A non-transitive relevant implication corresponding to classical logic consequence

  • Peter Verdée
  • Inge De Bal
  • Aleksandra Samonek

Abstract

In this paper we first develop a logic independent account of relevant implication. We propose a stipulative denition of what it means for a multiset of premises to relevantly L-imply a multiset of conclusions, where L is a Tarskian consequence relation: the premises relevantly imply the conclusions iff there is an abstraction of the pair <premises, conclusions> such that the abstracted premises L-imply the abstracted conclusions and none of the abstracted premises or the abstracted conclusions can be omitted while still maintaining valid L-consequence.
          Subsequently we apply this denition to the classical logic (CL) consequence relation to obtain NTR-consequence, i.e. the relevant CL-consequence relation in our sense, and develop a sequent calculus that is sound and complete with regard to relevant CL-consequence. We present a sound and complete sequent calculus for NTR. In a next step we add rules for an object language relevant implication to the
sequent calculus. The object language implication reflects exactly the NTR-consequence relation. One can see the resulting logic NTR-> as a relevant logic in the traditional sense of the word.
       By means of a translation to the relevant logic R, we show that the presented logic NTR is very close to relevance logics in the Anderson-Belnap-Dunn-Routley-Meyer tradition. However, unlike usual relevant logics, NTR is decidable for the full language, Disjunctive Syllogism (A and ~AvB relevantly imply B) and Adjunction (A and B relevantly imply A&B) are valid, and neither Modus Ponens nor the Cut rule are admissible.

Published
2019-02-03
How to Cite
VERDÉE, Peter; DE BAL, Inge; SAMONEK, Aleksandra. A non-transitive relevant implication corresponding to classical logic consequence. The Australasian Journal of Logic, [S.l.], v. 16, n. 2, p. 10-40, feb. 2019. ISSN 1448-5052. Available at: <https://ojs.victoria.ac.nz/ajl/article/view/5273>. Date accessed: 23 may 2019. doi: https://doi.org/10.26686/ajl.v16i2.5273.