At first blush these two topics may seem entirely unrelated. Stalnaker’s Thesis is that the probability of (If A then B) is the conditional probability of B given A. A Moore Statement (one that instantiates Moore’s Paradox) is a statement that could be true, but could not be believed.
But the two get closer when we replace the intuitive notion of belief with subjective probability. Then there are two kinds of Moore Statements to be distinguished:
An Ordinary Moore Statement is one that could be true, but cannot have probability one.
A Strong Moore Statement is one that could have positive probability, but could not have probability one.
When we introduce conditionals with Stalnaker’s Thesis there are Moore Statements in our language. I will give examples of both sorts, and indicate why they are important.
Example 1. Imagine the following situation:
1. The match is not struck
2. The match is wet
3. It is not the case that if the match is struck, it will burn.
Om the basis of lines 1. and 2 we can give we can give several warrants for line 3. Following Stalnaker, we could assert that if the match is struck, it will not burn, because it is wet. And then equivalently, it is not the case that if the match is struck then it will burn.
Obviously Lewis might reject this reasoning. However, Lewis would then say that the match might or might not burn if struck. But that also implies that it is not the case that the match will burn if struck.
Now define:
X = (the match is not struck, and it is not the case that if the match is struck, it will burn)
The above imaginary scenario shows that X could be true.
But X could not have probability one.
Let’s use P for probability as usual. If a conjunction has probability 1, so do its conjuncts. Thus if P(X) = 1 then P(the match is not struck) = 1, and P(the match is struck) = 0.
So then P(if the match is struck it will burn) = P(the match burns | the match is struck) is either 1 or undefined. (This depends just on the convention adopted for probability conditional on a proposition with probability 0.) Hence P(It is not the case that if the match is struck, it will burn) either equals 0 or is not defined. And accordingly, that is so for X as well: P(X) = 0 or P(X) is undefined.
Therefore X is an Ordinary Moore Statement.
To display and example of a Strong Moore Statement, we need to show that something can have positive probability. For this we can use a numerical example.
Example 2. Tosses with a fair die.
The basic statements involved are just about the outcome of a toss, and each outcome has probability 1/6. Define:
A = the outcome is either two or six. True in possibilities {2, 6}
~ A = the outcome is neither two nor six = the outcome is either odd or 4. True in possibilities {1, 3, 5, 4}
B = the outcome is six. True in possibilities {6}
Y = (~ A and it is not the case that if A then B)
The probability of ~A is 4/6.
By Stalnaker’s Thesis, the probability of the conditional (if A than B), equals P(B | A) = 1/2 = 3/6.
So the negation of that conditional also has probability three out of six: P(~(if A then B)) = 3/6.
The probability of the disjunction of ~ A and ~(if A then B) is the sum of the probabilities of their disjuncts minus P(Y). This cannot be greater than 1. So (3/6) +(4/6) – P(Y) is less than or equal to 1.
Since the probability of the two conjuncts of Y together cannot be more than 1, it follows that their conjunction (that is, Y itself) has a probability greater than or equal to 1/6.
By the same argument as in Example 1, mutatis mutandis, Y cannot have probability 1.
Therefore, Y is a Strong Moore Statement.
Remark 1, re belief.
In the scenario for Example 1 it is natural reaction to say that we can believe X. That seems right, and if so, shows that the intuitive notion of belief does not imply subjective probability 1. There are other reasons to suggest that belief takes only a “sufficiently high” subjective probability (cf. Eva, Shear, and Fitelson 2022). This does have the drawback that no single number is high enough for all examples (the lottery paradox), so that belief must be context-sensitive.
Remark 2, re closure under conditionalization.
In an earlier post (Conditionals, Probabilities, and ‘Or to If’ 12/07/2022) I presented the argument that,
for any given domain, the set of probability functions that satisfy Stalnaker’s Thesis is not closed under conditionalization.
The argument was rather abstract, and what it lacked were good, concrete examples. Examples 1. and 2. above fill that gap.
There was a similar situation with the Reflection Principle for subjective probability. A probabilistic Moore Statement is one that is not self-contradictory, one that can even have a positive probability, but you cannot conditionalize on it, because it cannot have probability 1. That there are such statements entails that the set of probability functions which satisfy the principle, on a given domain, is not closed under conditionalization.
For a discussion of how Moore’s Paradox is related to closure under conditionalization see also my post “A Brief Note on the Logic of Subjective Probability” (07/24/2019)
Remark 3, about triviality results
Lewis’ famous triviality result for Stalnaker’s Thesis assumed that the admissible probability functions on a model is closed under conditionalization, and indeed, that it should be. The above examples show that this assumption of Lewis’ precludes Stalnaker’s Thesis from the outset.
Similarly for the other triviality results that I have seen.
REFERENCES
Eva, Benjamin; Ted Shear and Branden Fitelson (2022) “Four Approaches to Supposition”. Phisci-archive.pitt.edu/18412/7/fats.pdf
Bas van Fraassen's philosophy blog 