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Journal of Logic and Computation Advance Access published online on June 26, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp030
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© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

Logics Preserving Degrees of Truth from Varieties of Residuated Lattices

Félix Bou and Francesc Esteva

Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.
E-mail: fbou{at}iiia.csic.es; esteva{at}iiia.csic.es

Josep Maria Font

Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.
E-mail: jmfont{at}ub.edu

Àngel J. Gil

Departament d’Economia i Empresa, Universitat Pompeu Fabra, Barcelona, Spain.
E-mail: angel.gil{at}upf.edu

Lluís Godo

Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.
E-mail: godo{at}iiia.csic.es

Antoni Torrens and Ventura Verdú

Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.
E-mail: atorrens{at}ub.edu; v.verdu{at}ub.edu

Received 3 March 2008.

Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e. 1 is the only truth value preserved by the inferences of the logic. In this article, we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth-preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.

Keywords: Substructural logic; many-valued logic; degrees of truth; residuated lattices; non-protoalgebraic logic; Gentzen system; Tarski-style condition



References

  1. Adillon R, Verdú V. On a contraction-less intuitionistic propositional logic with conjunction and fusion. Studia Logica (Special issue on Abstract Algebraic Logic, Part I) (2000) 65:11–30.
  2. Aglianó P, Montagna F. Varieties of BL-algebras. I. General properties. Journal of Pure and Applied Algebra (2003) 181:105–129.[CrossRef][Web of Science]
  3. Aguzzoli S, Gerla B, Haniková Z. Complexity issues in Basic Logic. Soft Computing (2005) 9:919–934.[CrossRef][Web of Science]
  4. Avron A. The semantics and proof theory of linear logic. Theoretical Computer Science (1988) 57:161–184.[CrossRef][Web of Science]
  5. Baaz M, Preining N, Zach R. First-order Gödel logics. Annals of Pure and Applied Logic (2007) 147:23–47.[CrossRef][Web of Science]
  6. Baaz M, Zach R. Compact propositional Gödel logics. In. In: 28th International Symposium on Multiple Valued Logic (1998) Fukuoka, Japan: IEEE Computer Society Press. 108–113.
  7. Blok W, Pigozzi D. Protoalgebraic logics. Studia Logica (1986) 45:337–369.[CrossRef]
  8. Blok W, Pigozzi D. (1989) Algebraizable Logics, Vol. 396 of Memories of the American Mathematical Society, A.M.S.
  9. Blok W, Raftery J. Assertionally equivalent quasivarieties. International Journal of Algebra and Computation (2008) 18:589–681.[CrossRef][Web of Science]
  10. Bou F. A first approach to the deduction-detachment theorem in logics preserving degrees of truth. Malaga L, Magdalena M, Ojeda-Aciego XXXX, Verdegay JL, eds. (2008) Torremolinos (Málaga). 1061–1067. In Proceedings of the 12th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems,
  11. Bou F, Esteva F, Font JM, Gil A, Godo L, Torrens A, Verdú V. T-norm based fuzzy logics preserving degrees of truth. Malaga L, Magdalena M, Ojeda-Aciego XXXX, Verdegay JL, eds. (2008) Torremolinos (Málaga). 1053–1060. In Proceedings of the 12th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems.
  12. Bou F, García-Cerdaña À, Verdú V. There are no Tarski style axiomatizations for logics above FLew that are not superintuitionistic. (2007) unpublished manuscript.
  13. Chang CC. Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society (1958) 88:467–490.[CrossRef]
  14. Cignoli R, Mundici D, D’Ottaviano I. (2000) Kluwer. Algebraic Foundations of Many-valued Reasoning, Vol. 7 of Trends in Logic - Studia Logica Library.
  15. Cignoli R, Torrens A. An algebraic analysis of product logic. Multiple-Valued Logic (2000) 5:45–65.
  16. Cignoli R, Torrens A. Hájek basic fuzzy logic and Lukasiewicz infinite-valued logic. Archive for Mathematical Logic (2003) 42:361–370.[CrossRef][Web of Science]
