Decidability and expressiveness for first-order logics of probability

Decidability and expressiveness issues for two first-order logics of probability are considered. In one the probability is on possible worlds, whereas 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. It is shown 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 Π₁² 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, Π_∞¹. Thus, the logic cannot be axiomatized in either case. When the probability is on the set of possible worlds, the validity problem is Π₁² complete with as little as one unary predicate in the language, even without equality. With equality, Π_∞¹ hardness with only a constant symbol is obtained. In many applications it suffices to restrict attention to domains of a bounded size; it is shown that the logics are decidable in this case