Axiomatizing the logic of comparative probability (Q976539)

The author considers the problem of axiomatizing a probabilistic extension of classical propositional logic. The aim of the paper is to give a ``Gabbay-style'' axiomatization that the author considers to be simple. After addressing the notion of a ``more simple rule'', the author fixes the formal language. This language is an extension of the classical propositional language which contains a binary operator \(\leq.\) This operator allows for the comparison of probabilities. The author then introduces six probabilistic axioms, A1 to A6. Next, a model \((U,\pi,V)\) of the logic is given, where \(U\) is a Boolean algebra, \(\pi\) a probability measure and \(V\) a valuation. The main part of the paper is devoted to show the soundness and completeness of the axioms of propositional logic and A1--A6 for the class of the above models. The axiomatization given here makes use of the Kraft-Pratt-Seidenberg theorem from measurement theory.