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A first approach to abstract modal logics

Published online by Cambridge University Press:  12 March 2014

Josep M. Font  and
Ventura Verdú
Josep M. Font
Affiliation:
Faculty of Mathematics, University of Barcelona, 08007 Barcelona, Spain
Ventura Verdú
Affiliation:
Faculty of Mathematics, University of Barcelona, 08007 Barcelona, Spain
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Abstract

The object of this paper is to make a study of four systems of modal logic (S4, S5, and their intuitionistic analogues IM4 and IM5) with the techniques of the theory of abstract logics set up by Suszko, Bloom, Brown, Verdú and others. The abstract concepts corresponding to such systems are defined as generalizations of the logics naturally associated to their algebraic models (topological Boolean or Heyting algebras, general or semisimple). By considering new suitably defined connectives and by distinguishing between having the rule of necessitation only for theorems or as a full inference rule (which amounts to dealing with all filters or with open filters of the algebras) we are able to reduce the study of a modal (abstract) logic L to that of two nonmodal logics L− and L+ associated with L. We find that L is “of IM4 type” if and only if L− and L+ are both intuitionistic and have the same theorems, and logics of type S4, IM5 or S5 are obtained from those of type IM4 simply by making classical L−, L+ or both. We compare this situation with that found in recent approaches to intuitionistic modal logic using birelational models or using higher-level sequent-systems. The treatment of modal systems with abstract logics is rather new, and in our way to it we find several general constructions and results which can also be applied to other modal systems weaker than those we study in detail.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 1042 - 1062
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1]Blok, W. J. and Kóhler, P., Algebraic semantics for quasi-classical modal logics, this Journal, vol. 48(1983), pp. 941–964.Google Scholar
[2]Blok, W. and Pigozzi, D., Alfred Tarski's work on general metamathematics, this Journal, vol. 53 (1988), pp. 36–50.Google Scholar
[3]Bloom, S. L., A note on Ψ-consequences, Reports on Mathematical Logic, vol. 8 (1977), pp. 3–9.Google Scholar
[4]Bloom, S. L. and Brown, D. J., Classical abstract logics, Dissertationes Mathejnaticae (Rozprawy Matematyczne), vol. 102 (1973), pp. 43–51.Google Scholar
[5]Božić, M. and Došen, K., Models for normal intuitionistic modal logics, Studia Logica, vol. 43 (1984) 217–245.CrossRefGoogle Scholar
[6]Brown, D. J. and Suszko, R., Abstract Logics. Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 102 (1973), pp. 9–42.Google Scholar
[7]Bull, R. A., Some modal calculi based on IC, Formal systems andrecursive functions (Crossley, J. N. and Dummett, M., editors), North-Holland, Amsterdam, 1965, pp. 3–7.CrossRefGoogle Scholar
[8]Chellas, B. F., Modal logic: an introduction, Cambridge University Press, Cambridge, 1980.CrossRefGoogle Scholar
[9]Czelakowski, J., Algebraic aspects of deduction theorems, Studia Logica, vol. 44 (1985), pp. 369–387.CrossRefGoogle Scholar
[10]Došen, K., Models for stronger normal intuitionistic modal logics, Studia Logica, vol. 44 (1985), pp. 39–70.CrossRefGoogle Scholar
[11]Došen, K., Sequent-systems for modal logic, this Journal, vol. 50 (1985), pp. 149–168.Google Scholar
[12]Došen, K., Higher-level sequent systems for intuitionistic modal logic, Publications de l'Institut Mathématique (Béograd), vol. 39 (1986), pp. 3–12.Google Scholar
[13]Dzik, W., On the content of the lattice of logics. II, Reports on Mathematical Logic, vol. 14 (1982), pp. 29–47.Google Scholar
[14]Fischer-Servi, G., Semantics for a class of intuitionistic modal calculi, Italian studies in the philosophy of science (Chiara, M. L. Dalla, editor), Reidei, Dordrecht, 1980, pp. 59–72.CrossRefGoogle Scholar
[15]Fischer-Servi, G., Axiomatizations for some intuitionistic modal logics, Rendiconti del Seminario Matematico, Università e Politecnico di Torino, vol. 42 (1984), pp. 179–194.Google Scholar
