Classical Logic with n Truth Values as a Symmetric Many-Valued Logic

Abstract

We introduce Boolean-like algebras of dimension n (n{mathrm {BA}}s) having n constants {{{mathsf {e}}}}_1,ldots ,{{{mathsf {e}}}}_n, and an (n+1)-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of n{mathrm {BA}}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The n{mathrm {BA}}s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, n{mathrm {CL}}, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in n{mathrm {CL}}, and, via the embeddings, in all the finite tabular logics.