All of Kris Brown's Comments + Replies

But if p is "It's a cat" and q is "It has four legs", and P describes our beliefs (or more precisely, say, my beliefs at 5 pm UTC October 20, 2024), then P(q|p)>P(q). Which surely means p is a materially good reason for q. But p⊬q, so the inference from p to q is still logically bad. So we don't have logicism about reasons in probability theory. Moreover, probability expressions are not invariant under substituting non-logical vocabulary. For example, if r is "It has two legs", and we substitute q with r, then P(r|p)<P(r). Which can only mean the inference from p to r is materially bad. I think the axioms of probability can be thought of as being relative to material conceptual relations. Specifically, the additivity axiom says that the probabilities of "mutually exclusive" statements can be added together to yield the probability of their disjunction. What does "mutually exclusive" mean? Logically inconsistent? Not necessarily. It could simply mean materially inconsistent. For example, "Bob is married" and "Bob is a bachelor" are (materially, though not logically) mutually exclusive. So their probabilities can be added to form the disjunction. (This arguably also solves the problem of logical omniscience, see here).