A Ranking-Theoretic Account of
Ceteris Paribus Conditions
Wolfgang Spohn
Presentation at the Workshop
Conditionals, Counterfactual and Causes
In Uncertain Environments
Düsseldorf, May 20 – 22, 2011
Contents
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Four Readings of Ceteris Paribus Conditions
Four Problems for Explaining Normal Conditions
Eight Interpretations of Normal Conditions (1) - (2)
Going for the Epistemic Interpretation
Very Brief Basics of Ranking Theory (1) - (5)
Explicating Normal Conditions: the Set-up (1) - (2)
Normal and Exceptional Conditions (1) - (3)
The Final Explication (1) - (3)
Concluding Remarks (1) - (3)
Bibliographical Note
Spohn, Ceteris Paribus Conditions
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Four Readings of Ceteris Paribus Conditions
That something – a claim or assertion, a law, etc. – holds ceteris
paribus may mean four things: that it holds
• other things being equal (equal to what?); then it means something
like the general law of causality: equal causes, equal effects, or
equal conditions, equal outcomes – whatever „equal“ means here,
• other things being absent; then it assumes that other relevant
variables have or take a natural zero value,
• other things being ideal; then the topic should rather be dealt with
under the heading „idealizations in science“ – or
• other things being normal; this is perhaps the most common reading
that finds the largest interest; in any case, it is the only one I will
consider here; so, for the purposes of this talk ceteris paribus =
normal conditions.
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Four Problems for Explaining Normal Conditions
Most explanations of cp (= normal) conditions founder at least at one of
the following problems: that according to the explanation
(1)
a cp claim is too vague to be useful,
(2)
a cp claim is too easily true,
(3)
in particular, both “cp, F’s are G” and “cp, F’s are non-G” are true,
(4)
a cp claim is too easily false.
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Eight Interpretations of Normal Conditions (1)
That something – a claim or assertion, a law, etc. – holds given normal
conditions has received at least the following eight interpretations:
• the natural interpretation: “cp, F’s are G” means “usually or mostly,
F’s are G”. This has problem (1).
• the statistical interpretation: tries to replace the vague “usually” by a
precise statistical probability. However, everybody rejects this idea;
cp claims are not statistical. Moreover, “usually” means “usually for
us” and is hence indexical and hardly amenable to statistics.
• the eliminativistic interpretation: “cp, F’s are G” is to be replaced by
“under conditions C, all F’s are G”. This has problem (4); under all
replacements actually proposed cp claims thereby turn out false.
There is only a hope that one day one might find the right conditions.
• the trivial interpretation: “cp, F’s are G” means “F’s are G, unless
they aren’t”. This has problem (2).
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Eight Interpretations of Normal Conditions (2)
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the existential interpretation: “cp, F’s are G” means “there are suitable conditions C under which all F’s are G”. Most efforts went into
this interpretation and in particular into specifying “suitable”. However, all proposals have problems (2) and (3).
the causal interpretation: “cp, F’s are G” means “being F is a partial
cause of, or plays a causal role for, being G”. Has again problems (2)
and (3). Also, one might say, if that‘s the right understanding, then
better stop trying to explaining cp and turn to causal analysis.
the conditional interpretation: that interprets the conditional in “cp,
F’s are G” as some kind of non-monotonic conditional (there are
various options).
the epistemic interpretation: the unusual or exceptional is the
unexpected, and the usual or normal is the not unexpected. Hence,
“cp, F’s are G” roughly means “under conditions which are not expected to be violated, all F’s are G”.
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Going for the Epistemic Interpretation
The epistemic interpretation avoids
• problem (1), as we shall see,
• problem (3), obviously,
• and problems (2) and (4), though apparently in too radical a way,
namely by refusing (objective) truth values to cp claims and rendering them as expressions of our beliefs or expectations. I shall
discuss the point at the end of my talk.
In the epistemic interpretation, cp clauses referring to normal conditions
somehow express, and are tied to, conditional belief. Therefore we
should first consider conditional belief.
