Algebraic Study of Sette's Maximal Paraconsistent Logic

• Published in: Studia logica Vol. 54; no. 1; pp. 89 - 128
• Main Author: Pynko, Alexej P.
• Format: Journal Article
• Language: English
• Published: Wroclaw, Poland Kluwer Academic Publishers 01.01.1995; Ossolineum
• Subjects: Abstract algebraic logic; Abstract logic; Abstract model theory; Algebra; Axioms; Boolean algebras; Logical theorems; Mathematical theorems; Morphisms; Universal algebra
• ISSN: 0039-3215; 1572-8730
• DOI: 10.1007/BF01058534

The aim of this paper is to study the paraconsistent deductive system Pⁱ within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse that Pⁱ is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebra S being the unique quasivariety semantics for Pⁱ. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated by S is not equivalent to any algebraizable deductive system. We also show that Pⁱ has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebra S. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension of Pⁱ is the classical deductive system PC. Throughout the paper we also study those abstract logics which are in a way similar to Pⁱ, and are called here abstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdú and the author.