Logic may be simple. Logic, congruence and algebra

Jean-Yves Béziau

DOI: http://dx.doi.org/10.12775/LLP.1997.009

Abstract


This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such notions as semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra).

Keywords


algebraic logic; paraconsistent logic; universal logic

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References


Andréka, H., T. Gergely and I. Németi, “On universal algebraic constructions of logic”, Studia Logica, 36 (1977), 9–47.

Barnes, D. W, and J. M. Mack, An Algebraic Introduction to Mathematical Logic, Springer-Verlag, Berlin, 1975.

de Barros, C., N. C. A. da Costa, and J. M. Abe, Tópicos de Teoria dos Sistemas Ordenados. Vol. II. Sistemas de Curry, Institute for Advanced Studies, São Paulo, 1995.

Béziau, J.-Y., “Nouveau regard et nouveaux r´ esultats sur la logique paraconsistante C1”, Logique et Analyse, 36 (1993), 45–58.

Béziau, J. Y., “Du Pont’s paradox and the problem of intensional logic”, in Proceedings of the 7th International Colloquium Logica’93, V. Svodoba and P. Kolar (eds.), Philosophia, Prague, 1994, p. 62–65.

Béziau, J. Y., “Universal Logic”, in Proceedings of the 8th International Colloquium Logica’94, T. Childers and O. Majer (eds.), Philosophia, Prague, 1995, p. 73–93.

Béziau, J. Y., Recherches sur la logique universelle, PhD, University of Paris 7, Paris, 1995.

Birkhoff, G., “Universal algebra”, in Comptes Rendus du Premier Congres Canadien de Math´ ematiques, University of Toronto Press, Toronto, 1946, p. 310–326.

Birkhoff, G., “The rise of modern algebra to 1936”, in Men and Institutions in American Mathematics, Graduate Studies, Texas Technical Studies, 13 (1976), p. 41–63.

Birkhoff, G., “The rise of modern algebra, 1936 to 1950”, in Men and Institutions in American Mathematics, Graduate Studies, Texas Technical Studies, 13 (1976), p. 65–85.

Birkhoff, G., “Universal algebra”, in Selected Papers on Algebra and Topology by Garrett Birkhoff, G.-C. Rota and J. S. Oliveira (eds.), Birkhäuser , Basel, 1987.

Blok, W. J., and D. Pigozzi, Alegbraizable Logics, Memoirs of the American MathematicaL Society, vol. 396 (1989).

Blok, W. J., and D.Pigozzi, “Introduction”, Studia Logica, 60 (1991), 365–374.

Bloom, W. J., “Roman Suszko: a reminiscence”, Studia Logica, 63 (1984), p. 313.

Bourbaki, N., “The architecture of mathematics”, American Mathematical Monthly 57 (1950), 221–232.

Bourbaki, N. Theory of Sets, Addison-Wesley, 1968.

Cartan, H., “Sur le fondement logique des math´ ematiques”, Revue Scientifique, 1 (1943), 3–11.

Cohn, P. M., Universal Algebra, Harper and Row, London, 1965.

da Costa, N. C. A., “Calculs propositionnels pour les systémes formles inconsistants”, Comptes Rendus de l’Académie des Sciences de Paris, 257 (1963), 3790–3793.

da Costa, N. C. A., Algebras de Curry, University of São Paulo, 1966.

da Costa, N. C. A., “0p´ erations non monotones dans les treillis”, Comptes Rendijes de l’Académie des Sciences de Paris, 263 (1966), 429-432.

da Costa, N. C. A., and E. H. Alves, “A semantical analysis of the calculi Cn”, Notre Dame Journal of FormaL Logic, 18 (1977), 621–630.

da Costa, N. C. A., and J.-Y. Béziau, “La théorie de la valuation en question”, in Proceedings of the IX Latin American Symposium on Mathematical Logic (Part 2), Universidad Nacional del Sur, Bahia Blanca, 1994, p. 95–104.

da Costa, N. C. A., and J.-Y. Béziau, “Théorie de la valuation”, Logique et Analyse, (1994), 145–146.

da Costa, N. C. A., J.-Y. Béziau and O. A. S. Bueno, “Aspects of paraconsistent logic”, Bulletin of the IGPL, 4 (1995), 597–614.

da Costa, N. C. A., J.-Y. Béziau and O. A. S. Bueno, “Malinowski and Suszko on many-valued logics: on the reduction of many-valuedness to two-valuedness”, Modern Logic, 6 (1996), 272–299.

da Costa, N. C. A., and M. Guillaume, “Sur les calculs Cn”, Anais da Academia Brasileira de Ciências, 36 (1964), 379–382.

Curry, H. B., “On the definition of substitution, replacement and allied notions in an abstract formal system”, Revue Philosophique de Louvain, 50 (1952), 251–269.

Curry, H. B., Letçons de Logique Algébrique, Gauthier-Villars/Nauwelaerts, Paris/Louvain, 1952.

Dieudonné, J., “The work of Bourbaki during the last thirty years”, Notice of the American Mathematical Society, 29 (1982), 618–623.

