A PERSPECTIVE ON THE ALGEBRA OF LOGIC

作者：Raftery J G^*

来源：Quaestiones Mathematicae, 2011, 34(3): 275-325.

DOI：10.2989/16073606.2011.622856

• Algebra of logic
• relation algebra
• residuation
• substructural logic
• modal logic
• deductive system
• algebraizable logic
• Leibniz operator
• reduced matrix
• Leibniz hierarchy

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.

• 出版日期2011