Algebraic study of Sette's maximal paraconsistent logic

Studia Logica 54 (1):89 - 128 (1995)
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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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Citations of this work

On Priest's logic of paradox.Alexej P. Pynko - 1995 - Journal of Applied Non-Classical Logics 5 (2):219-225.
Paraconsistency and Sette’s calculus P1.Janusz Ciuciura - 2015 - Logic and Logical Philosophy 24 (2).

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References found in this work

Algebraizable Logics.W. J. Blok & Don Pigozzi - 2022 - Advanced Reasoning Forum.
An algebraic approach to non-classical logics.Helena Rasiowa - 1974 - Warszawa,: PWN - Polish Scientific Publishers.
The mathematics of metamathematics.Helena Rasiowa - 1963 - Warszawa,: Państwowe Wydawn. Naukowe. Edited by Roman Sikorski.
Introduction to mathematical logic.Elliott Mendelson - 1964 - Princeton, N.J.,: Van Nostrand.
Theory of Logical Calculi: Basic Theory of Consequence Operations.Ryszard Wójcicki - 1988 - Dordrecht, Boston and London: Kluwer Academic Publishers.

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