Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

Studia Logica 97 (1):31 - 60 (2011)
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Abstract

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa's school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the "core" of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency

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Arnon Avron
Tel Aviv University

Citations of this work

Inferences and Metainferences in ST.Pablo Cobreros, Paul Egré, David Ripley & Robert van Rooij - 2020 - Journal of Philosophical Logic 49 (6):1057-1077.
Ideal Paraconsistent Logics.O. Arieli, A. Avron & A. Zamansky - 2011 - Studia Logica 99 (1-3):31-60.
Four-Valued Paradefinite Logics.Ofer Arieli & Arnon Avron - 2017 - Studia Logica 105 (6):1087-1122.
Remarks on naive set theory based on lp.Hitoshi Omori - 2015 - Review of Symbolic Logic 8 (2):279-295.

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References found in this work

On the theory of inconsistent formal systems.Newton C. A. Costa - 1972 - Recife,: Universidade Federal de Pernambuco, Instituto de Matemática.
On the theory of inconsistent formal systems.Newton C. A. da Costa - 1974 - Notre Dame Journal of Formal Logic 15 (4):497-510.
Natural 3-valued logics—characterization and proof theory.Arnon Avron - 1991 - Journal of Symbolic Logic 56 (1):276-294.
Paraconsistent extensional propositional logics.Diderik Batens - 1980 - Logique and Analyse 90 (90):195-234.

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