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Relational semantics and a relational proof system for full Lambek calculus

Published online by Cambridge University Press:  12 March 2014

Wendy MacCaull
Wendy MacCaull*
Affiliation:
Department of Mathematics, Computing and Information Sciences, St. Francis Xavier University, P.O. Box 5000, Antigonish, NS, B2G 2W5, Canada, E-mail: wmaccaul@stfx.ca
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Abstract

In this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

Keywords

full Lambek calculusrelational semanticsproof theoryternary relationsKripke semanticstableauxcountermodel
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 2 , June 1998 , pp. 623 - 637
Copyright
Copyright © Association for Symbolic Logic 1998

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