Post relation algebras and their proof system

Abstract

A class of nonclassical relation algebras that correspond to Post logics is introduced and a method of algebraization of those logics is proposed. Relational semantics for Post logics leads to a Rasiowa-Sikorski style proof system for Post logics. A logic LPo intended to provide a formal tool to verify equations in Post relation algebras is defined. Two kinds of rules for the relational logic are defined: decomposition rules enabling the decomposition of relational formulas into some simpler formulas, depending on symbols of relational operations occurring in the formulas; and specific rules, which correspond to semantical postulates assumed in the models of the relational logic. The rules apply to finite sequences of formulas. As a result of application of a rule, a family of new sequences is obtained.<>