Paraconsistency, self-extensionality, modality

Arnon Avron*, Anna Zamansky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new (paraconsistent) negation as φ = Def ∼ φ (where ∼ is the classical negation). We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from most other C-systems in having the important replacement property (which means that equivalence of formulas implies their congruence). We further show that B is a very robust C-system in the sense that almost any axiom which has been considered in the context of C-systems is either already a theorem of B or its addition to B leads to a logic that is no longer paraconsistent. There is exactly one notable exception, and the result of adding this exception to B leads to the other logic studied here, S5.

Original languageEnglish
Pages (from-to)851-880
Number of pages30
JournalLogic Journal of the IGPL
Issue number5
StatePublished - 1 Oct 2020


  • B
  • KTB
  • modal logic
  • paraconsistent logic


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