My quarrel with the conditional interpretation is this: it is burdened with,
but remains indeterminate about the mysteries of the interpretation
of the conditional, and if it determinately takes the epistemic side, it
does not use the most preferable account of conditional belief.
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Very Brief Basics of Ranking Theory (1)
For, the most preferable account of conditional belief is offered by
ranking theory. Here are the required basics:
Ranking theory is a theory of degrees of belief and thus of inductive
inference or defeasible reasoning. In contrast to probability theory it
is able to adequately represent not only belief or acceptance, but
also conditional belief or acceptance. Thereby it is able to provide a
full dynamic theory of belief.
These basic features have widely ramifying consequences, and one
will be in the analysis of normal conditions, as I will be going to
explain.
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Very Brief Basics of Ranking Theory (2)
Let W be a possibility space or a set of possible worlds and A be a
complete algebra of propositions over W.
κ is a (negative) ranking function iff κ is a function from W into N+ = N ∪
{∞} such that κ(w) = 0 for some w ∈ W.
Alternatively, we can define κ as a set function. Then κ is a (negative)
ranking function for A iff κ is a function from A into N+ such that the
following axioms hold:
(a) κ(W) = 0 and κ(∅) = ∞,
(b) κ(A ∪ B) = min (κ(A), κ(B)),
[the law of disjunction]
(b') for any B ⊆ A κ(∪B) = min {κ(A) | A ∈ B}
[the law of (infinite) disjunction],
(c) κ(A) = 0 or κ(~A) = 0
[the law of negation].
If the point function is A-measurable, the two definitions come to the
same via the relation: κ(A) = min {κ(w) | w ∈ A} for all A ∈ A.
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Very Brief Basics of Ranking Theory (3)
Ranking functions are gradings of disbelief.
κ(A) > 0 says that A is disbelieved or taken to be false (to some
degree),
κ(A) = 0 says that A is not disbelieved (and perhaps believed).
κ(A) = 0 and κ(~A) = 0 represents suspense of judgment w.r.t. A.
κ(~A) > 0 says that A is believed or taken to be true.
Axioms (a) and (b) entail that belief is consistent and deductively closed.
One might also define a positive ranking function β by β(A) = κ(~A)
expressing belief directly. However, negative ranking functions are
the theoretically most fruitful version.
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Very Brief Basics of Ranking Theory (4)
If κ(A) < ∞, the conditional rank of w given A and of B given A is defined
as κ(w | A) = κ(w) – κ(A) if w ∈ A
and = ∞ otherwise, and as
κ(B | A) = min {κ(w | A) | w ∈ B} = κ(A ∩ B) – κ(A).
Thus, κ(~B | A) > 0 represents the conditional belief in B given A.
This definition is the crucial advance of ranking theory over its
predecessors (such as Shackle’s functions of potential surprise as
also used by Levi or Cohen’s Baconian probability) and in my view
the adequate representation of conditional belief entailing all the
further strengths of ranking theory.
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Very Brief Basics of Ranking Theory (5)
The definition of conditional ranks is equivalent to
(d)
κ(A ∩ B) = κ(A) + κ(B | A)
[the law of conjunction].
One might also axiomatize ranking theory by that law of conjunction and
(e)
κ(B | A) = 0 or κ(~B | A) = 0 [the conditional law of negation].
This shows that ranking theory assumes nothing but conditional
consistency besides the definition of conditional ranks.
If one recalls how successfully the dynamics of subjective probabilities
can be explained on the basis of the notion of conditional probability, then it is plausible that an equally successful dynamics of belief
can be explained on the basis of the notion of conditional ranks.
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Explicating Normal Conditions: the Set-up (1)
Although cp talk seems to speak about open-ended possibilities consisting of don’t know-whats’, we need to refer to fixed conceptual frame,
a fixed set W of possibilities (= small worlds) and a well-defined algebra A over W; otherwise theorizing cannot start. More precisely, let A
be generated by (random) variables on W, as is familiar from probability theory.
Specifically, we will have to refer to two target variables X and Y and one
(comprehensive) background variable Z (that may decompose in
many component variables). So, we may conceive of small worlds
as triples 〈x, y, z〉 of possible values of X, Y and Z.
