www.lesswrong.com/posts/TyusAoBMjYzGN3eZS/why-i-m-not-a-bayesian
1 correction found
In order for credences to obey the axioms of probability, all the logical implications of a statement must be assigned the same credence.
Probability coherence does not require giving the same credence to every proposition implied by a statement. It only requires equal credence for logically equivalent statements; if A implies B, the probability axioms generally require P(A) ≤ P(B), not P(A) = P(B).
Full reasoning
This sentence confuses logical implication with logical equivalence.
Under standard probability axioms, if two propositions are logically equivalent, they must receive the same probability. But if one proposition merely implies another, they do not have to get the same credence. Instead, probability is monotone: when (A \Rightarrow B), the event for (A) is a subset of the event for (B), so a coherent probability assignment must satisfy (P(A) \le P(B)).
A simple counterexample shows the claim is false. Let:
- (A): “It is raining and windy.”
- (B): “It is raining.”
Then (A) logically implies (B), but a perfectly coherent agent can assign, for example, (P(A)=0.2) and (P(B)=0.5). Those credences obey the axioms of probability. So it is incorrect to say that all logical implications of a statement must have the same credence.
What probability coherence does require is sameness for logically equivalent reformulations, not for every weaker consequence of a statement.
2 sources
- Logic and Probability (Stanford Encyclopedia of Philosophy)
It can easily be shown that if P satisfies these constraints, then ... P(φ) = P(ψ) for all formulas φ, ψ that are logically equivalent.
- Proofs Probability (University of Washington CSE 547 / STAT 548)
Other useful properties of probability measure: Let (Ω,F,P) be a probability space. If A ⊂ B, then P(A) ≤ P(B).