Decidability and Expressiveness for First-Order Logics of Probability

We consider decidability and expressiveness issues for two first-order logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show that when the probability is on the domain, if the language contains only unary predicates then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is ?21 complete, as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, ?1∞ hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is ?21 complete with as little as one unary predicate in the language, even without equality. With equality, we get ?1∞ hardness with only a constant symbol. We then turn our attention to an analysis of what causes this overwhelming complexity. For example, we show that if we require rational probabilities then we drop from ?21 to ?11. In many contexts it suffices to restrict attention to domains of bounded size; fortunately, the logics are decidable in this case. Finally, we show that, although the two logics capture quite different intuitions about probability, there is a precise sense in which they are equi-expressive.