X, Y, and Z are only about a given single case, a specific application. Of
course, we can generalize by repeating the set-up (just as in probability theory, where we generalize from a single throw of a die to an infinite series of throws). Here, we may focus only on the single case.
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Explicating Normal Conditions: the Set-up (2)
We have, i.e., (tentatively) believe in a hypothesis H (a law) about the relation between the target variables X and Y. That is, H is a proposition that is X-Y-measurable (i.e. only about X and Y), but neither Xnor Y-measurable (i.e. it makes no claims about single variables).
We also think that the relation between X and Y somehow depends on
the background variable Z. We will have to study how.
In any case, our beliefs or rather our doxastic attitude concerning this
set-up are represented by some ranking function ξ for A.
That the hypothesis H is conjectured according to the ranking function ξ
means that H is the strongest proposition about X and Y believed
according to ξ, i.e., ξ(~H) > 0 and for no X-Y-measurable H' ⊂ H
ξ(~H') > 0.
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Normal and Exceptional Conditions (1)
The basic observation now is this:
Even if the hypothesis H is unconditionally believed in ξ, i.e. ξ(~H) > 0, it
need not be believed under all conditions. There will be a background condition, i.e., a Z-measurable proposition N that is unsurprising or not excluded; that is, ξ(N) = 0. In this case, we still have
ξ(~H | N) > 0; the hypothesis H is still held under such a condition
N, which we may provisionally call a normal condition.
However, there might also be a condition E about the background Z,
given which the hypothesis H about X and Y is no longer held, i.e.,
such that ξ(~H | E) = 0. This entails ξ(E) = ξ(~H ∩ E) ≥ ξ(~H); i.e.,
such a background E must be at least as disbelieved as the violation of H. We may therefore provisionally call E an exceptional
condition with which we do not reckon according to ξ.
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Normal and Exceptional Conditions (2)
So far I have suggested:
(1)
The condition N for the variable Z is normal iff ξ(N) = 0,
(2)
the condition E on Z is exceptional iff ξ(~H | E) = 0.
•
One should think that E and ~E cannot both be exceptional. This is
implied by (2).
•
One should allow that N and ~N are both normal. (1) does this.
•
One should think that E is exceptional iff ~E is normal. (1) and (2)
do not satisfy this. Therefore one might suggest to replace (1) by:
(3)
The condition N on Z is normal iff ξ(~H | N) > 0.
•
However, this closely resembles the trivial interpretation of normal
conditions.
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Normal and Exceptional Conditions (3)
Moreover, we are used to speaking of the normal conditions and the
exceptional conditions. This might mean, respectively, the weakest
normal and the weakest exceptional condition. Here, we find that (2)
and (3) imply:
(4)
if N and N' are normal, N ∪ N' is normal, too,
(5)
if E and E' are both exceptional, then E ∪ E' is exceptional, too.
It might also mean, respectively, the strongest normal and the strongest
exceptional condition. However, (2) and (3) do not guarantee that:
•
if N and N' are normal, N ∩ N' is normal, too,
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if E and E' are both exceptional, then E ∩ E' is exceptional, too.
Rather, we might have ξ(~H | E ∩ E') > 0. Then, E ∩ E' would be normal
according to (3), while intuition would like to call E ∩ E' doubly
exceptional, because the judgment about H is reversed twice.
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The Final Explication (1)
These considerations allow me to introduce the explication I want to
finally endorse:
First, we may use (5) to define:
E ≥ 1 = ∪ {E | E is exceptional} as the (weakest) exceptional condition, and correspondingly,
(7) N0 = ~E ≥ 1 as the normal condition (still to be justified).
Next, I have suggested:
(6)
E ⊆ E ≥ 1 is doubly exceptional iff ξ(~H | E) > 0, i.e., iff the hypothesis H is in turn supported by E.
Then, we may use (4) to define:
(8)
E ≥ 2 = ∪ {E ⊆ E ≥ 1 | E is doubly exceptional} as the (weakest)
doubly exceptional condition.
And so on.
