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A study of Kripke-type models for some modal logics by Gentzen’s sequential method. (English) Zbl 0405.03013


MSC:

03B45 Modal logic (including the logic of norms)
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03F05 Cut-elimination and normal-form theorems
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[1] Bass, H., Finite monadic algebras, Proc. Amer. Math. Soc., 5 (1958), 258-268. · Zbl 0089.01904 · doi:10.2307/2033149
[2] Cresswell, M. J., Frames and models in classical modal logic, in: Algebra and logic, Lecture Notes in Math., 450 (1975), 63-86, Springer, Berlin-Heidelberg- New York. · Zbl 0315.02029
[3] Fitting, M. C., Intuitionistic logic model theory and forcing, North-Holland, Amsterdam, 1969. · Zbl 0188.32003
[4] Gentzen, G., Untersuchungen iiber das logische Schliessen I, II, Math, Z., 39 (1935), 176-210, 405-431.
[5] 9 Investigations into logical deduction (English translation of [4]), in: Szabo, M. E., ed., The collected papers of Gerhard Gentzen, 68-131, North- Holland, Amsterdam, 1969.
[6] Godel, K., Die Vollstandigkeit der Axiome des logischen Funktionenkalkiils, Monatsh. Math. Phys., 37 (1930), 349-360.
[7] Gratzer, G., Universal algebra, Van Nostrand, Princeton, 1968. · Zbl 0182.34201
[8] Hayashi, T., Notes on K and KI, private communication, 1975.
[9] 9 Disjunctive property in McCarthy’s prepositional knowledge system, private communication, 1976.
[10] Henkin, L., The completeness of the first-order functional calculus, /. Symbolic Logic, 14 (1949), 159-166. · Zbl 0034.00602 · doi:10.2307/2267044
[11] Hintikka, J., Knowledge and Belief, an introduction to the logic of the two no- tions, Cornell University Press, Ithaca and London, 1962.
[12] Itoh, M., On the relation between the modal sentential logic and monadic predicate calculus I, II, III (in Japanese), /. Japan Assoc. for Philosophy of Sciences, 3, 4, 6 (1955-56), 40-43, 14-19, 18-25.
[13] Kreisel, G., A survey of proof theory, /. Symbolic Logic, 33 (1968), 321-388. · Zbl 0177.01002 · doi:10.2307/2270324
[14] 9 A survey of proof theory II, in: Festad, J. E., ed., Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971.
[15] Kripke, S., Semantical analysis of modal logic I - normal modal prepositional calculi, Z. Math. Logik Grundlagen Math., 9 (1963), 67-96. · Zbl 0118.01305 · doi:10.1002/malq.19630090502
[16] , Semantical analysis of intuitionistic logic I, in: Formal systems and recursive functions, North-Holland, Amsterdam, 1965.
[17] Lemmon, E. J., Algebraic semantics for modal logics I, II, /. Symbolic Logic, 31 (1966), 46-65, 192-218. · Zbl 0147.24805 · doi:10.2307/2270619
[18] Lemmon, E. J. and Scott, D. S., Intensional logic, preliminary draft of initial chapters by E. J. Lemmon, mimeographed, Stanford University, 1966.
[19] Lyndon, R. C, Notes on Logic, Van Nostrand, Princeton, 1966. · Zbl 0168.00301
[20] Maehara, S., A general theory of completeness proofs, Ann. of the Assoc. for Philosophy of Science, 3 (1970), 242-256. · Zbl 0226.02032
[21] McCarthy, J., private communication, 1975.
[22] , An axiomatization of knowledge and the example of the wise man puzzle, private communication, 1976.
[23] Mitchell, B., Theory of categories, Academic Press, New York and London, 1965. · Zbl 0136.00604
[24] Ohnishi, M. and Matsumoto, K., Gentzen method in modal calculi, Osaka Math. J., 9 (1957), 113-130 and 11 (1959), 115-120. · Zbl 0080.00701
[25] Prawitz, D., Natural deduction, a proof-theoretical study, Almqvist & Wiksell, Stockholm, 1965. · Zbl 0173.00205
[26] , Ideas and results in proof theory, in: Fenstad, J. E., ed., Proceed- ings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971. · Zbl 0226.02031
[27] , Comments on Gentzen-type procedures and the classical notion of truth, in: Proof Theory Symposion, Kiel 1974, Lecture Notes in Math., 500 (1975), 290-319, Springer, Berlin-Heidelberg-New York. · Zbl 0342.02022
[28] Rasiowa, H., An algebraic approach to non-classical logics, North-Holland, Amsterdam, 1974. · Zbl 0299.02069
[29] Rasiowa, H. and Sikorski, R., The mathematics of metamathematics, Mono- grafie Mathemtyczne 41, Warszawa, 1963. · Zbl 0122.24311
[30] Sato, M., Kripke-type models for McCarthy’s prepositional knowledge system, unpublished memo, 1975.
[31] Schutte, K., Vollstdndige Systeme modaler und intuitionistischer Logik, Er- gebnisse der Mathematik und ihrer Grenzgebiete, Band 42, Springer, Berlin- Heidelberg-New York, 1968. · Zbl 0157.01602
[32] Scott, D. S., Continuous lattices, in: Toposes, algebraic geometry and logic, Lecture Notes in Math., 274 (1972), 97-136, Springer, Berlin-Heidelberg-New York.
[33] , Data types as lattices, in: Logic conference, Kiel 1974, Lecture Notes in Math., 499 (1975), 579-651, Springer, Berlin-Heidelberg-New York.
[34] Segerberg, K., An essay in classical modal logic, Filosofiska Studier, Uppsala University, 1971. · Zbl 0311.02028
[35] Smullyan, R. M., First-order logic, Ergebnisse der Mathmatik und ihrer Grenz- gebiete, Band 43, Springer, Berlin-Heidelberg-New York, 1968.
[36] Sonobe, O., A note on the modal logic S5, private communication, 1975.
[37] Takahashi, M., A system of simple type theory of Gentzen style with inference on extensionality and the cut-elimination in it, Comm. Math. Univ. Sancti Pauli, 18 (1970), 129-147. · Zbl 0223.02021
[38] Takeuti, G., Proof theory, North-Holland, Amsterdam, 1975.
[39] Zucker, J., The correspondence between cut-elimination and normalization, Ann. Math. Logic, 7 (1974), 1-155. · Zbl 0298.02023 · doi:10.1016/0003-4843(74)90010-2
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