The NSn x m - propositional calculus

Abstract

In 2000, we introduced n × m – valued Lukasiewicz algebras with negation (or NS n × m –algebras). These algebras were investigated in different papers and they constitute a non-trivial generalization of n – valued Lukasiewicz-Moisil algebras. In this article, we introduce the NS n × m – propositional calculus, denoted by S n × m, in terms of the binary connectives (implication), → (weak implication), ∧ (con- junction), ∨ (disjunction) and the unary ones ∼(negation) and σij, 1 ≤ i ≤ n − 1, 1 ≤ j ≤ m − 1. We prove that S n × m belongs to the class of standard systems of implicative extensional propositional cal- culi as defined by H. Rasiowa (H. Rasiowa, An Algebraic Approach to Non-Classical Logics , North-Holland, Amsterdam, London, New York, 1974). Besides, we show that the definitions of NS n × m –algebra and algebra associated with S n × m are equivalent. Finally, we conclude that the completeness theorem for S n × m holds.