Concerning measures in first order calculi

Abstract

~0. Introduction. The idea of treating probability as a real valued function defined on sentences is an old one (see ['6] and [7], where other references can be found). Carnap's at tempt to set up a theory of probabili ty which will have a logical status analogous to that o f two valued logic, is closely connected with it, ef. [1]. So far the sentences were used mainly from a algebraic point of view, that is, the operations that were involved were those of the sentential calculus. (Th e work of Carnap and his collaborators does, however, touch on probabilities which are defined for special cases of first order monadic sentences.) A measure on a sentential calculus which assigns real values to sentences is essentially the same as a measure on the Lindenbaum-Tarski algebra of that calculus, thus its investigation falls under the study of measures on Boolean algebras. These were studied quite a lot; see [3, 5] were other references are given. In this work the notions of a measure on a first order calculus, and of a measuremodel, are introduced and investigated. This is done not from a point of view concerning the foundations of probability but with an eye to mathematical logic and measure theory; the concepts with which we shall deal form a natural generalization of the concepts of a theory and a model in the usual sense. In §1 the not ion of a measure on a first order calculus is introduced. In §2 the notion of a measure-model is defined and a theorem analogous to the completeness theorem is proved. In §3 the case of a calculus with an equality is treated. §4 is concerned with measure-models in which the measure is invariant under permutations of the individuals, and §5 contains a specific example of such a model. Whereas the propositions of §§1, 2 are analogous to similar ones concerning theories and models (in the usual sense), §§4, 5 deal with situations which are typical to measures and measure-models and have no analogous counterpart.