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Beyond Rasiowa's Algebraic Approach to Non-classical Logics

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Abstract

This paper reviews the impact of Rasiowa's well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generalized allows us to survey some issues in the development of the field in the last twenty to thirty years. The last part of the paper reviews some recent lines of research that in some way transcend Rasiowa's approach. I hope in this way to give the reader a general view of Rasiowa's key position in the evolution of Algebraic Logic during the twentieth century.

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References

  1. BABYONYSHEV, S., ‘Fully Fregean logics’, Reports on Mathematical Logic, 37 (2003), 59–78.

  2. BARTOL, W., E. ORŁOWSKA, AND A. SKOWRON, ‘Helena Rasiowa, 1917–1994’, Modern logic, 5 (1995), 231–247.

    Google Scholar 

  3. BELARDINELLI, F., P. JIPSEN, AND H. ONO, ‘Algebraic aspects of cut elimination’, Studia Logica, 77 (2004), 209–240.

    Article  Google Scholar 

  4. BETH, E. W., ‘Review of [68]’, Mathematical Reviews, MR22:1510.

  5. BLOK, W., AND B. JÓNSSON, ‘Equivalence of consequence operations’, Studia Logica (Special Issue in memory of Willem Blok), (2006). To appear.

  6. BLOK, W., AND D. PIGOZZI, ‘Protoalgebraic logics’, Studia Logica, 45 (1986), 337–369.

    Google Scholar 

  7. BLOK, W., AND D. PIGOZZI, Algebraizable logics, vol. 396 of Mem. Amer. Math. Soc., A. M. S., Providence, 1989.

  8. BLOK, W., AND D. PIGOZZI, ‘Local deduction theorems in algebraic logic’, in H. Andréka, J. D. Monk, and I. Németi, (eds.), Algebraic Logic, vol. 54 of Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 1991, pp. 75–109.

    Google Scholar 

  9. BLOK, W., AND D. PIGOZZI, ‘Algebraic semantics for universal Horn logic without equality’, in A. Romanowska, and J. D. H. Smith, (eds.), Universal Algebra and Quasigroup Theory, Heldermann, Berlin, 1992, pp. 1–56.

    Google Scholar 

  10. BLOK, W., AND J. RAFTERY, ‘Assertionally equivalent quasivarieties’, 2004. Manuscript.

  11. BLOK, W., AND J. REBAGLIATO, ‘Algebraic semantics for deductive systems’, Studia Logica (Special Issue on Abstract Algebraic Logic, Part II), 74 (2003), 153–180.

    Google Scholar 

  12. BOU, F., Implicación estricta y lógicas subintuicionistas, Master Thesis, University of Barcelona, 2001.

  13. BROWN, D. J., AND R. SUSZKO, ‘Abstract logics’, Dissertationes Math. (Rozprawy Mat.), 102 (1973), 9–42.

  14. CELANI, S., AND R. JANSANA, ‘A closer look at some subintuitionistic logics’, Notre Dame Journal of Formal Logic, 42 (2001), 225–255.

    Google Scholar 

  15. CINTULA, P., ‘Weakly implicative (fuzzy) logics, I’, Archive for Mathematical Logic. To appear.

  16. CZELAKOWSKI, J., ‘Reduced products of logical matrices’, Studia Logica, 39 (1980), 19–43.

    Article  Google Scholar 

  17. CZELAKOWSKI, J., ‘Equivalential logics, I, II’, Studia Logica, 40 (1981), 227–236 and 355–372.

    Google Scholar 

  18. CZELAKOWSKI, J., Protoalgebraic logics, vol. 10 of Trends in Logic - Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2001.

  19. CZELAKOWSKI, J., ‘The Suszko operator. Part I’, Studia Logica (Special Issue on Abstract Algebraic Logic, Part II), 74 (2003), 181–231.

