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An abstract algebraic logic approach to tetravalent modal logics

Published online by Cambridge University Press:  12 March 2014

Josep Maria Font  and
Miquel Rius
Josep Maria Font
Affiliation:
Faculty of Mathematics, University of Barcelona, Gran Via 585, E-08007 Barcelona, Spain, E-mail: font@mat.ub.es
Miquel Rius
Affiliation:
Faculty of Mathematics, University of Barcelona, Gran Via 585, E-08007 Barcelona, Spain, E-mail: font@mat.ub.es
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Abstract

This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

Keywords

Abstract logicgeneralized matrixtetravalent modal algebrafour-valued modal algebraDe Morgan algebrathree-valued Łukasiewicz algebrafour-valued logicmodal logicalgebraizable logicfull modelstrongly adequate Gentzen calculus
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 2 , June 2000 , pp. 481 - 518
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Balbes, R. and Dwinger, P., Distributive lattices, University of Missouri Press, Columbia (Missouri), 1974.Google Scholar
[2]Becchio, D., Sur les définitions des algèbres trivalentes de Łukasiewicz données par A. Monteiro, Logique et Analyse, vol. 16 (1978), pp. 339–344.Google Scholar
[3]Belnap, N. D., How a computer should think, Contemporary aspects of philosophy (Ryle, G., editor), Oriel Press, Boston, 1976, pp. 30–56.Google Scholar
[4]Belnap, N. D., A useful four-valued logic, Modern uses of multiple-valued logic (Dunn, J. M. and Epstein, G., editors), Reidel, Dordrecht-Boston, 1977, pp. 8–37.Google Scholar
[5]Białinicki-Birula, A. and Rasiowa, H., On the representation of quasi-Boolean algebras, Bulletin de l'Académie Polonaise des Sciences, Classe III, vol. 5 (1957), pp. 259–261.Google Scholar
[6]Blok, W J. and Pigozzi, D., Abstract algebraic logic and the deduction theorem, The Bulletin of Symbolic Logic, to appear.Google Scholar
[7]Blok, W J., Protoalgebraic logics, Stadia Logica, vol. 45 (1986), pp. 337–369.CrossRefGoogle Scholar
[8]Blok, W J., Algebraizable logics, Memoirs of the American Mathematical Society, vol. 396 (1989), A.M.S., Providence.Google Scholar
[9]Blok, W J., The deduction theorem in algebraic logic, manuscript, 1989.Google Scholar
[10]Blok, W J., Local deduction theorems in algebraic logic, Algebraic logic (Andréka, H., Monk, J. D., and Németi, I., editors), Colloquia Mathematica Societatis János Bolyai, vol. 54, North-Holland, Amsterdam, 1991, pp. 75–109.Google Scholar
[11]Brown, D. J. and Suszko, R., Abstract logics, Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 102 (1973), pp. 9–42.Google Scholar
[ 12]Czelakowski, J., Protoalgebraic logics, forthcoming.Google Scholar
[13]Czelakowski, J., Reduced products of logical matrices, Studio Logica, vol. 39 (1980), pp. 19–43.CrossRefGoogle Scholar
[14]Czelakowski, J., Equivalential logics, I, II, Studio Logica, vol. 40 (1981), pp. 227-236 and 355–372.CrossRefGoogle Scholar
[15]Figallo, A., On the congruences in four-valued modal algebras, Portugaliae Mathematica, vol. 49 (1992), pp. 249–261.Google Scholar
[16]Figallo, A. and Landini, P., On generalized I-algebras and 4-valued modal algebras, Reports on Mathematical Logic, vol. 29 (1995), pp. 3–18.Google Scholar
[17]Figallo, A. and Ziliani, A., Symmetric tetra-valued modal algebras, Notas de la Sociedad de Matematica de Chile, vol. 10 (1991), no. 1, pp. 133–141.Google Scholar
[18]Fitting, M., Bilattices and the theory of truth, Journal of Philosophical Logic, vol. 18 (1989), pp. 225–256.CrossRefGoogle Scholar
