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Kripke models for linear logic

Published online by Cambridge University Press:  12 March 2014

Gerard Allwein  and
J. Michael Dunn
Gerard Allwein
Affiliation:
Computer Science Department, Center for Innovative Computer Applications, Indiana University, Bloomington, Indiana 47405, E-mail: gtall@cs.indiana.edu
J. Michael Dunn
Affiliation:
Philosophy Department, Computer Science Department, Indiana University, Bloomington, Indiana 47405, E-mail: dunn@cs.indiana.edu
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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. Commutativity and associativity 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 residuated with Linear Logic's cotensor 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 2 , June 1993 , pp. 514 - 545
Copyright
Copyright © Association for Symbolic Logic 1993

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References

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