Algebraic study of Sette's maximal paraconsistent logic |
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Authors: | Alexej P Pynko |
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Institution: | (1) Department 100, V.M. Glushkov Institute of Cybernetics, Academy of Sciences of Ukraine, Glushkov prosp. 40, 252207 Kiev, Ukraine |
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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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Keywords: | Primary — 03B22 03B53 03G25 Secondary — 03C05 08B15 08B26 08C15 |
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