Eytan, M., “Tableaux de Smullyan, ensembles de Hintikka et toutça: un point de vue algébrique”, Mathéematiques et Sciences Humaines, 48 (1974), 21–27.

Font, J. M., and R. Jansana, “A general algebraic semantics for sentential logics”, preprint, Barcelonna, 1993.

Font, J. M., and R. Jansana, “Sobre álgebras de Tarski-Lindenbaum”, preprint, Barcelonna, 1994.

Font, J. M., and R. Jansana, “On the sentential logics associated with strongly nice and semi-nice general logic”, to appear in the Bulletin of the IGPL.

Girard, J.-Y., “On the unity of logic”, Annals of Pure and Applied Logic, 59 (1993), 201–217.

Glivenko, V., Th´ eorie G´ en´ erale des Structures, Hermann, Paris, 1938.

Gödel, K., “Eine Eigenschaft der Realisierung des Aussagenkalküls”, Ergebnisse Eines Mathematischen Kolloquiums, 3 (1932), 20–21.

Henkin, L., La Structure Algébrique des Théories Mathématiques, Gauthier-Villars/Nauwelaerts, Paris/Louvain, 1965.

Johnstone, P. T., Stone Spaces, CUP, Cambridge, 1982.

Kleene, S. C., Introduction to Metamathematics, Amsterdam, 1952.

Lewin, R. A., I. F. Mikenberg and M. G. Schwarze, “C1 is not algebraizable”, Notre Dame Journal of Formal Logic, 32 (1991), 609–611.

Łoś, J., “An algebraic proof of the completeness for the two-valued prepositional calculus”, Académie Polonaise des Sciences, 12 (1951), 236–240.

Łoś, J., and R. Suszko, “Remarks on sentential logics”, Indigationes Mathematicae, 20 (1958), 177–183.

MacLane, S., Abgekürzte Beweise im Logikkalkül, PhD (Inaugural-dissertation), Göttingen, 1934.

MacLane, S., “History of abstract algebra: origin, rise and decline of a movement”, in Men and Institutions in American Mathematics, Graduate Studies, Texas Technical Studies, 13 (1981), p. 3–35.

Mortensen, C., “Every quotient algebra for C1 is trivial”, Notre Dame Journal of Formal Logic, 21 (1980), 694–700.

Mortensen, C., “Paraconsistency and C1”, in Paraconsistent Logic: Essays on the Inconsistent, G. Priest, R. Routley and J. Norman (eds.), Philosophia, Münich, 1989, p. 289–305.

Ore, O., “On the foundation of abstract algebra”, Annals of Mathematics, 36 (1935), 406–437.

Papert, S., “Structures et cat´ egories”,in Logique et Connaissance Scientifique, J. Piaget (ed.), Gallimard, p. 487–511.

Porte, J., Recherches sur la Théorie Générale des Systèmes Fornels et sur les Systèmes Connectifs, Gauthier-Villars/Nauwelaerts, Paris/Louvain, 1965.

Quine, W. V. O., Philosophy of Logic, Englewood Cliffs, Prentice-Hall, 1970.

Scott, D. S., “Completeness and axiomatizability in many-valued logic”, in Proceedings of the Tarski Symposium, L. Henkin (ed.), American Mathematical Society, Providence, 1974, p. 411–435.

Shoesmith, D. J., and T. J. Smiley, Multiple-Conclusion Logic, CUP, Cambridge, 1978.

Surma, S. J., “On the origin and subsequent applications of the concept of Lindenbaum algebra”, in Logic, Methodology and Philosophy of Science VI, L. J. Cohen, J. Łoś, H. Pfeiffer and K.-D. Podewski (eds.), PWN/North-Holland, Warsaw/Amsterdam, 1982.

Surma, S. J., “Alternatives to the consequence-theoretic approach to metalogic”, in Proceedings of the IX Latin American Symposium on Mathematical Logic (Part 2), Universidad Nacional del Sur, Bahia Blanca, 1994, p. 1–30.

Suszko, R., “Remarks on Łukasiewicz’s three-valued logic”, Bulletin of the Section of Logic, 4 (1975).

Suszko, R., “Abolition of the Fregean axiom”, in Logic Colloquium, R. Parikh (ed.), Springer-Verlag, Berlin, 1975, p. 169–239.

Suszko, R., “The Fregean axiom and Polish mathematical logic in the 1920s’”, Studia Logica, 36 (1977), 377–380.

Sylvan, R., “Variations on da Costa C systems and dual-intuitionistic logics – I. Analyses of Cω and CCω”, Studia Logica, 69 (1990), 47–65.

Tarski, A., “Grundzüge des Systemankalküls, Erster Teil”, Fundamenta Mathematicae, 25 (1935), 503–526.

Urbas, I., “Dual-intuitionistic logic”, Notre Dame Journal of Formal Logic, 37 (1996), 440–451.

Whitehead, A. N., A Treatise on Universal Algebra, Cambridge University Press, Cambridge, 1898.

Wójcicki, R., Theory of Logical Calculi, Kluwer, Dordrecht, 1988.

Zygmunt, J., An Essay in Matrix Semantics for Consequence Relations, Wydawnictwo Uniwersytetu Wrocławskiego, Wrocław, 1984.








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