Matrices and Algebraizability

Chapter
First Online:
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 40)

Abstract

This chapter deals with matrices and algebraizability and their consequences, investigating in particular, the question of characterizability by finite matrices, as well as the algebraizability of (extensions of) mbC. Some negative results, in the style of the well-known Dugundji’s theorem for modal logics, are proved for several extensions of mbC.

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References

  1. 1.
    Blok, Willem J., and Don Pigozzi. 1989. Algebraizable Logics, vol. 77(396) of Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI, USA.Google Scholar
  2. 2.
    Anellis, Irving. 2004. The genesis of the truth-table device. Russell: The Journal of Bertrand Russell Studies 24: 55–70.Google Scholar
  3. 3.
    Wójcicki, Ryszard. 1984. Lectures on propositional calculi. Ossolineum, Wroclaw, Poland. http://www.ifispan.waw.pl/studialogica/wojcicki/papers.html.
  4. 4.
    Dugundji, James. 1940. Note on a property of matrices for Lewis and Langford’s calculi of propositions. The Journal of Symbolic Logic 5(4): 150–151.CrossRefGoogle Scholar
  5. 5.
    Chagrov, Alexander V., and Michael Zakharyaschev. 1997. Modal logic, vol. 35. Oxford Logic Guides Oxford: Oxford University Press.Google Scholar
  6. 6.
    Coniglio, Marcelo E., and Newton M. Peron. 2014. Dugundji’s theorem revisited. Logica Universalis 8(3–4): 407–422. doi: 10.1007/s11787-014-0106-4.CrossRefGoogle Scholar
  7. 7.
    Carnielli, Walter A., Marcelo E. Coniglio, and João Marcos. 2007. Logics of Formal Inconsistency. In Handbook of Philosophical Logic (2nd. edition), eds. Dov M. Gabbay and Franz Guenthner, vol. 14, 1–93. Springer. doi: 10.1007/978-1-4020-6324-4_1.
  8. 8.
    Boole, George. 1847. The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning. Cambridge: Macmillan, Barclay, & Macmillan. Reprinted by Basil Blackwell, Oxford.Google Scholar
  9. 9.
    Boole, George. 1854. An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. London: Macmillan. Reprinted by Dover, 1958.Google Scholar
  10. 10.
    De Morgan, August. 1847. Formal Logic: Or, the Calculus of Inference, Necessary and Probable. London: Taylor and Walton. Reprinted by The Open Court Company, London, 1926.Google Scholar
  11. 11.
    Jevons, William S. 1864. Pure logic: Or, the logic of quality apart from quantity. London: E. Stanford. https://archive.org/details/purelogicorlogi00jevogoog.
  12. 12.
    Peirce, Charles S. 1870. Description of a notation for the logic of relatives, resulting from an amplification of the conceptions of Boole’s calculus of logic. Memoirs of the American Academy 9: 317–378. Reprinted in vol. III of [Hartshorne, Charles, and Paul Weiss, eds. Collected Papers of Charles Sanders Peirce, Vols. 1–6. Cambridge: Harvard University Press, 1931–1935].Google Scholar
  13. 13.
    Peirce, Charles S. 1880. On the algebra of logic. Chapter I: Syllogistic. Chapter II: The logic of non-relative terms. Chapter III: The logic of relatives. American Journal of Mathematics 3: 15–57. Reprinted in vol. III of [Hartshorne, Charles, and Paul Weiss, eds. Collected Papers of Charles Sanders Peirce, Vols. 1–6. Cambridge: Harvard University Press, 1931–1935].Google Scholar
  14. 14.
    Peirce, Charles S. 1885. On the algebra of logic; a contribution to the philosophy of notation. American Journal of Mathematics 7(2): 180–202. Reprinted in vol. III of [Hartshorne, Charles, and Paul Weiss, eds. Collected Papers of Charles Sanders Peirce, Vols. 1–6. Cambridge: Harvard University Press, 1931–1935].Google Scholar
  15. 15.
