Borel probabilistic and quantitative logic

Abstract

The present paper introduces the notion of the probabilistic truth degree of a formula by means of Borel probability measures on the set of all valuations, endowed with the usual product topology, in classical two-valued propositional logic. This approach not only overcomes the limitations of quantitative logic, which require the probability measures on the set of all valuations to be the countably infinite product of uniform probability measures, but also remedies the drawback that probability logic behaves only locally. It is proved that the notions of truth degree, random truth degree in quantitative logic and the probability of formulas in probability logic can all be brought as special cases into the unified framework of the probabilistic truth degree. Thus quantitative logic and probability logic are unified. It also proves a one-to-one correspondence between deductively closed theories and topologically closed subsets of the space of all valuations, and a one-to-one correspondence between probabilistic truth degree functions and Borel probability measures on the space of all valuations. The second part of the present paper proposes an axiomatic definition of the probabilistic truth degree, and it is finally proved that each probabilistic truth degree function is represented by a unique Borel probability measure on the space of all valuations in the way given in the first part. Thus a theory which we call probabilistic and quantitative logic in the framework of classical propositional logic is established.