Dugundji’s theorem revisited

sem informaçãoIn 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness (for the systems KH and VB) showed that not every modal logic can be characterized by Kripke frames. Besides, the creation of non-classical modal logics puts the problem of characterization of finite matrices very far away from the original scope of Dugundji’s result. In this sense, we will show how to update Dugundji’s result in order to make precise the scope and the limits of many-valued matrices as semantic of modal systems. A brief comparison with the useful Chagrov and Zakharyaschev’s criterion of tabularity for modal logics is provided.In 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness (for the systems KH and VB) showed that not every modal logic can be characterized by Kripke frames. Besides, the creation of non-classical modal logics puts the problem of characterization of finite matrices very far away from the original scope of Dugundji’s result. In this sense, we will show how to update Dugundji’s result in order to make precise the scope and the limits of many-valued matrices as semantic of modal systems. A brief comparison with the useful Chagrov and Zakharyaschev’s criterion of tabularity for modal logics is provided.8407422sem informaçãosem informaçãosem informaçãohttp://plato.stanford.edu/archives/win2010/entries/logic-modal-origins/, Ballarin, R.: Modern origins of modal logic. In: The Stanford Encyclopedia of Philosophy, Winter 2010 edition. (2010)Béziau, J.Y., A new four-valued approach to modal logic (2011) Log. Anal, 54 (213), pp. 109-121Bueno-Soler, J., (2009) Multimodalidades anódicas e catódicas: a negação controlada em lógicas multimodais e seu poder expressivo (Anhodic and cathodic multimodalities: controlled negation in multimodal logics and their expressive power, in Portuguese). PhD thesis, Instituto de Filosofia e Ciências Humanas (IFCH), Universidade Estadual de Campinas, , Unicamp, Campinas:Carnielli, W.A., Pizzi, C., Modalities and multimodalities (2008) Logic, Epistemology, and the Unity of Science, vol, p. 12. , Springer-Verlag, New York:Chagrov, A.V., Zakharyaschev, M., Modal logic (1997) Oxford Logic Guides, vol, p. 35. , Oxford University Press, Oxford:Creswell, M.J., Hughes, G.E., (1996) A New Introduction to Modal Logic, , Routledge, London:Dugundji, J., Note on a property of matrices for Lewis and Langford’s calculi of propositions (1940) J. Symb. Log, 5 (4), pp. 150-151Esakia, L., Meskhi, V., Five critical modal systems (1977) Theoria, 43 (1), pp. 52-60Gödel, K.: Eine intepretation des intionistischen Aussagenkalkül. Ergebnisse eines mathematischen Kolloquiums 4, 6–7 (1933) (English translation in [13], pp. 300–303)Gödel, K.: Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematischen Kolloquiums 4, 34–38 (1933) (English translation in [13], pp. 222–225)Gödel, K., Kurt Godel, Collected Works: Publications 1929–1936. Oxford University Press (1986) CaryHenkin, L., Fragments of the proposicional calculus (1949) J. Symb. Log, 14 (1), pp. 42-48Lewis, C.I., Langford, C.H., (1932) Symbolic Logic, , Century, New York:Lemmon, E.J., New foundations for Lewis modal systems (1957) J. Symb. Log, 22 (2), pp. 176-186Lemmon, E.J., Algebraic semantics for modal logics I (1966) J. Symb. Log, 31 (1), pp. 44-65Łukasiewicz,J.: O logice trójwartościowej. Ruch Filozoficzny 5, 170–171 (1920) (English translation in [19] pp. 87–88)Łukasiewicz, J., (1970) Selected Works. Studies in Logic, , North-Holland Publishing Company, London:McKinsey, J.C.C., A reduction in number of the postulates for C. I. Lewis’ system of strict implication (1934) Bull. (New Ser.) Am. Math. Soc, 40, pp. 425-427Magari, R., Representation and duality theory for diagonalizable algebras (1975) Stud. Log, 34 (4), pp. 305-313Scroggs, S.J., Extensions of the Lewis system S5 (1951) J. Symb. Log, 16 (2), pp. 112-120Sobociński, B., Family K of the non-Lewis modal systens. Notre Dame (1964) J. Formal Log, V (4), pp. 313-318Zeman, J.J., Modal Logic: The Lewis Systems. Clarendon Press (1973) U

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