Skip to main content Accessibility help

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

  • D. M. Gabbay (a1)


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.



Hide All
[1]Amati, G. and Pirri, F., Uniform tableaux methods for intuitionistic modal logic I, to appear, in Studio Logica.
[2]Blackburn, P. and de Rijke, M., Why combine logics?, to appear.
[3]Blok, W. J. and Pigozzi, D., Algebraizable logics, Memoirs of AMS (1989), no. 396.
[4]Božić, M. and Došen, K., Models for normal intuitionistic modal logics, Studio Logica, vol. 43 (1984), pp. 217245.
[5]Bull, R. A., A modal extension of intuitionist logic, Notre Dame Journal of Formal Logic, vol. 6 (1965).
[6]Bull, R. A., MIPC as the formalization of an intuitionistic concept of modality, this Journal, vol. 31 (1966), pp. 609616.
[7]Czelakowski, J., Algebraizability of logic and the deduction theorem, Lecture notes, Colchester, 08 1992, Fourth European Summer school in Logic, Language and Information.
[8]Delgrande, J. P., An approach to default reasoning based on a first order conditional logic, Artificial Intelligence, vol. 36 (1988), pp. 6390.
[9]Dörre, J., Gabbay, D., and König, E., Fibred semantics for feature based grammar logic, Journal of Logic, Language and Information, to appear.
[10]Došen, K., Models for stronger normal intuitionistic modal logics, Studia Logica, vol. 44 (1985), pp. 3970.
[11]Eiben, A. E., Jánossy, A., and Kurucz, A., Combining logics, Amsterdam, 12 1992.
[12]Ewald, W. B., Intuitionistic tense and modal logic, this Journal, vol. 51 (1986), pp. 166179.
[13]Fagin, R., Halpern, J. Y., and Vardi, M. Y., A nonstandard approach to the logical omniscience problem, Reasoning about knowledge, TARK 1990 (Parikh, R., editor), Morgan Kaufmann, 1990, pp. 4155.
[14]Fine, K. and Schurz, G., Transfer theorems for stratified multimodal logics, to appear.
[15]Finger, M. and Gabbay, D., Combining temporal logic systems, Notre Dame Journal of Formal Logic, to appear.
[16]Finger, M. and Gabbay, D. M., Adding a temporal dimension to a logic, Journal of Logic, Language and Information, vol. 1 (1992), pp. 203233.
[17]Fischer-Servi, G., On modal logic with an intuitionistic base, StudiaLogica, vol. 36 (1977).
[18]Fischer-Servi, G., Semantics for a class of intuitionistic modal calculi, Italian studies in the philosophy of science (Chiara, M. L. Dalla, editor), D. Reidel, 1980, pp. 5971.
[19]Fischer-Servi, G., Axiomatizations for some intuitionistic modal logics, Rend. Sem. Mat. Univ. Politecn. Torino, vol. 42 (1984), pp. 179194.
[20]Fitch, F. B., Intuitionistic modal logic with quantifiers, Portugalia Mathematica, vol. 7, (1948), pp. 113118.
[21]Fitting, M., Tableaux for many valued modal logic, Report, 01 25, 1994.
[22]Fitting, M., Logics with several modal operators, Theoria, vol. 35 (1969), pp. 259266.
[23]Fitting, M., Many valued modal logics, Fundamenta Informatica, vol. 15 (1991), pp. 235254.
[24]Fitting, M., Many valued modal logics II, Fundamenta Informaticae, vol. 17 (1992), pp. 5573.
[25]Font, J. M., Modality and possibility in some intuitionistic modal logic, Notre Dame Journal of Formal logic, vol. 27 (1986), pp. 533546.
[26]Gabbay, D. M., Fibred semantics and the combination of logics, paper delivered at Logic Colloquium 1992, Veszprém, Hungary, 08 1992,; a version of the notes is published as a Technical Report No 36, by the University of Stuttgart, Sonderforschungbereich 340, Azenbergstr 12, 70174 Stuttgart, Germany, 1993.
[27]Gabbay, D. M., Conditional implication and nonmonotonic consequence, Views on conditionals (del Cerro, L. al., editors), 1995, OUP, pp. 347369.
[28]Gabbay, D. M., Fibred semantics and the weaving of logic, part 3, How to make your logic fuzzy, Imperial College, 1995, draft.
[29]Gabbay, D. M., Fibred semantics and the weaving of logics, part 2, Logic Colloquium 92 (Czermak, L., Gabbay, D., and de Rijke, M., editors), 1995, SILLI/CUP, pp. 95113.
[30]Gödel, K., Eine Interpretation des intuitistischen aussagenkalküls, Ergebnisse eines Mathematischen Kolloquiums, vol. 4 (1933), pp. 3940.
[31]Goranko, V. and Passy, S., Using the universal modality: Gains and questions, Journal of Logic and Computation, vol. 2 (1992), pp. 530.
[32]Jones, A. J. I., Towards a formal theory of defeasible deontic conditionals, Annals of Mathematics and Artificial Intelligence, to appear.
[33]Jones, A. J. I. and Pörn, I., Ideality, subideality and denontic logic, vol. 65, 1985.
[34]Jones, A. J. I. and Pörn, I., ‘Ought’ and ‘must’, Synthese, vol. 66 (1986).
[35]Kracht, M. and Wolter, F., Properties of independently axiomatizable bimodal logics, this Journal, vol. 56 (1991), pp. 14691485.
[36]Lakemeyer, G., Tractable meta-reasoning in propositional logics of belief, IJCAI-87, 1987, pp. 402408.
[37]Levesque, H., A logic of implicit and explicit belief, AAI-84, 1984, pp. 198202.
[38]Ono, H., On some intuitionistic modal logic, Publications Research Institute Mathematical Science, vol. 13 (1977), pp. 687722.
[39]Ono, H., Some problems in intermediate predicate logics, Reports on Mathematical Logic, vol. 21 (1987), pp. 5568.
[40]Pfalzgraf, J., A note on simplexes as geometric configurations, Archiv der Mathematik, vol. 49 (1987), pp. 134140.
[41]Pfalzgraf, J., Logical fiberings and polycontextural systems, Fundamentals of artificial intelligence research (Jorrand, Ph. and Kelemen, J., editors), Springer-Verlag, 1991, LNCS 535.
[42]Pfalzgraf, J. and Stokkermans, K., On robotics scenarios and modeling with fibered structures, Automated practical reasoning: Algebraic approaches (Pfalzgraf, J. and Wang, D., editors), Springer series texts and Monographs in Symbolic Computation, Springer-Verlag, 1994.
[43]Prior, A. N., Time and modality, Oxford University Press, 1957.
[44]Suzuki, N. Y., An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics, Studia Logica, vol. 48 (1988), pp. 141155.
[45]Suzuki, N. Y., Kripke bundles for intermediate predicate logics and Kripke frames for intuitionistic modal logics, Studia Logica, vol. 49 (1990), pp. 289306.
[46]Wijesekera, D., Constructive modal logic 1, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 271301.

Related content

Powered by UNSILO

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

  • D. M. Gabbay (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.