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 be any reasonable initial credence function, any
proposition within the domain of objective chances, any time,
and any proposition which is admissible at . Now, let be
the proposition that the objective chance of at is . Then
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 and are necessarily equivalent, then the objective chance
of always equals the objective chance of . More formally, for
all times (where denotes the objective chance distribution
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 is actually true. Then the proposition is necessarily equivalent to (I use as the ‘actually’-operator). For, if we evaluate at a counterfactual world , will be true at just in case is true at , since is actually true. Now, as an obvious instance of the contingent a priori, we should always be certain about . However, we should not always be certain about any proposition 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 and
such that the subjective probability of should not always
equal the subjective probability of . More formally ( being a
reasonable credence function),
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 be the proposition that the coin will come down heads tomorrow, and let be today. As above, is the objective chance distribution at time (and, of course, at the actual world). The crucial observation is that the following two statements are a priori equivalent:
To see this, note that the logic of ‘actually’ implies the following theorem:
We have already argued for this principle. If is true at the actual world, then is true exactly at the -worlds, and if is false at the actual world, then is true exactly at the -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 is either the one of or the one of ; 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 is or the objective chance of is ; since both disjuncts are equivalent, (1) follows.
Now, the Principal Principle makes the following prediction:
But this is wrong. We should assign credence to 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 should undermine our certainty in . Even though I do not take this idea to be a live option (we should always be certain about a logical truth such as !), one can demonstrate the coherency of the epistemic state described by
relative to the coherency of another state. Clearly, the following epistemic state is rational:
My thinking that the objective chance of the coin landing heads is should not undermine my certainty that the coin will land heads just in case it will actually land heads. However, since is a priori equivalent to (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.