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MODAL LOGIC WITHOUT CONTRACTION IN A METATHEORY WITHOUT CONTRACTION

Published online by Cambridge University Press:  13 February 2019

PATRICK GIRARD  and
ZACH WEBER
PATRICK GIRARD*
Affiliation:
School of Humanities, University of Auckland
ZACH WEBER*
Affiliation:
Department of Philosophy, University of Otago
*
*PHILOSOPHY, SCHOOL OF HUMANITIES UNIVERSITY OF AUCKLAND 14A SYMONDS ST, AUCKLAND CENTRAL AUCKLAND 1010, NEW ZEALAND E-mail: p.girard@auckland.ac.nz
†DEPARTMENT OF PHILOSOPHY UNIVERSITY OF OTAGO PO BOX 56, DUNEDIN 9054 NEW ZEALAND E-mail: zach.weber@otago.ac.nz
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Abstract

Standard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using a nonclassical substructural logic as the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with a noncontractive logic in the background. This sheds light on which modal principles are invariant under changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics.

Keywords

03B4703B45non-classical logicmodal logicsubstructural logiccontraction-free logicdefinability theoryclassical recapture
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 4 , December 2019 , pp. 685 - 701
Copyright
Copyright © Association for Symbolic Logic 2019 

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