  17. Cornish WH. Varieties generated by finite BCK-algebras. Bulletin of the Australian Mathematical Society (1980) 22:411–430.[CrossRef]
  18. Craig W. On axiomatizability within a system. The Journal of Symbolic Logic (1953) 18:30–32.[CrossRef]
  19. Czelakowski J. (2001) Kluwer Academic Publishers. Protoalgebraic logics, Vol. 10 of Trends in Logic - Studia Logica Library.
  20. Czelakowski J, Jansana R. Weakly algebraizable logics. The Journal of Symbolic Logic (2000) 65:641–668.[CrossRef]
  21. Czelakowski J, Pigozzi D. Fregean logics. Annals of Pure and Applied Logic (2004) 127:17–76.[CrossRef][Web of Science]
  22. Czelakowski J, Pigozzi D. Fregean logics with the multiterm deduction theorem and their algebraization. Studia Logica (2004) 78:171–212.[CrossRef]
  23. Nola ADi, Esteva F, Godo L, Montagna F. Varieties of BL-algebras. Soft Computing (2005) 9:875–888.[CrossRef][Web of Science]
  24. Dosen K, Schroeder-Heister P, eds. (1993) Oxford Science Publications. Substructural Logics, Vol. 2 of Studies in Logic and Computation.
  25. Dummett M. A propositional calculus with denumerable matrix. The Journal of Symbolic Logic (1959) 24:97–106.[CrossRef]
  26. Esteva F, Gispert J, Godo L, Noguera C. Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results. Fuzzy Sets and Systems (2007) 158:597–618.[CrossRef][Web of Science]
  27. Esteva F, Godo L. Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems (2001) 124:271–288.[CrossRef][Web of Science]
  28. Esteva F, Godo L, Montagna F. Equational characterization of the subvarieties of BL generated by t-norm algebras. Studia Logica (2004) 76:161–200.[CrossRef]
  29. Esteva F, Godo L, Noguera C. On rational weak nilpotent minimum logics. Journal of Multiple-Valued Logic and Soft Computing (2006) 12:9–32.
  30. Font JM. An abstract algebraic logic view of some multiple-valued logics. In. In: Beyond two: Theory and applications of multiple-valued logic—Fitting M, Orlowska E, eds. (2003) Physica-Verlag. 25–58. vol. 114 of Studies in Fuzziness and Soft Computing.
  31. Font JM. Generalized matrices in abstract algebraic logic. In. In: Trends in Logic. 50 years of Studia Logica—Hendriks VF, Malinowski J, eds. (2003) Kluwer. 57–86. Vol. 21 of Trends in Logic - Studia Logica Library.
  32. Font JM. Beyond Rasiowa's algebraic approach to non-classical logics. Studia Logica (2006) 82:172–209.
  33. Font JM. On substructural logics preserving degrees of truth. Bulletin of the Section of Logic (2007) 36:117–130.
  34. Font JM. Taking degrees of truth seriously. Studia Logica (Special issue on Truth Values, Part I) (2009) 91:383–406.
  35. Font JM, Gil A, Torrens A, Verdú V. On the infinite-valued £ukasiewicz logic that preserves degrees of truth. Archive for Mathematical Logic (2006) 45:839–868.[CrossRef][Web of Science]
  36. Font JM, Guzmán F, Verdú V. Characterization of the reduced matrices for the {{wedge}, {vee}}- fragment of classical logic. Bulletin of the Section of Logic (1991) 20:124–128.
  37. Font JM, Jansana R. Leibniz filters and the strong version of a protoalgebraic logic. Archive for Mathematical Logic (2001) 40:437–465.[CrossRef][Web of Science]
  38. Font JM, Jansana R. (2009) First edition 1996. Association for Symbolic Logic. A General Algebraic Semantics for Sentential Logics. Second revised edition, Vol. 7 of Lecture Notes in Logic. Electronic version freely available through Project Euclid at projecteuclid.org/euclid.lnl/1235416965.
  39. Font JM, Jansana R, Pigozzi D. Fully adequate Gentzen systems and the deduction theorem. Reports on Mathematical Logic (2001) 35:115–165.
  40. Font JM, Jansana R, Pigozzi D. A survey of abstract algebraic logic. Studia Logica (Special issue on Abstract Algebraic Logic, Part II) (2003) 74:13–97. with an ‘Update’ in 91, 125–130, 2009.
  41. Font JM, Jansana R, Pigozzi D. On the closure properties of the class of full g-models of a deductive system. Studia Logica (Special issue in memory of Willem Blok) (2006) 83:215–278.
