A Counterexample to the Principal Principle

25 01 2008

In his paper ‘A Subjectivist’s Guide to Objective Chance’, Lewis proposes an intimate connection between subjective probabilities and objective chances: the Principal Principle. In Lewis’s eyes, this principle captures almost all there is to know about our conception of objective chances.

In a forthcoming paper entitled ‘Knowledge and Objective Chance’, Hawthorne and Lasonen mention in passing a counterexample to the Principal Principle (a draft of which you can find here). Essentially, they think that instances of the contingent a priori provide a source of potential counterexamples. This idea stands in an interesting relation to a recent paper of Williamson (‘Indicative versus Subjunctive, Congruential versus Non-Hyperintensional Contexts’), in which the modal status of statements involving objective and subjective probabilities is discussed (a draft of which you can find there). It seems to me that Williamson’s considerations may provide a deeper reason to explain why we should not expect something like the Principal Principle to hold. In the following note, I will present a structurally similar counterexample by way of relating it to Williamson’s claims about the modal nature of the two kinds of probability.

Posted by Moritz.

The Principal Principle can partly be motivated by way of examples. Suppose a fair coin is going to be tossed tomorrow. How likely should we think it to be true that it will come down heads? 1/2, of course. Why? Because its present objective chance of coming down heads is 1/2. The Principal Principal generalises this pattern of reasoning. It states that in the absence of evidence which bears more directly on a proposition A, we should adjust our credence in A to what we take to be the objective chance of A. More precisely, the Principal Principal can be stated as follows:

(The Principal Principle)
Let C be any reasonable initial credence function, A any
proposition within the domain of objective chances, t any time,
and E any proposition which is admissible at t. Now, let X be
the proposition that the objective chance of A at t is x. Then
C(A | X \& E)=x.

A few remarks may be in order. The reference to times is needed, since objective chances obtain relative to times. Today the chances of the coin coming down heads may be 50%. Tomorrow, after the coin was tossed, the chances will be either 1 or 0, depending on whether it comes down heads or not. Now to the idea of admissible evidence. It requires a great deal of work to specify the admissible propositions. As a first approximation, one can start by saying that information is admissible if it is solely concerned with the history up to time t. For instance, every proposition which is solely about the history up to now will be admissible for the proposition that the coin will come down heads tomorrow. Since the question of which propositions are admissible will not play any role in the argument, I will leave it at that.

Let me start by pointing to the fact, highlighted in Williamson’s paper, that statements about objective chances do not generate hyperintensional contexts. A sentential operator D is said to be hyperintensional if there are necessarily equivalent sentences A and B such that D(A) is true but D(B) is false. The idea that objective chances do not give rise to hyperintensional contexts can be put like this:

(Objective Chances Are Not Hyperintensional)
If \small A and B are necessarily equivalent, then the objective chance
of A always equals the objective chance of B. More formally, for
all times t (where P_t denotes the objective chance distribution
at t):
\models \Box (A \equiv B) \supset (P_t(A)=P_t(B)) .

The argument for this thesis is straightforward. Objective chances measure objective possibilities. Thus, the objective chance of a proposition or sentence to be true derives from its modal properties. But if two propositions or sentences are necessarily equivalent, they have the same modal profile and therefore the same modal properties. Hence, the expression of objective chances does not constitute a hyperintensional context.

Subjective probabilities, on the other hand, seem to give rise to hyperintensional contexts. Consider the following example. Suppose a proposition p is actually true. Then the proposition @ p \equiv p is necessarily equivalent to p (I use @ as the ‘actually’-operator). For, if we evaluate @ p \equiv p at a counterfactual world w, @ p \equiv p will be true at w just in case p is true at w, since p is actually true. Now, as an obvious instance of the contingent a priori, we should always be certain about @ p \equiv p. However, we should not always be certain about any proposition p which happens to be true. For instance, we should not be certain that the coin will come down heads tomorrow (even if in fact the coin will come down heads tomorrow). But, of course, we should be certain that the coin will come down heads just in case the coin will actually come down heads. Hence, subjective probabilities constitute a hyperintensional context:

(Subjective Probabilities Are Hyperintensional)

There are necessarily equivalent propositions or sentences A and
B such that the subjective probability of A should not always
equal the subjective probability of B. More formally (C being a
reasonable credence function),
\not \models \Box (A \equiv B) \supset C(A) = C(B) .

