Classical logic with n truth values as a symmetric many-valued logic

Bucciarelli, Antonio;Ledda, Antonio;Paoli, Francesco;Salibra, Antonino

2023-01-01

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Abstract

We introduce Boolean-like algebras of dimension n (nBA s) having n constants e1, … , en, 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 nBA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The nBA 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, nCL , and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in nCL , and, via the embeddings, in all the finite tabular logics.