    Schröder, Ernst. 1890–1910. Vorlesungen über die Algebra der Logik, Vols. I–III (in German). Leipzig: B.G. Teubner. Reprints: Chelsea, 1966; Thoemmes Press, 2000.Google Scholar
  16. 16.
    Burris, Stanley. 2013. The algebra of logic tradition. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. Summer 2013 edition.Google Scholar
  17. 17.
    Czelakowski, Janusz. 2001. Protoalgebraic Logics. Trends in Logic Series, vol. 10. Dordrecht: Kluwer Academic Publishers.Google Scholar
  18. 18.
    Font, Josep Maria, Ramón Jansana, and Don Pigozzi. 2003. A survey of abstract algebraic logic. Studia Logica 74(1–2): 13–97.Google Scholar
  19. 19.
    Font, Josep Maria, and Ramón Jansana. 2009. A General Algebraic Semantics for Sentential Logics. Vol. 7 of Lecture Notes in Logic, 2nd ed. Ithaca, NY, USA: Association for Symbolic Logic.Google Scholar
  20. 20.
    Font, Josep Maria. 2016. Abstract algebraic logic: An introductory textbook. Vol. 60 of Mathematical Logic and Foundations Series. London: College Publications.Google Scholar
  21. 21.
    da Costa, Newton C.A., Jean-Yves Béziau, and Otávio Bueno. 1995. Aspects of paraconsistent logic. Bulletin of the IGPL 3(4): 597–614.Google Scholar
  22. 22.
    Carnielli, Walter A., and João Marcos. A taxonomy of C-systems. In [Carnielli, Walter A., Marcelo E. Coniglio, and Itala M. L. D’Ottaviano, eds. 2002. Paraconsistency: The Logical Way to the Inconsistent. Proceedings of the 2nd World Congress on Paraconsistency (WCP 2000), Vol. 228 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York], pp. 1–94.Google Scholar
  23. 23.
    Halldén, Sören. 1949. The logic of Nonsense. Uppsala: Uppsala Universitets Årsskrift.Google Scholar
  24. 24.
    Jaśkowski, Stanisław. 1948. Rachunek zdań dla systemów dedukcyjnych sprzecznych (in Polish). Studia Societatis Scientiarun Torunesis–Sectio A I(5): 57–77. Translated to English as “A propositional calculus for inconsistent deductive systems”. Logic and Logical Philosophy 7: 35–56, 1999. Proceedings of the Stanisław Jaśkowski’s Memorial Symposium, held in Toruń, Poland, July 1998.Google Scholar
  25. 25.
    Corbalán, María I. 2012. Conectivos de Restauração Local (Local Restoration Connectives, in Portuguese). Masters thesis, IFCH, State University of Campinas. http://www.bibliotecadigital.unicamp.br/document/?code=000863780&opt=4&lg=Den_US.
  26. 26.
    Bochvar, Dmitri A. 1938. Ob odnom trechzna čnom isčislenii i ego primenenii k analizu paradoksov klassiceskogo funkcional’nogo isčislenija (in Russian). Matématičeskij Sbornik, 46(2): 287–308. Translated to English by M. Bergmann 1981 “On a Three-valued Logical Calculus and Its Application to the Analysis of the Paradoxes of the Classical Extended Functional Calculus”. History and Philosophy of Logic 2: 87–112.Google Scholar
  27. 27.
    Segerberg, Krister. 1965. A contribution to nonsense-logics. Theoria 31(3): 199–217.CrossRefGoogle Scholar
  28. 28.
    D’Ottaviano, Itala M.L., Newton C.A. da Costa. 1970. Sur un problème de Jaśkowski (in French). Comptes Rendus de l’Académie de Sciences de Paris (A-B) 270: 1349–1353.Google Scholar
  29. 29.
