Axiomatizing the logic of comparative probability. (English) Zbl 1193.03044

Burgess, John P.

Notre Dame J. Formal Logic 51, No. 1, 119-126 (2010).

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.

Reviewer: Jürgen Landes (Narbonne)

MSC:

03B48	Probability and inductive logic
03A05	Philosophical and critical aspects of logic and foundations
03A10	Logic in the philosophy of science

Keywords:

probability logic; qualitative probability; axiomatization