Maximum Entropy Inference with Quantified Knowledge

We investigate uncertain reasoning with quantified sentences of the predicate calculus treated as the limiting case of maximum entropy inference applied to finite domains. Motivation and notation In this modest note we consider one possible approach to the following problem P: Suppose that my subjective beliefs in some sentences θ1, θ2, . . . , θm of a predicate language are constrained to satisfy a certain set K, say, of linear constraints. In that case what belief should I assign to some other sentence φ ? Following Johnson [14] and Carnap et al [4], [5] we shall limit ourselves to the well studied case where the overlying predicate language L contains just finitely many unary predicate symbols, P1, P2, . . . , Pt and denumerably many constant symbols a1, a2, a3, . . . , the intention here being that these constants are distinct and exhaust the universe. In particular the language does not have equality nor any functions symbols. Then according to ideas of de Finetti [6], Gaifman [10], ∗Supported by a UK Engineering and Physical Sciences Research Council (EPSRC) Research Studentship