    Google Scholar 

  20. CZELAKOWSKI, J., AND R. JANSANA, ‘Weakly algebraizable logics’, The Journal of Symbolic Logic, 65 (2000), 641–668.

    Google Scholar 

  21. CZELAKOWSKI, J., AND D. PIGOZZI, ‘Fregean logics’, Annals of Pure and Applied Logic, 127 (2004), 17–76.

    Article  Google Scholar 

  22. CZELAKOWSKI, J., AND D. PIGOZZI, ‘Fregean logics with the multiterm deduction theorem and their algebraization’, Studia Logica, 78 (2004), 171–212.

    Article  Google Scholar 

  23. DUNN, J. M., AND G. M. HARDEGREE, Algebraic methods in philosophical logic, vol. 41 of Oxford Logic Guides, Oxford Science Publications, Oxford, 2001.

  24. FEFERMAN, S., ‘Review of [66]’, The Journal of Symbolic Logic XVII:72a, 1952.

    Google Scholar 

  25. FITTING, M., ‘A tribute to professor Helena Rasiowa’, in E. Orłowska, (ed.), Logic at work, Physica Verlag, 1998, pp. v–viii.

  26. FONT, J. M., ‘Belnap's four-valued logic and De Morgan lattices’, Logic Journal of the IGPL, 5 (1997), 413–440.

    Google Scholar 

  27. FONT, J. M., ‘On special implicativefilters’, Mathematical Logic Quarterly, 45 (1999), 117–126.

    Google Scholar 

  28. FONT, J. M., ‘An abstract algebraic logic view of some multiple-valued logics’, in M. Fitting, and E. Orlowska, (eds.), Beyond two: Theory and applications of multiplevalued logic, vol. 114 of Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg-Berlin-New York, 2003, pp. 25–58.

    Google Scholar 

  29. FONT, J. M., ‘Generalized matrices in abstract algebraic logic’, in V. F. Hendriks, and J. Malinowski, (eds.), Trends in Logic. 50 years of Studia Logica, vol. 21 of Trends in Logic - Studia Logica Library, Kluwer, Dordrecht, 2003, pp. 57–86.

    Google Scholar 

  30. FONT, J. M., A. GIL, A. TORRENS, AND V. VERDÚ, ‘On the infinite-valued Lukasiewicz logic that preserves degrees of truth’, Archive for Mathematical Logic. To appear.

  31. FONT, J. M., F. GUZMÁN, AND V. VERDÚ, ‘Characterization of the reduced matrices for the {∧ ∨}-fragment of classical logic’, Bulletin of the Section of Logic, 20 (1991), 124–128.

    Google Scholar 

  32. FONT, J. M., AND R. JANSANA, A general algebraic semantics for sentential logics, vol. 7 of Lecture Notes in Logic, Springer-Verlag, 1996. 135 pp. Presently distributed by the Association for Symbolic Logic.

  33. FONT, J. M., R. JANSANA, AND D. PIGOZZI, ‘Fully adequate Gentzen systems and the deduction theorem’, Reports on Mathematical Logic, 35 (2001), 115–165.

    Google Scholar 

  34. FONT, J. M., R. JANSANA, AND D. PIGOZZI, ‘A survey of abstract algebraic logic’, Studia Logica (Special Issue on Abstract Algebraic Logic, Part II), 74 (2003), 13–97.

    Google Scholar 

  35. FONT, J. M., R. JANSANA, AND D. PIGOZZI, ‘On the closure properties of the class of full models of a deductive system’, Studia Logica (Special Issue in memory of Willem Blok), (2006). To appear.

  36. FONT, J. M., AND G. RODRÍGUEZ, ‘Algebraic study of two deductive systems of relevance logic’, Notre Dame Journal of Formal Logic, 35 (1994), 369–397.

    Google Scholar 

  37. FONT, J. M., AND V. VERDÚ, ‘Algebraic logic for classical conjunction and disjunction’, Studia Logica (Special Issue on Algebraic Logic), 50 (1991), 391–419.

    Google Scholar 

  38. FONT, J. M., Estudi algebraic d’algunes lògiques intuïcionistes modals [An algebraic study of some intuitionistic modal logics], Ph. D. Dissertation, University of Barcelona, 1981.