Of course, this does not hold if one identifies propositions with sets of possible worlds as, for instance, Lewis (1980) did. But the foregoing example shows that this way of individuating propositions is too coarse grained to be sensitive to the distinctive features of our epistemic lifes.
The observation is thus that subjective possibilities measure epistemic possibilities which are, as examples of the contingent a priori and the necessary a posteriori show, incomparable in strength with metaphysical possibilities. Since subjective probabilities fall on the epistemic side of this distinction, it takes no wonder that they give rise to hyperintensional contexts.

Now to the counterexample. Let A be the proposition that the coin will come down heads tomorrow, and let t be today. As above, P_t is the objective chance distribution at time t (and, of course, at the actual world). The crucial observation is that the following two statements are a priori equivalent:

(1) P_t(A)=P_t(\neg A)= 1/2,
(2) P_t(@A \equiv A)=1/2.

To see this, note that the logic of ‘actually’ implies the following theorem:

(3) \Box ((@A \equiv A) \equiv A) \vee \Box ((@A \equiv A) \equiv \neg A).

We have already argued for this principle. If A is true at the actual world, then @A \equiv A is true exactly at the A-worlds, and if A is false at the actual world, then @A \equiv A is true exactly at the \neg A-worlds.

From (3), the equivalence of (1) and (2) follows by using the non-hyperintensionality of objective chances. For the direction from (1) to (2): By (3) and the non hyperintensionality, we get that the objective chance of @A \equiv A is either the one of A or the one of \neg A; since both are the same, (2) follows. For the direction from (2) to (1): By (3) and the non hyperintensionality of objective chances, it follows from (2) that either the objective chance of A is 1/2 or the objective chance of \neg A is 1/2; since both disjuncts are equivalent, (1) follows.

Now, the Principal Principle makes the following prediction:

(4) C(@A \equiv A | P_t(@A \equiv A)=1/2)=1/2.

But this is wrong. We should assign credence 1 to @A \equiv A no matter what, since we can always be certain that the coin will land heads just in case it will actually land heads. Hence, there seems to be a counterexample to the Principal Principle.

One may think that conditionalizing on P_t(@A \equiv A)=1/2 should undermine our certainty in @A \equiv A. Even though I do not take this idea to be a live option (we should always be certain about a logical truth such as @ A \equiv A!), one can demonstrate the coherency of the epistemic state described by

(5) C(@A \equiv A | P_t(@A \equiv A)=1/2)=1.

relative to the coherency of another state. Clearly, the following epistemic state is rational:

(6) C(@A \equiv A | P_t(A)=1/2)=1.

My thinking that the objective chance of the coin landing heads is 1/2 should not undermine my certainty that the coin will land heads just in case it will actually land heads. However, since P_t(A)=1/2 is a priori equivalent to P_t(@A \equiv A)=1/2 (as we have seen in arguing for the equivalence of (1) and (2) above), (6) is a rational epistemic state just in case (5) is. So, we have shown the relative coherency of an epistemic state described by (5): it is coherent just in case (6) desribes a coherent state. So, there is a counterexample to the Principle Principal if (6) is correct. And, as I have argued, (6) is correct.

What is the source of the counterexample to the Principal Principal? An answer suggests itself: the counterexample derives from the fact that subjective chances are hyperintensional whereas objective chances are not.




8 responses

23 02 2008

Hi Moritz, cool stuff!

Without having read Williamson’s paper, your explanation why the objective chance operator is not hyperintensional leaves me puzzled: what does it mean to ‘measure objective possibility’? Offhand, I would have thought that the objective chance that, say, this will actually land heads can be 1/2. On Williamson’s view, it always has to be either 0 or 1 (due to the rigidification), right?

It seems to me that examples such as yours and Hawthorne’s illustrate why Lewis expresses the Principle in terms of propositions (classes of worlds) and not sentences. What proposition does “@A ≡ A” express? That depends on what is meant by “expressing”: if we go by modal profile, “@A ≡ A” expresses the class of A-worlds; if we go by (conventional) information content, it expresses the universal class of all worlds. Either way, we don’t get a counterexample: your credence in the world being an A-world given that the objective chance for that is 1/2 should be 1/2. Your credence in the world being one of all worlds given that the objective chance for that is 1/2 should be undefined, as you should be absolutely certain that the objective chance for the universal proposition is not 1/2.

