Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T07:45:02.182Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Allwein, Gerard and Dunn, Michael, Kripke models for linear logic, this Journal, vol. 58 (1993), pp. 514–545.Google Scholar
[2] Andréka, Hajnal and Mikulás, Szabolcs, Lambek calculus and its relational semantics: completeness and incompleteness, Journal of Logic, Language and Information, vol. 3 (1994), pp. 1–38.10.1007/BF01066355Google Scholar
[3] Buszkowski, Wojciech and Orlowska, Ewa, Relational formalization of dependencies in information systems, to appear in Rough Sets (Orlowska, E., editor).Google Scholar
[4] D'Agostino, Marcello and Gabbay, Dov, A generalization of analytic deduction via labelled deductive systems I: basic substructural logics, Journal of Automated Reasoning, vol. 13 (1994), pp. 243–281.10.1007/BF00881958Google Scholar
[5] Došen, Kosta, A brief survey of frames for the Lambek calculus, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 38 (1992), pp. 179–187.10.1002/malq.19920380113Google Scholar
[6] Gabbay, Dov, A general theory of structured consequence relations, Substructural logics (Schroeder-Heister, P. and Došen, K., editors), Oxford, 1993, pp. 109–152.Google Scholar
[7] Luz-Filho, Saturnino, Grammar specifications in categorial logics and theorem proving, Proceedings of 13th conference on automated deduction (McRobbie, M. and Slaney, J. K., editors), Lecture Notes in Computer Science, vol. 1104, Springer-Verlag, Berlin, 1996, pp. 703–712.10.1007/3-540-61511-3_123Google Scholar
[8] MacCaull, Wendy, Relational tableaux for tree models, language models and information networks, to appear in Logic at work: Essays dedicated to the memory of Helena Rasiowa (Orlowska, E., editor), Physica.Google Scholar
[9] MacCaull, Wendy, Relational proof theory for linear and other substructural logics, Journal of the Interest Group of Pure and Applied Logic, vol. 5 (1987), pp. 673–697.Google Scholar
[10] Mulvey, Chris and Wick-Pelletier, Joan, The quantization of the calculus of relations, Canadian Mathematical Society Conference Proceeding, vol. 13, American Mathematical Society, 1992.Google Scholar
[11] Ono, Hiroakira, Semantics for substructural logics, Substructural logics (Schroeder-Heister, P. and Došen, K., editors), Oxford, 1993, pp. 259–291.Google Scholar
[12] Ono, Hiroakira and Komori, Yuichi, Logics without the contraction rule, this Journal, vol. 50 (1985), pp. 169–201.Google Scholar
[13] Orlowska, Ewa, Relational interpretation of modal logics, Algebraic logic (Andréka, H. et al., editors), North-Holland, 1991.Google Scholar
[14] Orlowska, Ewa, Relational proof system for relevant logics, this Journal, vol. 57 (1992), pp. 1425–1440.Google Scholar
[15] Orlowska, Ewa, Relational semantics for nonclassical logics: formulas are relations, Philosophical logic in Poland (Wolenski, Jan, editor), Kluwer, 1994, pp. 167–186.10.1007/978-94-015-8273-5_11Google Scholar
[16] Orlowska, Ewa, Relational proof systems for modal logics, Proof theory for modal logics (Wansing, H., editor), Kluwer, 1996, pp. 55–77.10.1007/978-94-017-2798-3_5Google Scholar
[17] Roorda, Dirk, Resource logics: Proof theoretical investigations, Ph.d. dissertation , University of Amsterdam, 1990.Google Scholar
[18] Routley, Richard and Meyer, Robert, The semantics of entailment, Truth, syntax and modality (LeBlanc, H., editor), North-Holland, 1973.Google Scholar