[19]Fitting, M., Bilattices and the semantics of logic programming, Journal of Logic Programming, vol. 11 (1991), pp. 91–116.CrossRefGoogle Scholar
[20]Font, J. M., On the Leibniz congruences, Algebraic methods in logic and in computer science (Rauszer, C., editor), Banach Center Publications, vol. 28, Polish Academy of Sciences, Warszawa, 1993, pp. 17–36.Google Scholar
[21]Font, J. M., Belnap's four-valued logic and De Morgan lattices, Logic Journal of the I.G.P.L., vol. 5 (1997), no. 3, pp. 413–440.Google Scholar
[22]Font, J. M. and Jansana, R., A general algebraic semantics for sentential logics, Lecture Notes in Logic, vol. 7, Springer-Verlag, 1996.CrossRefGoogle Scholar
[23]Font, J. M. and Rius, M., A four-valued modal logic arising from Monteiro's last algebras, Proceedings of the 20th international symposium on multiple-valued logic (Charlotte), The IEEE Computer Society Press, 1990, pp. 85–92.CrossRefGoogle Scholar
[24]Font, J. M. and Verdú, V., Abstract characterization of a four-valued logic, Proceedings of the 18th international symposium on multiple-valued logic (de Mallorca, Palma), The IEEE Computer Society Press, 1988, pp. 389–396.Google Scholar
[25]Font, J. M., A first approach to abstract modal logics, This Journal, vol. 54 (1989), no. 3, pp. 1042–1062.Google Scholar
[26]Font, J. M., Algebraic logic for classical conjunction and disjunction, Studio Logica, vol. 50 (1991), pp. 391–419.CrossRefGoogle Scholar
[27]Ginsberg, M. L., Multivalued logics: A uniform approach to inference in artificial intelligence, Computational Intelligence, vol. 4 (1988), pp. 265–316.CrossRefGoogle Scholar
[28]Loureiro, I., Álgebras modais tetravalentes, Ph. D. Thesis, Faculdade de Ciências de Lisboa, 1983.Google Scholar
[29]Loureiro, I., Finitely generated free tetravalent modal algebras, Discrete Mathematics, vol. 46 (1983), pp. 41–48.CrossRefGoogle Scholar
[30]Loureiro, I., Prime spectrum of a tetravalent modal algebra, Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 389–394.CrossRefGoogle Scholar
[31]Loureiro, I., Principal congruences of tetravalent modal algebras, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 76–80.CrossRefGoogle Scholar
[32]Monteiro, A., Algebras De Morgan (Curso de Algebra de la Lógica III), unpublished manuscript, 74 pp., Bahía Blanca (Argentina), 1966.Google Scholar
[33]Monteiro, L., Axiomes indépendants pour les algèbres de Łukasiewicz trivalentes, Bulletin de la Societé des Sciences Mathématiques et Physiques de la R. P. Roumanie, Nouvelle Série, vol. 7 (1963), pp. 199–202.Google Scholar
[34]Monteiro, L., Les algèbres de Heyting et de Łukasiewicz trivalentes, Notre Dame Journal of Formal Logic, vol. 11 (1970). pp. 453–466.CrossRefGoogle Scholar
[35]Nowak, M., Logics preserving degrees of truth, Studio Lógica, vol. 49 (1990), pp. 483–499.CrossRefGoogle Scholar
[36]Gomes, A. Pereira, O regresso de António Monteiro a Portugal de 1977 a 1979, Portugaliae Mathematica, vol. 39 (1980), pp. xxxiii–xxxviii.Google Scholar
[37]Verdú, V., Logics projectively generated from [M] = (F4, [{1}]) by a set of homomorphisms, Zeitschrift für Mathematische Logik und Grundlagen des Mathematik, vol. 33 (1987), pp. 235–241.CrossRefGoogle Scholar
[38]Visser, A., Four-valued semantics and the liar, Journal of Philosophical Logic, vol. 13 (1984), pp. 181–212.CrossRefGoogle Scholar
[39]Wójcicki, R., Matrix approach in the methodology of sentential calculi, Studia Logica, vol. 32 (1973), pp. 7–37.CrossRefGoogle Scholar
[40]Wójcicki, R., Theory of logical calculi. Basic theory of consequence operations, Synthese Library, vol. 199, Reidel, Dordrecht, 1988.CrossRefGoogle Scholar