23 02 2008

oops, the software swallowed my biconditionals. By “@A A” I meant “@A iff A”. And there’s a “coin” missing in the first paragraph. Sorry, should reread before submitting..

23 02 2008

Hi wo, I inserted some ‘≡’s in your comment above, I hope that is fine with you. Benjamin.

23 02 2008

I guess the following is related to wo’s point:

It is prima facie plausible that a sentence S in the present progressive and ‘now’ ∩ S express the same proposition in every context (where ‘∩’ means concatenation).
Similarly, some philosophers find the view attractive that a simple sentence S and the corresponding sentence ‘actually’ ∩ S express the same proposition with respect to every context of utterance.

However, if you apply temporal operators to S and ‘now’ ∩ S respectively, the results may differ in truth-value. Even though it is raining, it will not always be the case that is raining. But since it is raining now, it will always be the case that is raining now.
Similarly, if you apply modal operators to S and ‘actually’ ∩ S respectively, the results may differ in truth-value. Even though snow is white, it is not necessary that snow is white. But since snow is actually white, it is necessary that snow is actually white.

Then, two sentences may express the same proposition and yet embed differently within modal contexts. If this is correct, then it is not clear that the above example affect Lewis’ principle which is stated in terms of propositions. On the view described, ‘@p ≡ p’ and ‘p ≡ p’ do express the same proposition.

So, if the view described holds, what can we then learn from your examples? (No rhetorical question!)

24 02 2008

Hey Benjamin, thanks for the ‘≡’s!

I agree with your analysis: if ‘S’ and ‘@S’ express the same proposition, then one doesn’t get a counterexample. Even if they express different propositions, as long as both occurrences of ‘@A ≡ A’ stand for the same proposition in the alleged counterexample, the Principle goes through.

One of the things I guess these examples show is that when rigidification is involved, one can’t just blindly insert the same sentence A at the two places in C(A|P(A)=x)=x, because the proposition which P() ends up evaluating will depend on rigidification devices in A that are redundant in the scope of C().

In this respect, the Principal Principle resembles the claim that if something is logically true, it couldn’t possibly be false. This seems to me obviously correct, if properly understood. But of course if you express it as a schema: ‘if |= A then [] A’, and then plug in a suitable instance of ‘@A ≡ A’, you get false instances. The reason, as before, is that the box [] now ends up evaluating the proposition that A, rather than the trivial proposition expressed by ‘@A ≡ A’.

24 02 2008

Hi there,

I think that’s an interesting option. What the example shows is that we got at least an apparent counterexample, namely on the level of sentences. A rational person’s credences may be such that she is certain about
(1) The objective chance that the coin lands heads just in case it actually lands heads is 1/2,
but she assigns credence 1 to what she would express by
(2) The coin lands heads just in case it actually lands heads.

Now, one might say, as suggested by the two of you, that the proposition expressed by ‘The coin lands heads just in case it actually lands heads’ is not the same in (1) and (2) (I am not quite sure whether this is the best way to put it).

I think what’s still interesting is the observation that Lewis’s principle is of a kind quite similar to the claim that a priori truths are necessary (by the way, note that one can reproduce the counterexample for any coningent a priori truths concerning the future). This is probably why one can avoid taking the apparent counterexamples to be real ones by employing the same kind of strategies which have been used to argue that there are (deep down) no contingent a priori (and/or logical) truths. At the end, ones attitude towards the question whether there are contingent a priori truths will probably determine ones stance towards the apparent counterexample.

17 08 2008

Hi Moritz,
I am new to the subject of Philosophy of science and am a student. I was trying to make head or tail out of the lewis paper and came across your postings. Can you put me to simple explanations of the paper and its critique…

17 08 2008

Hi Deepali,

I don’t know about any simple stuff on the Principal Principle, and I agree that Lewis’s original paper is pretty dense. Perhaps you would enjoy reading Dorothy Edingtons 2004 paper “Two Kinds of Possibility”, which was published in the Aristotelian Society sup. Vol. It isn’t mainly about the Principal Principle, but it may provide quite an easy route to Lewis’s paper.

%d bloggers like this: