Abstract
The aim of this paper is to study the paraconsistent deductive systemP 1 within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP 1 is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP 1. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable deductive system. We also show thatP 1 has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebraS. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension ofP 1 is the classical deductive systemPC. Throughout the paper we also study those abstract logics which are in a way similar toP 1, and are called hereabstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdú and the author.
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The author should like to express his special thanks to Laboratory 852 of Space Research Institute of Russian Academy of Sciences and personally to Professor W.I. Borisienko for their crucial support in preparing the manuscript of the present paper.
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Pynko, A.P. Algebraic study of Sette's maximal paraconsistent logic. Stud Logica 54, 89–128 (1995). https://doi.org/10.1007/BF01058534
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DOI: https://doi.org/10.1007/BF01058534