    D’Ottaviano, Itala M.L. 1982. Sobre uma Teoria de Modelos Trivalente (On a three-valued model theory, in Portuguese). Ph.D. thesis, IMECC, State University of Campinas, Brazil.Google Scholar
  30. 30.
    D’Ottaviano, Itala M.L. 1985. The completeness and compactness of a three-valued first-order logic. Revista Colombiana de Matemáticas XIX(1–2): 77–94.Google Scholar
  31. 31.
    D’Ottaviano, Itala M.L. 1985. The model extension theorems for J3-theories. In Methods in Mathematical Logic. Proceedings of the 6th Latin American Symposium on Mathematical Logic held in Caracas, Venezuela, August 1–6, 1983, Lecture Notes in Mathematics, ed. Carlos A. Di Prisco, vol. 1130, 157–173. Berlin: Springer.Google Scholar
  32. 32.
    D’Ottaviano, Itala M.L. 1987. Definability and quantifier elimination for J3-theories. Studia Logica 46(1): 37–54.Google Scholar
  33. 33.
    Michael Dunn, J. 1979. A theorem in 3-valued model theory with connections to number theory, type theory, and relevant logic. Studia Logica 38(2): 149–169.Google Scholar
  34. 34.
    Carnielli, Walter A., João Marcos, and Sandra de Amo. 2000. Formal inconsistency and evolutionary databases. Logic and Logical Philosophy 8: 115–152.CrossRefGoogle Scholar
  35. 35.
    de Amo, Sandra, Walter A. Carnielli, and João Marcos. 2002. A logical framework for integrating inconsistent information in multiple databases. In Foundations of Information and Knowledge Systems. Proceedings of the Second International Symposium, FoIKS 2002, held in Salzau Castle, Germany, February 20-23, 2002, Lecture Notes in Computer Science, eds. Thomas Eiter and Klaus-Dieter Schewe, vol. 2284, 67–84. Berlin: Springer.Google Scholar
  36. 36.
    Batens, Diderik, and Kristof De Clercq. 2004. A rich paraconsistent extension of full positive logic. Logique et Analyse 185–188: 227–257.Google Scholar
  37. 37.
    Schütte, Kurt. 1960. Beweistheorie (in German). Berlin: Springer.Google Scholar
  38. 38.
    Batens, Diderik. 1980. Paraconsistent extensional propositional logics. Logique et Analyse 90–91: 195–234.Google Scholar
  39. 39.
    Silvestrini, Luiz H. 2011. Uma Nova Abordagem Para A Noção De Quase-Verdade (A new approach to the Notion of Quasi-Truth, in Portuguese). Ph.D. thesis, IFCH, State University of Campinas, Brazil. http://www.bibliotecadigital.unicamp.br/document/?code=000788964&opt=4&lg=en_US.
  40. 40.
    Coniglio, Marcelo E., and Luiz H. Silvestrini. 2014. An alternative approach for quasi-truth. Logic Journal of the IGPL 22(2): 387–410. doi: 10.1093/ljigpal/jzt026.CrossRefGoogle Scholar
  41. 41.
    Mikenberg, Irene, Newton C.A. da Costa, and Rolando Chuaqui. 1986. Pragmatic truth and approximation to truth. The Journal of Symbolic Logic 51(1): 201–221.Google Scholar
  42. 42.
    Löwe, Benedikt, and Sourav Tarafder. 2015. Generalized algebra-valued models of set theory. The Review of Symbolic Logic 8(1): 192–205. doi: 10.1017/S175502031400046X.CrossRefGoogle Scholar
  43. 43.
    Łukasiewicz, Jan. 1920. O logice trójwartościowej (in Polish). Ruch Filozoficzny 5: 170–171. Translated to English as “On three-valued logic”. In Jan Łukasiewicz Selected Works, ed. Ludwik Borkowski. North Holland, 87–88, 1990.Google Scholar
  44. 44.
