Paraconsistency, paracompleteness, Gentzen systems, and trivalent semantics

Arnon Avron*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form, or of the form, and all the active formulas of its premises belong to the set. In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence criterion for such systems, and show that a quasi-canonical system of the type we investigate is coherent iff it is strongly paraconsistent or strongly paracomplete (in a sense defined in the paper), iff it has a trivalent, non-deterministic semantics of a special type (also defined in the paper) for which it is sound and complete. Finally, we determine when a system of this sort admits cut-elimination, and provide a simple procedure for transforming one which does not into one which does.

Original languageEnglish
Pages (from-to)12-34
Number of pages23
JournalJournal of Applied Non-Classical Logics
Issue number1-2
StatePublished - 2 Jan 2014


FundersFunder number
Israel Science Foundation280-10


    • Gentzen-type systems
    • cut-admissibility
    • non-deterministic semantics
    • paracomplete logics
    • paraconsistent logics
    • three-valued semantics


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