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Models for normal intuitionistic modal logics. (English) Zbl 0634.03014
Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal system $${\mathbb{K}}$$ based on Heyting’s propositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for an investigation of intuitionistic analogues of systems stronger than $${\mathbb{K}}$$. A brief survey is given of the existing literature on intuitionistic modal logic.
Reviewer: K.Došen

##### MSC:
 03B45 Modal logic (including the logic of norms)
##### Keywords:
modal logic; intuitionistic logic; Kripke-style models
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##### References:
 [1] J. L. Bell andM. Machover,A Course in Mathematical Logic, North-Holland, Amsterdam 1977. · Zbl 0359.02001 [2] G. Boolos,The Unprovability of Consistency: An Essay in Modal Logic, Cambridge University Press, Cambridge 1979. · Zbl 0409.03009 [3] M. Božić,Positive logic with double negation,Publications de l’Institut Mathématique, Beograd, 35 (49) (1984), pp. 21–31. [4] R. A. Bull,Some modal calculi based on IC, in J. N. Crossley and M. Dummett (eds.),Formal Systems and Recursive Functions, North-Holland, Amsterdam 1965, pp. 3–7. · Zbl 0137.24904 [5] R. A. Bull,A modal extension of intuitionist logic,Notre Dame Journal of Formal Logic 6 (1965), pp. 142–146. · Zbl 0137.24905 [6] R. A. Bull,MIPC as the formalisation of an intuitionist concept of modality,The Journal of Symbolic Logic 31 (1966), pp. 609–616. · Zbl 0192.03001 [7] B. F. Chellas,Modal Logic: An Introduction, Cambridge University Press, Cambridge 1980. · Zbl 0431.03009 [8] H. B. Curry,A Theory of Formal Deducibility, University of Notre Dame Press, Notre Dame (Indiana), 1950. · Zbl 0041.34807 [9] H. B. Curry,Foundations of Mathematical Logic, McGraw-Hill, New York 1963. · Zbl 0163.24209 [10] K. Došen,Models for stronger normal intuitionistic modal logics,Studia Logica (to appear). [11] K. Došen,Intuitionistic double negation as a necessity operator,Publications de l’Institut Mathématique, Beograd, 35 (49) (1984), pp. 15–20. · Zbl 0555.03012 [12] K. Došen,Negative modal operators in intuitionistic logic,Publications de l’Institut Mathématique, Beograd, 35 (49) (1984), pp. 3–14. [13] K. Došen,Negation as a modal operator (to appear). · Zbl 0626.03006 [14] G. Fischer Servi,On modal logic with an intuitionistic basis,Studia Logica 36 (1977), pp. 141–149. · Zbl 0364.02015 [15] G. Fischer Servi,The finite model property for MIPQ and some consequences,Notre Dame Journal of Formal Logic 19 (1978), pp. 687–692. · Zbl 0385.03019 [16] G. Fischer Servi,Semantics for a class of intuitionistic modal calculi, in M. L. Dalla Chiara (ed.),Italian Studies in the Philosophy of Science, Reidel, Dordrecht 1980, pp. 59–72. [17] F. B. Fitch,Intuitionistic modal logic with quantifiers,Portugaliae Mathematica 7 (1948), pp. 113–118. · Zbl 0034.15303 [18] M. C. Fitting,Intuitionistic Logic Model Theory and Forcing, North-Holland, Amsterdam 1969. · Zbl 0188.32003 [19] M. C. Fitting,Logics with several modal operators,Theoria 35 (1969), pp. 259–266. · Zbl 0188.31801 [20] D. M. Gabbay,Semantical Investigations in Heyting’s Intuitionistic Logic, Reidel, Dordrecht 1981. · Zbl 0453.03001 [21] G. Gargov andK. Kirov,The logic of ”strong box” in IGL is IS4Grz,Proceedings of the 11th Spring Conference of the Union of Bulgarian Mathematicians, Bulgarian Academy of Sciences, Sofia, 1982, pp. 154–159. · Zbl 0521.03043 [22] S. A. Kripke,Semantical analysis of modal logic I: Normal modal prepositional calculi,Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9 (1963), pp. 67–96. · Zbl 0118.01305 [23] M. Mihajlova,Reduction of modalities in several intuitionistic modal logics,Comptes rendus de l’Académie bulgare des Sciences 33 (1980), pp. 743–745. · Zbl 0453.03019 [24] G. E. Mints, O nekotorykh ischisleniiakh modal’noi logiki,Trudy Matematicheskogo Instituta V. A. Steklova 98 (1968), pp. 88–111. [25] D. Prawitz,Natural Deduction: A Proof-Theoretical Study, Almqvist & Wiksell, Stockholm 1965. · Zbl 0173.00205 [26] A. N. Prior,Time and Modality, Oxford University Press, Oxford 1957. · Zbl 0079.00606 [27] M. K. Rennie,Models for multiply modal systems,Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 16 (1970), pp. 175–186. · Zbl 0193.30001 [28] K. Segerberg,Modal logics with linear alternative relations,Theoria 36 (1970), pp. 301–322. · Zbl 0235.02019 [29] V. B. Šehtman,A remark on M. K. Rennie’s paper ”Models for multiply modal systems”,Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 23 (1977), pp. 555–558. · Zbl 0382.03016 [30] V. Sotirov,Modal theories with intuitionistic logic,6th Balkan Mathematical Congress (Summaries), University of Sofia, 1977, p. 91. [31] V. Sotirov,Ne finitno approksimiruemye intuitsionistskie modal’nye logiki,Matematicheskie Zametki 27 (1980), pp. 89–94. [32] A. Ursini,A modal calculus analogous to K4W, based on intuitionistic propositional logic, I0,Studia Logica 38 (1979), pp. 297–311. · Zbl 0423.03014 [33] D. Vakarelov,Simple examples of incompl’ete logics,Comptes rendus de l’Academie bulgare des Sciences 33 (1980), pp. 587–589. · Zbl 0435.03013 [34] D. Vakarelov,Intuitionistic modal logics incompatible with the Law of the Excluded Middle,Studia Logica 40 (1981), pp. 103–111. · Zbl 0469.03009
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