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

Ofer Arieli, Arnon Avron, Anna Zamansky

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)31-60
Number of pages30
JournalStudia Logica
Issue number1
StatePublished - Feb 2011
Externally publishedYes

Bibliographical note

Funding Information:
Acknowledgements. We thank João Marcos for his helpful comments. This research was supported by The Israel Science Foundation (grant No 809-06).


  • non-deterministic semantics
  • paraconsistent logics
  • three-valued logics

ASJC Scopus subject areas

  • Logic
  • History and Philosophy of Science


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