On real-valued evaluation of propositional formulas

Abstract

Arguably, [0,1]-valued evaluation of formulas is dominant form of representation of uncertainty, believes, preferences and so on despite some theoretical issues - most notable one is incompleteness of any unrestricted finitary formalization. We offer an infinitary propositional logic (formulas remain finite strings of symbols, but we use infinitary inference rules with countably many premises, primarily in order to address the incompleteness issue) which is expressible enough to capture finitely additive probabilistic evaluations, some special cases of truth functionality (evaluations in Lukasiewicz, product, Gödel and logics) and the usual comparison of such evaluations. The main technical result is the proof of completeness theorem (every consistent set of formulas is satisfiable).