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Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics

  • D. M. Gabbay (a1)
Abstract
Abstract

This is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systems Li, iI, to form a new system LI. The methodology ‘fibres’ the semantics i of Li into a semantics for LI, and ‘weaves’ the proof theory (axiomatics) of Li into a proof system of LI. There are various ways of doing this, we distinguish by different names such as ‘fibring’, ‘dovetailing’ etc, yielding different systems, denoted by etc. Once the logics are ‘weaved’, further ‘interaction’ axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics.

The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property.

Some results on combining logics (normal modal extensions of K) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20–30 years. We hope our methodology will help organise the field systematically.

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[3] W. J. Blok and D. Pigozzi , Algebraizable logics, Memoirs of AMS (1989), no. 396.

[4] M. Božić and K. Došen , Models for normal intuitionistic modal logics, Studio Logica, vol. 43 (1984), pp. 217245.

[5] R. A. Bull , A modal extension of intuitionist logic, Notre Dame Journal of Formal Logic, vol. 6 (1965).

[8] J. P. Delgrande , An approach to default reasoning based on a first order conditional logic, Artificial Intelligence, vol. 36 (1988), pp. 6390.

[10] K. Došen , Models for stronger normal intuitionistic modal logics, Studia Logica, vol. 44 (1985), pp. 3970.

[16] M. Finger and D. M. Gabbay , Adding a temporal dimension to a logic, Journal of Logic, Language and Information, vol. 1 (1992), pp. 203233.

[18] G. Fischer-Servi , Semantics for a class of intuitionistic modal calculi, Italian studies in the philosophy of science ( M. L. Dalla Chiara , editor), D. Reidel, 1980, pp. 5971.

[22] M. Fitting , Logics with several modal operators, Theoria, vol. 35 (1969), pp. 259266.

[25] J. M. Font , Modality and possibility in some intuitionistic modal logic, Notre Dame Journal of Formal logic, vol. 27 (1986), pp. 533546.

[31] V. Goranko and S. Passy , Using the universal modality: Gains and questions, Journal of Logic and Computation, vol. 2 (1992), pp. 530.

[34] A. J. I. Jones and I. Pörn , ‘Ought’ and ‘must’, Synthese, vol. 66 (1986).

[38] H. Ono , On some intuitionistic modal logic, Publications Research Institute Mathematical Science, vol. 13 (1977), pp. 687722.

[40] J. Pfalzgraf , A note on simplexes as geometric configurations, Archiv der Mathematik, vol. 49 (1987), pp. 134140.

[41] J. Pfalzgraf , Logical fiberings and polycontextural systems, Fundamentals of artificial intelligence research ( Ph. Jorrand and J. Kelemen , editors), Springer-Verlag, 1991, LNCS 535.

[44] N. Y. Suzuki , An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics, Studia Logica, vol. 48 (1988), pp. 141155.

[45] N. Y. Suzuki , Kripke bundles for intermediate predicate logics and Kripke frames for intuitionistic modal logics, Studia Logica, vol. 49 (1990), pp. 289306.

[46] D. Wijesekera , Constructive modal logic 1, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 271301.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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