[16]Font, J. M., Implication and deduction in some intuitionistic modal logics, Reports on Mathematical Logic, vol. 17 (1984), pp. 27–38.Google Scholar
[17]Font, J. M., Monadicity in topological pseudo-Boolean algebras, Models and sets (Müller, G. and Richter, M., editors), Lecture Notes in Mathematics, vol. 1103, Springer-Verlag, Berlin, 1984, pp. 169–192.CrossRefGoogle Scholar
[18]Font, J. M., Modality and possibility in some intuitionistic modal logics, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 533–546.CrossRefGoogle Scholar
[19]Font, J. M., On some congruence lattices of a topological Heyting lattice, Contributions to general algebra 5 (Czermak, J.et al., editors), Verlag Hölder-Pichler-Tempski, Wien, and Teubner, Stuttgart, 1987, pp. 129–137.Google Scholar
[20]Font, J. M. and Verdú, V., Abstract characterization of a four-valued logic, Proceedings of the 18th international symposium on multiple-valued logic (Palma de Mallorca, 1988), The Computer Society Press, Washington, D.C., 1988, pp. 389–396.Google Scholar
[21]Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalkuls, Ergebnisse eines Mathematisches Kolioquiums, vol. 4 (1932), pp. 39–40.Google Scholar
[22]Halmos, P. R., Algebraic Logic. I: Monadic Boolean algebras, Compositio Mathematica, vol. 12 (1955), pp. 217–249.Google Scholar
[23]Monteiro, A., La semi-simplicité des algèbres de Boole topologiques et les systèmes déductifs, Revista de la Union Matemàtica Argentina, vol. 25 (1971), pp. 417–448.Google Scholar
[24]Ono, H., On some intuitionistic modal logics, Publications of the Research Institute for the Mathematical Sciences, Kyoto University, vol. 13 (1977), pp. 687–722.CrossRefGoogle Scholar
[25]Ono, H. and Suzuki, N.-Y., Relations between intuitionistic modal logics and intermediate predicate logics, Reports on Mathematical Logic, vol. 22 (1988) (to appear).Google Scholar
[26]Porte, J., Axiomatization and independence in S4 and in S5, Reports on Mathematical Logic, vol. 16 (1982), pp. 23–35.Google Scholar
[27]Porte, J., Fifty years of deduction theorems, Proceedings of the Herbrand symposium (Stern, J., editor), North-Holland, Amsterdam, 1982, pp. 243–250.CrossRefGoogle Scholar
[28]Prawitz, D., Natural deduction. A proof-theoretical study, Almqvist & Wikseil, Stockholm, 1965.Google Scholar
[29]Rasiowa, H., An algebraic approach to non-classical logics, North-Holland, Amsterdam, 1974.Google Scholar
[30]Rasiowa, H. and Sikorski, R., The mathematics of metamathematics, PWN, Warsaw, 1970.Google Scholar
[31]Sotirov, V. I., Modal theories with intuitionistic logic, Proceedings of the conference on mathematical logic (Sofia, 1980), Bulgarian Academy of Sciences, Sofia, 1984, pp. 139–172.Google Scholar
[32]Suzuki, N.-Y., An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics (to appear).Google Scholar
[33]Tarski, A., Über einige fundamentale Bregriffe der Metamathematik, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), pp. 22–29.Google Scholar
[34]Verdú, V., Contribution to the study of some classes of abstract logics, Ph. D. Dissertation, University of Barcelona, Barcelona, 1978 (Catalan).Google Scholar
[35]Verdú, V., Some algebraic structures determined by closure operators, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 31 (1985), pp. 275–278.CrossRefGoogle Scholar
[36]Verdú, V., Logics projectively generated from [M] = (F4, [{1}]) by a set of homomorphisms, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), pp. 235–241.CrossRefGoogle Scholar
[37]Verdú, V. and Font, J. M., Abstract logics related to classical conjunction and disjunction, Studio Logica (to appear).Google Scholar
[38]Wójcicki, R., Lectures on propositional calculi, Ossolineum, WrocŁaw, 1984.Google Scholar
[39]Yokota, S., General characterization results on intuitionistic modal propositional logics, Commentarii Mathematici Universitatis Sancti Pauli, vol. 34 (1985), pp. 177–199.Google Scholar
[40]Yokota, S., Modal Heyting algebras with comodal operator, Commentarii Mathematici Universitatis Sancti Pauli, vol. 35 (1986), pp. 59–64.Google Scholar
[41]Yokota, S., Some intuitionistic versions of the modal propositional logic K(S), Commentarii Mathematici Universitatis Sancti Pauli, vol. 35 (1986), pp. 65–67.Google Scholar