Probability Spaces for First-Order Logic

null, Kenneth Presting (University of St. Andrews, Rutgers University)

Abstract: Probability for first-order languages has previously been studied from the perspective of Bayesian subjective probability (Hailperin) or abstract measure functions on sentences (Gaifman).  This paper defines standard probability spaces in the Kolmogorov style as a measure space (*O, *F, mu ), where *O will be defined as a product space and mu() as a product of measures.. We show how to interpret 1^o formulas as extensions which are measurable sets. Modern probability spaces have been typically used with  propositional (Boolean) or set-theoretic expressions denoting events as elements of the sigma-field *F, ie measurable subsets of the domain set *O. For 1^o expressions we define the domain as a product space of two sample types, combining the two basic modes of sampling from a population, viz, sampling with replacement and without replacement. These spaces are an extension of cylindrical algebra (Monk) or polyadic algebra (Halmos). The constructed probability space is interpreted into first-order model theory, by showing events in the probability space to be extensions of expressions in the 1^o language. Probability calculations are demonstrated for a variety of expressions, some basic and some from contemporary research. One interesting conclusion is that Ramsey’s conjecture on conditional probability can be proved for quantified conditionals, P[(x)(Ax→Bx)] = P[(x)Bx | (x)Ax].