  39. GALATOS, N., AND C. TSINAKIS, ‘Equivalence of consequence relations: an ordertheoretic and categorial perspective’, Manuscript in preparation.

  40. HALMOS, P. R., Algebraic logic, Chelsea Publ. Co., New York, 1962.

    Google Scholar 

  41. HALMOS, P. R., I want to be a mathematician. An automathography, Springer-Verlag, Heidelberg and New York, 1985.

    Google Scholar 

  42. HENKIN, L., ‘The completeness of the first order functional calculus’, The Journal of Symbolic Logic, 14 (1949), 159–166.

    Google Scholar 

  43. HENKIN, L., J. D. MONK, AND A. TARSKI, Cylindric algebras, North-Holland, Amsterdam, 1971 (part I) and 1985 (part II).

  44. HERRMANN, B., Equivalential logics and definability of truth, Ph. D. Dissertation, Freie Universität Berlin, 1993, 61 pp.

  45. JANSANA, R., ‘Selfextensional logics in abstract algebraic logic: a brief survey’, in J.-Y. Béziau, A. Costa-Leite, and A. Facchini, (eds.), Aspects of Universal Logic, no. 17 in Cahiers de Logique, Centre de Recerches Sémiologiques, Université de Neuchâtel, Neuchâtel, 2004, pp. 32–65.

  46. Jansana, R., ‘Selfextensional logics with implication’, in J.-Y. Béziau, (ed.), Logica Universalis, Birkhäuser Verlag, Basel, 2005, pp. 65–88.

    Google Scholar 

  47. Jansana, R., ‘Selfextensional logics with a conjunction’, Studia Logica. To appear.

  48. JANSANA, R., AND A. PALMIGIANO, ‘Referential semantics: duality and applications’, Reports on Mathematical Logic (Special Issue in memory of Willem Blok), (2006). To appear.

  49. KREISEL, G., ‘Review of [69]’, Mathematical Reviews, MR29:1149.

  50. LOŚ, J., O matrycach logicznych, vol. 19 of Prace Wrocławskiego Towarzystwa Naukowege, Ser. B, University of Wrocław, 1949.

  51. LOŚ, J., AND R. SUSZKO, ‘Remarks on sentential logics’, Indag. Math., 20 (1958), 177–183.

    Google Scholar 

  52. MAEHARA, S., ‘Lattice-valued representation of the cut-elimination theorem’, Tsukuba Journal of Mathematics, 15 (1991), 509–521.

    Google Scholar 

  53. MAREK, W., ‘The foundations of mathematics in Poland after World War II’, in R. Gandy, and M. Hyland, (eds.), Logic Colloquium 76, North-Holland, Amsterdam, 1977, pp. 129–138.

    Google Scholar 

  54. MAREK, W., ‘Mathematical logic in Warsaw in the 60's and 70's, or the interaction of logic and life’, in Conference Trends in Logic III, Polish-Japanese Institute of Information Technology, Publishing House, 2005, pp. 2–17.

  55. MERTON, R. K., On the shoulders of giants, University of Chicago Press, 1993.

  56. MURAWSKI, R., ‘Remarks on A. Mostowski's philosophy of mathematics’, 2005. Talk given at the conference Trends in Logic III (Warsaw/Ruciane-Nida, September 2005).

  57. NOWAK, M., ‘Logics preserving degrees of truth’, Studia Logica, 49 (1990), 483–499.

    Article  Google Scholar 

  58. ONO, H., ‘Substructural logics and residuated lattices - an introduction’, in V. F. Hendriks, and J. Malinowski, (eds.), Trends in Logic. 50 years of Studia Logica, vol. 21 of Trends in Logic - Studia Logica Library, Kluwer, Dordrecht, 2003, pp. 193–228.