  42. Galatos N, Jipsen P, Kowalski T, Ono H. (2007) Elsevier. Residuated Lattices: an Algebraic Glimpse at Substructural Logics, Vol. 151 of Studies in Logic and the Foundations of Mathematics.
  43. Galatos N, Ono H. Algebraization, parameterized local deduction theorem and interpolation for substructural logics over FL. Studia Logica (Special Issue in Memory of Willem Blok) (2006) 83:279–308.
  44. Gil AJ. Sistemes de Gentzen multidimensionals i lògiques finitament valorades. Teoria i aplicacions (1996) University of Barcelona. Ph.D. Dissertation.
  45. Gispert J. Axiomatic extensions of the nilpotent minimum logic. Reports on Mathematical Logic (2003) 37:113–123.
  46. Gottwald S. (2001) Research Studies Press. A Treatise on Many-valued Logics, Vol. 9 of Studies in Logic and Computation.
  47. Hájek P. (1998) Kluwer. Metamathematics of Fuzzy Logic, Vol. 4 of Trends in Logic - Studia Logica Library.
  48. Höhle U. Commutative, residuated l-monoids. In. In: Non-classical Logics and their Applications to Fuzzy Subsets—Höhle U, Klement EP, eds. (1995) Kluwer Academic Publishers. 53–106.
  49. Horcík R, Noguera C, Petrík M. On n-contractive fuzzy logics. Mathematical Logic Quarterly (2007) 53:268–288.[CrossRef][Web of Science]
  50. Jansana R. Selfextensional logics with a conjunction. Studia Logica (2006) 84:63–104.[CrossRef]
  51. Montagna F, Noguera C, Horcík R. On weakly cancellative fuzzy logics. Journal of Logic and Computation (2006) 16:423–450.[Abstract/Free Full Text]
  52. Mostert PS, Shields AL. On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics (1957) 65:117–143.[CrossRef][Web of Science]
  53. Noguera C, Esteva F, Gispert J. On triangular norm based axiomatic extensions of the weak nilpotent minimum logic. Mathematical Logic Quarterly (2008) 54:387–409.[CrossRef][Web of Science]
  54. Novák V. On the syntactico-semantical completeness of first-order fuzzy logic, parts I and II. Kybernetika (1990) 26:47–66. 134–154.[Web of Science]
  55. Nowak M. Logics preserving degrees of truth. Studia Logica (1990) 49:483–499.[CrossRef]
  56. Ono H. Substructural logics and residuated lattices - an introduction. In. In: Trends in Logic. 50 years of Studia Logica—Hendriks VF, Malinowski J, eds. (2003) Kluwer. 193–228. Vol. 21 of Trends in Logic - Studia Logica Library.
  57. Ono H, Komori Y. Logics without the contraction rule. The Journal of Symbolic Logic (1985) 50:169–201.[CrossRef]
  58. Pavelka J. On fuzzy logic I, II, III. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (1979) 25:45–52. 119–134, 447–464.[CrossRef][Web of Science]
  59. Pigozzi D. Fregean algebraic logic. In. In: Algebraic Logic—Andréka H, Monk JD, Németi I, eds. (1991) North-Holland. 473–502. Vol. 54 of Colloq. Math. Soc. János Bolyai.
  60. Raftery J. Correspondences between Gentzen and Hilbert systems. The Journal of Symbolic Logic (2006) 71:903–957.[CrossRef]
  61. Raftery J. The equational definability of truth predicates. Reports on Mathematical Logic (Special issue in memory of Willem Blok) (2006) 41:95–149.
  62. Rasiowa H. (1974) North-Holland. An Algebraic Approach to Non-classical Logics, vol. 78 of Studies in Logic and the Foundations of Mathematics.
  63. Scott D. Background to formalisation. In. In: Truth, syntax and modality—Leblanc H, ed. (1973) North-Holland. 244–273.
  64. Scott D. Completeness and axiomatizability in many-valued logic. In. Henkin L, et al, eds. (1974) American Mathematical Society. 411–436. Proceedings of the Tarski Symposium, Vol. 25 of Proceedings of Symposia in Pure Mathematics.
  65. Spinks M, Veroff R. Constructive logic with strong negation is a substructural logic, II. Studia Logica (2008) 89:401–425.[CrossRef]
  66. Wójcicki R. Theory of Logical Calculi. Basic Theory of Consequence Operations (1988) Reidel. Vol. 199 of Synthese Library.

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