear Kantian glasses and, in any
case consider the whole Kantian business fraudulent, especially as false
assumptions can be enforced in this fashion, and we hope we can avoid classical thuggery, for lesser political means are not going to move us.
The viability of dialectical logic has more corollaries than just the
unseating of the consistency dogma, and curbing of classical logical imperialism.
Several other classical projects and accounts are also upset.
Firstly,
for example, the classical idea, deriving from Hilbert, of absolute, i.e. nonrelative, simple consistency proofs by way of a modelling in the world has to
be adjusted. Non-triviality may be so established, but to establish negation
consistency it has also to be shown that a consistent subsituation of the
world is selected. (Alternatively such proofs could be reconstrued as relative consistency proofs, showing simple consistency relative to T, for what
that is worth.)
Secondly, the familiar account of rationality which makes simple consistency a necessary condition of rationality in beliefs, theories, or actions
has to be discarded. There is nothing in principle to prevent a dialectician's
beliefs from being perfectly rational and reasoned, and a dialectical theory
may be completely logical.
The two issues, of rational inconsistent theories
and rationally-held inconsistent beliefs are of course intricately connected:
2
for (so Routley 75 argues) an animal's beliefs constitute its theory as to
how things are, i.e. as to part of T.
The "framework of rational inquiry"
thus extends beyond the consistent to dialectical theories and practices.
Consistency imposes no Kantian (or Strawsonian) bound on rational inquiries.
The same holds for intelligibility: intelligibility likewise is not bounded
by consistency (see further 5.6).
The rational intelligent person is not one
who adheres to the consistency hypothesis but one who allows for the possibility at least that the world, though hopefully rational and intelligible
1
Forcing CH to be analytic takes us around the meaning-of-negation circuit
that we shall go around in 2.8 and 6.5.
Briefly, however, natural language negation is not so restricted in sense as to render CH analytic,
and does not conform to classical exclusion principles.
66
1.7 THE PARACONSISTENT REQUIREMENT ON AN AVEQUATE THEORY Of ENTAILMENT
but possibly neither, is inconsistent, and who realises that inconsistency
implies neither irrationality nor unintelligibility nor total disorganisation.
To cope with the wide range of dialectical reasoning, both everyday
and technical, an adequate theory of entailment, and likewise an adequate
theory of conditionality, should be paraconsistent.
Let us call this the
paraconsistency requirement.
It is a corollary that adequate theories
cannot be simply extensions of classical logic, e.g. modal logics; and it
will emerge that the requirement wipes out various apparent alternatives
to relevant logics, e.g. certain conceptivist and non-transitivist positions.
Since classical and modal logics are rejected as inadequate, the question is:
what are the new logics of entailment and conditionality to be like?
67