A perspective on the algebra of logic

Abstract

Equations are the most basic formulas of algebra, and the logical rules for manipulating them are so intuitive that they are seldom formalized. Consequently, non-algebraic deductive systems (or ‘logics’) are very often interpreted in equational languages—although this is not always possible. For the optimal transfer of algebraic techniques, we require invertible interpretations that respect the structure of substitution; they should also induce isomorphism between the extension lattice of a system and that of its algebraic counterpart. The successful resolution of concrete logical problems in the presence of such an  isomorphism has inspired (1) a robust general notion of equivalence between deductive systems, (2) a precise account of ‘algebraizable’ logics (pioneered by Blok and Pigozzi) and (3) a stock of ‘bridge theorems’ between logic and algebra. Moreover, an algebraic invariant in the theory of equivalence— called the Leibniz operator—has given rise to (4) a classification of deductive systems, analogous to the Maltsev classification of varieties in universal algebra. The present paper is a selective exposition of these developments.

Keywords: Algebra of logic, relation algebra, residuation, substructural logic, modal logic, deductive system, algebraizable logic, Leibniz operator, reduced matrix, Leibniz hierarchy.

Quaestiones Mathematicae 34(2011), 275–325.