(9)
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The Final Explication (2)
So, the general proposal is this:
(10) We set E ≥ 0 = W, the whole possibility space considered. For odd n
we define E ≥ n = ∪ {E ⊆ E ≥ n-1 | ξ(~H | E) = 0}, for even n we set
E ≥ n = ∪ {E ⊆ E ≥ n-1 | ξ(~H | E) > 0}, and we call E ≥n the exceptional condition of degree n.
(11) E n = E ≥ n – E ≥ n+1. Thus, N0 defined in (7) is the same as E0.
The sequence E0 = N0, E1, E2, … forms a partition of W.
Let [z] denote the proposition that the background variable Z takes the
specific value z. [z] may be called a maximal condition. The
sequence thus provides a classification of maximal conditions: the
maximal condition [z] is exceptional to degree n (and normal if n =
0) iff [z] ⊆ En.
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The Final Explication (3)
Thereby, we finally arrive at a classification of all background conditions:
(12) A proposition or condition E about Z is exceptional to degree n iff n
is the minimal degree of exceptionality among the maximal
conditions [z] ⊆ E. If n is even, we have ξ(~H | E) > 0, and if n is
odd, we have ξ(~H | E) = 0. A condition N that is exceptional to
degree 0 is normal.
Thus, the normal condition N0 is believed in any case, while a normal
condition N may even be disbelieved. Moreover, we have:
(13) [z] ⊆ N0 iff ξ([z]) < ξ(~H).
(14) 0 = ξ(E ≥ 0) ≤ ξ(E ≥ 1) ≤ ξ(E ≥ 2) ≤ ..., and
for even n: ξ(E n+1) > ξ(E n),
for odd n: ξ(E n+1) ≥ max {ξ([z] | [z] ⊆ E n} ≥ ξ(E n).
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Concluding Remarks (1)
•
This proposal is an explication of the epistemic interpretation of
ceteris paribus or normal conditions.
•
It escapes problem (1), i.e., it is precise, because the underlying
formal epistemology is precise.
•
It avoids problem (3), since the hypothesis H and its negation
cannot be held true at the same time.
•
It avoids the trivial interpretation. The normal condition N0 does not
simply contain all the maximal conditions [z] under which H is held
true. It is stronger by excluding the doubly, fourfold etc. exceptional
maximal conditions under which H is also held true.
•
Indeed, this proposal is, as far as I know, the only one that can deal
with the phenomenon of multiply exceptional conditions.
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Concluding Remarks (2)
•
If we would refer the other interpretations, in particular the trivial,
the eliminativistic, the existential, or the causal one, to a fixed setup, i.e., a fixed background Z, their inadequacy is salient. They derive their prima facie plausibility from their reference to open background conditions, which, however, are theoretically unmanageable.
By contrast, the proposed epistemic interpretation has something
substantial and reasonable to say already given any fixed background (that may then be assumed to hold for open backgrounds
as well).
•
The epistemic interpretation entails that a ceteris paribus law does
not make a specific claim, i.e. has no objective truth condition; the
notion of normality it involves is epistemically relativized. This need
not make it disrespectable. Indeed, I think this is how it is with laws
in general (but this is another issue).
Spohn, Ceteris Paribus Conditions
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Concluding Remarks (3)
•
Still, there are certain beliefs associated with a ceteris paribus law
or with normal conditions. In the set-up X, Y, Z, ξ, we first have
ξ(~N0) > 0, i.e., the normal condition is held true. The second characteristic belief is ξ(~H | N0) > 0, i.e., the conditional belief in the hypothesis given the normal condition. Both entail the belief in H & N0.
•
Moreover, the belief in N0 can be true or false, the normal condition
may actually obtain or not. The conditional belief in H given N0 is
usually not something that can be true or false. There is, however,
an objectification theory for ranking functions, which explains the
extent to which conditional beliefs can be (objectively) true or false.
Hence, the proposed account of ceteris paribus laws need not be
stuck in the epistemic relativization.
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Bibliographical Note
The talk is taken from chapter 13 of my book The Laws of Belief. Ranking
Theory and Its Philosophical Applications (ca. 770 pp.), to appear at
OUP.
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