Semi-implication: A Chapter in Universal Logic

Arnon Avron*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

We introduce a general notion of semi-implication which generalizes both the implications used in classical, intuitionistic, and many other logics, as well as those used in relevance logics. It is mainly based on the relevant deduction property (RDP)—a weak form of the classical-intuitionistic deduction theorem which has motivated the design of the intensional fragments of the relevance logic R. However, CL , the pure equivalential fragment of classical logic, also enjoys the RDP with respect to ↔. We show that in the language of → this is the only exception. This observation leads to an adequate definition of semi-implication, according to which a finitary logic L has a semi-implication → iff L has a strongly sound and complete Hilbert-type system which is an extension by axiom schemas of HR (the standard Hilbert-type system for the implicational fragment of R). We also show that in the presence of a conjunction, or a disjunction, or an implication, a connective → of a logic L is a semi-implication iff it is an implication (i.e. it satisfies the classical-intuitionistic deduction theorem), and the same is true if L is induced by a matrix which has a single designated value or a single non-designated value.

Original languageEnglish
Title of host publicationThe road to universal logic
Subtitle of host publicationFestschrift for 50th Birthday of Jean-Yves Béziau Volume I
EditorsArnold Koslow , Arthur Buchsbaum
PublisherBirkhäuser/Springer, Cham
Pages59-72
Number of pages14
Edition1
ISBN (Electronic)978-3-319-10193-4
ISBN (Print)978-3-319-10192-7
DOIs
StatePublished - 2015

Publication series

NameStudies in Universal Logic
ISSN (Print)2297-0282
ISSN (Electronic)2297-0290

Keywords

  • Biconditional
  • Classical logic
  • Deduction theorems
  • Implication
  • Relevance logics
  • Semi-implication

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