    Google Scholar 

  59. ORŁOWSKA, E., AND A. SKOWRON, ‘Helena Rasiowa’, Studia Logica, 54 (1995), 1–2.

    Google Scholar 

  60. PIGOZZI, D., ‘Abstract algebraic logic: past, present and future. A personal view’, in J. M. Font, R. Jansana, and D. Pigozzi, (eds.), Workshop on Abstract Algebraic Logic, vol. 10 of Quaderns, Centre de Recerca Matemàtica, Bellaterra, 1998, pp. 122–138.

  61. PRUCNAL, T., AND A. WROŃSKI, ‘An algebraic characterization of the notion of structural completeness’, Bulletin of the Section of Logic, 3 (1974), 30–33.

    Google Scholar 

  62. RAFTERY, J., ‘Correspondences between Gentzen and Hilbert systems’, 2005. Submitted manuscript.

  63. RAFTERY, J., ‘The equational definability of truth predicates’, Reports on Mathematical Logic (Special Issue in memory of Willem Blok), (2006). To appear.

  64. RASIOWA, H., ‘On the ε-theorems’, Fundamenta Mathematicae, 43 (1956), 156–165.

    Google Scholar 

  65. RASIOWA, H., An algebraic approach to non-classical logics, vol. 78 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1974.

  66. RASIOWA, H., AND R. SIKORSKI, ‘A proof of the completeness theorem of Gödel’, Fundamenta Mathematicae, 37 (1950), 193–200.

    Google Scholar 

  67. RASIOWA, H., AND R. SIKORSKI, ‘Algebraic treatment of the notion of satisfability’, Fundamenta Mathematicae, 40 (1953), 62–95.

    Google Scholar 

  68. RASIOWA, H., AND R. SIKORSKI, ‘On the Gentzen theorem’, Fundamenta Mathematicae, 48 (1959), 57–69.

    Google Scholar 

  69. RASIOWA, H., AND R. SIKORSKI, The mathematics of metamathematics, P. W. N., Warszawa, 1963. (3rd edition, 1970).

  70. REBAGLIATO, J., AND V. VERDÚ, ‘On the algebraization of some Gentzen systems’, Fundamenta Informaticae (Special Issue on Algebraic Logic and its Applications), 18 (1993), 319–338.

    Google Scholar 

  71. REBAGLIATO, J., AND V. VERDÚ, ‘Algebraizable Gentzen systems and the deduction theorem for Gentzen systems’, Mathematics Preprint Series 175, University of Barcelona, 1995.

  72. ROBINSON, A., ‘Review of [64]’, Mathematical Reviews, MR18:711g.

  73. TARSKI, A., ‘Über einige fundamentale Begrifie der Metamathematik’, C. R. Soc. Sci. Lettr. Varsovie, Cl. III, 23 (1930), 22–29.

  74. TORRENS, A., ‘Algebraic models of multidimensional deductive systems’, 1991. Manuscript.

  75. VERDÚ, V., Contribució a l’estudi de certs tipus de lògiques abstractes [Contribution to the study of some types of abstract logics], Ph. D. Dissertation, University of Barcelona, 1978.

  76. WÓJCICKI, R., ‘Logical matrices strongly adequate for structural sentential calculi’, Bulletin de l’Académie Polonaise des Sciences, Classe III, XVII (1969), 333–335.

    Google Scholar 

  77. WÓJCICKI, R., ‘Matrix approach in the methodology of sentential calculi’, Studia Logica, 32 (1973), 7–37.

    Google Scholar 

  78. WÓJCICKI, R., ‘Referential matrix semantics for propositional calculi’, Bulletin of the Section of Logic, 8 (1979), 170–176.

    Google Scholar 

  79. WÓJCICKI, R., Lectures on propositional calculi, Ossolineum, Wrocław, 1984.

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  1. Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Gran Via 585, E-08007, Barcelona, Spain

    Josep Maria Font

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Correspondence to Josep Maria Font.

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This paper is an extended version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.

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Font, J.M. Beyond Rasiowa's Algebraic Approach to Non-classical Logics. Stud Logica 82, 179–209 (2006). https://doi.org/10.1007/s11225-006-7196-9

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