    Tomova, Natalya. 2012. A lattice of implicative extensions of regular Kleene’s logics. Reports on Mathematical Logic 47: 173–182.Google Scholar
  45. 45.
    Blok, Willem J., and Don Pigozzi. 2001. Abstract algebraic logic and the deduction theorem. Preprint. http://www.math.iastate.edu/dpigozzi/papers/aaldedth.pdf.
  46. 46.
    da Costa, Newton C.A. 1974. On the theory of inconsistent formal systems (Lecture delivered at the first Latin-American Colloquium on Mathematical Logic, held at Santiago, Chile, July 1970). Notre Dame Journal of Formal Logic 15(4): 497–510.Google Scholar
  47. 47.
    Sette, Antonio M.A. 1973. On the propositional calculus P \(^{1}\). Mathematica Japonicae 18(13): 173–180.Google Scholar
  48. 48.
    Lewin, Renato A., Irene Mikenberg, and Maria G. Schwarze. 1990. Algebraization of paraconsistent logic P \(^{1}\). The Journal of Non-Classical Logic 7(1/2): 79–88. http://www.cle.unicamp.br/jancl/.
  49. 49.
    Coniglio, Marcelo E., Francesc Esteva, and Lluís Godo. 2016. Maximal logics in the lattice of degree preserving logics of Ł\(_n\). To appear.Google Scholar
  50. 50.
    Kleene, Stephen C. 1952. Introduction to Metamathematics. Amsterdam: North-Holland.Google Scholar
  51. 51.
    Asenjo, Florencio G. 1966. A calculus for antinomies. Notre Dame Journal of Formal Logic 16(1): 103–105.CrossRefGoogle Scholar
  52. 52.
    Priest, Graham. 1979. The logic of paradox. Journal of Philosophical Logic 8(1): 219–241.CrossRefGoogle Scholar
  53. 53.
    Priest, Graham. 2007. Paraconsistency and Dialetheism. In Handbook of the History of Logic, (The Many Valued and Nonmonotonic Turn in Logic), eds. Dov M. Gabbay and John H. Woods, vol. 8, 129–204. North Holland. doi: 10.1016/S1874-5857(07)80006-9.
  54. 54.
    Priest, Graham. 2006. In contradiction: A study of the transconsistent, 2nd ed. Oxford University Press.Google Scholar
  55. 55.
    Priest, Graham, and Francesco Berto. 2013. Dialetheism. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, Summer 2013 edition.Google Scholar
  56. 56.
    Marcos, João. 2000. 8K solutions and semi-solutions to a problem of da Costa. Unpublished draft.Google Scholar
  57. 57.
    Ribeiro, Márcio M., and Marcelo E. Coniglio. 2012. Contracting logics. In Logic, Language, Information and Computation. Proceedings of WoLLIC 2012, Buenos Aires, Argentina, September 3-6, 2012, Lecture Notes in Computer Science, eds. Luke Ong and Ruy de Queiroz, vol. 7456, 268–281. Springer. doi: 10.1007/978-3-642-32621-9_20.
  58. 58.
    Arieli, Ofer, Arnon Avron, and Anna Zamansky. 2010. Maximally paraconsistent three-valued logics. In Proceedings of the 12th International Conference on the Principles of Knowledge Representation and Reasoning (KR 2010), Toronto, Ontario, Canada, May 9-13, 2010, eds. Fangzhen Lin, Ulrike Sattler, and Miroslaw Truszczynski, 310–318. AAAI Press.Google Scholar
  59. 59.
    Carnielli, Walter A., Marcelo E. Coniglio, Rodrigo Podiacki, and Tarcísio Rodrigues. 2014. On the way to a wider model theory: Completeness theorems for first-order logics of formal inconsistency. The Review of Symbolic Logic 7(3): 548–578. doi: 10.1017/S1755020314000148.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Philosophy and Centre for Logic, Epistemology and the History of Science (CLE)University of Campinas (UNICAMP)CampinasBrazil

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