Carnap's foundation of probability theory

Abstract

In 1945, in a paper in "Philosophy and Phenomenological Research", Rudolf Carnap made a distinction between two concepts of probability. One of these, called by him "probability2", is based in one form or another on frequency quotients of observed phenomena, whereas the other one, called "probabilityi", deals with a concept like "rational degree of belief", which doubtless most of the classical probabilists from Jacob Bernoulli onwards, and several of the modern ones, had in mind. The present work is a gigantic effort to make this concept precise, and to give, as it is stated on the cover "A clarification of the notion of probability ? and the construction of a new and exact theory of probability on a logically sound basis". The author tries to do this by basing the concept of probability on the semantics of a given object-language. As such he choses (p. 65) a language of an extremely simple type. It contains seven signs (viz. W , V , Y ' =*, Y, '(Y)' ; 't' stands for "tautology") a finite number of predicates of any finite order, an infinite sequence of individual variables, and either a finite number N or an infinite sequence of individual constants. In the former case the language is called "Sn"> i*1 the latter "Soc". Out of these signs "atomic sentences" are formed, which apply any one of the predicates, say of order n, to any n of the individual constants. A "state-description" ('3'; p. 70) is a conjunction, having as components one out of each "basic pair", consisting of an atomic sentence and its negation. (Hence the state-descriptions correspond one to one with the subsets of the set of all atomic sentences). Two state-descrip tions are called "isomorphic" (p. 109) if they can be obtained from each other by a permutation of the individual constants, and the dis junction of a class of all state-descriptions, isomorphic with one of them, is called a "structure<lescription" ('?ir', p. 116). If e (evidence) and h (hypothesis) are sentences, then the "degree of