Kripke models for linear logic

Journal of Symbolic Logic 58 (2):514-545 (1993)

Abstract

We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, similar to Curry's subtraction operator, which is resituated with Linear Logic's contensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic

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Author's Profile

Jon Michael Dunn
PhD: University of Pittsburgh; Last affiliation: Indiana University, Bloomington

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Citations of this work

Logical Consequence and the Paradoxes.Edwin Mares & Francesco Paoli - 2014 - Journal of Philosophical Logic 43 (2-3):439-469.
What Is an Inconsistent Truth Table?Zach Weber, Guillermo Badia & Patrick Girard - 2016 - Australasian Journal of Philosophy 94 (3):533-548.
Positive Modal Logic.J. Michael Dunn - 1995 - Studia Logica 55 (2):301 - 317.
Some Concerns Regarding Ternary-Relation Semantics and Truth-Theoretic Semantics in General.Ross T. Brady - 2017 - IfCoLog Journal of Logics and Their Applications 4 (3):755--781.

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