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International Workshop on Logic, Rationality and Interaction

LORI 2017: Logic, Rationality, and Interaction pp 3–16Cite as

A Logical Framework for Graded Predicates

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10455))

Abstract

In this position paper we present a logical framework for modelling reasoning with graded predicates. We distinguish several types of graded predicates and discuss their ubiquity in rational interaction and the logical challenges they pose. We present mathematical fuzzy logic as a set of logical tools that can be used to model reasoning with graded predicates, and discuss a philosophical account of vagueness that makes use of these tools. This approach is then generalized to other kinds of graded predicates. Finally, we propose a general research program towards a logic-based account of reasoning with graded predicates.

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Notes

  1. 1.

    This classification is a modification of that presented by Paoli in [26, 27].

References

  1. Běhounek, L.: In which sense is fuzzy logic a logic for vagueness? In: Lukasiewicz, T., Peñaloza, R., Turhan, A.Y. (eds.) Proceedings of the First Workshop on Logics for Reasoning about Preferences, Uncertainty, and Vagueness (PRUV 2014), Vienna, pp. 26–39 (2014)

    Google Scholar 

  2. Běhounek, L., Cintula, P.: Fuzzy logics as the logics of chains. Fuzzy Sets Syst. 157(5), 604–610 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blok, W.J., Pigozzi, D.L.: Algebraizable Logics, Memoirs of the American Mathematical Society, vol. 396. American Mathematical Society, Providence (1989). http://orion.math.iastate.edu/dpigozzi/

  4. Casari, E.: Comparative logics and Abelian \(\ell \)-groups. In: Logic Colloquium 1988, Padova. Studies in Logic and the Foundations of Mathematics, vol. 127, pp. 161–190. North-Holland, Amsterdam (1989)

    Google Scholar 

  5. Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft. Comput. 4(2), 106–112 (2000)

    Article  Google Scholar 

  6. Cintula, P., Fermüller, C.G., Hájek, P., Noguera, C. (eds.): Handbook of Mathematical Fuzzy Logic (in three volumes). Studies in Logic, Mathematical Logic and Foundations, vols. 37, 38, and 58. College Publications (2011, 2015)

    Google Scholar 

  7. Cintula, P., Horčík, R., Noguera, C.: Non-associative substructural logics and their semilinear extensions: axiomatization and completeness properties. Rev. Symbolic Logic 6(3), 394–423 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cintula, P., Horčík, R., Noguera, C.: The quest for the basic fuzzy logic. In: Montagna, F. (ed.) Petr Hájek on Mathematical Fuzzy Logic. OCL, vol. 6, pp. 245–290. Springer, Cham (2015). doi:10.1007/978-3-319-06233-4_12

    Google Scholar 

  9. Cintula, P., Noguera, C.: Implicational (semilinear) logics I: a new hierarchy. Arch. Math. Logic 49(4), 417–446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cintula, P., Noguera, C.: A Henkin-style proof of completeness for first-order algebraizable logics. J. Symb. Log. 80(1), 341–358 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dummett, M.: A propositional calculus with denumerable matrix. J. Symb. Log. 24(2), 97–106 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  12. Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124(3), 271–288 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Esteva, F., Godo, L., García-Cerdaña, À.: On the hierarchy of t-norm based residuated fuzzy logics. In: Fitting, M.C., Orlowska, E. (eds.) Beyond Two: Theory and Applications of Multiple-Valued Logic, Studies in Fuzziness and Soft Computing, vol. 114, pp. 251–272. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  14. Fine, K.: Vagueness, truth and logic. Synthese 30, 265–300 (1975). Reprinted with corrections in [22], pp. 119–150

    Google Scholar 

  15. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  16. Galatos, N., Ono, H.: Cut elimination and strong separation for substructural logics: an algebraic approach. Ann. Pure Appl. Logic 161(9), 1097–1133 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hájek, P.: Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4. Kluwer, Dordrecht (1998)

    Book  MATH  Google Scholar 

  18. Hájek, P.: Fuzzy logics with noncommutative conjunctions. J. Log. Comput. 13(4), 469–479 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Stud. Logica. 70(2), 183–192 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jenei, S., Montagna, F.: A proof of standard completeness for non-commutative monoidal t-norm logic. Neural Netw. World 13(5), 481–489 (2003)

    Google Scholar 

  21. Keefe, R.: Theories of Vagueness. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  22. Keefe, R., Smith, P. (eds.): Vagueness: A Reader. MIT Press, Cambridge (1997)

    Google Scholar 

  23. Lambek, J.: On the calculus of syntactic types. In: Jakobson, R. (ed.) Structure of Language and Its Mathematical Aspects, pp. 166–178. American Mathematical Society, Providence (1961)

    Chapter  Google Scholar 

  24. Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, cl. III 23(iii), 30–50 (1930)

    Google Scholar 

  25. Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symb. Log. 72(3), 834–864 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Paoli, F.: Truth degrees, closeness, and the sorites. In: Bueno, O., Abasnezhad, A. (eds.) On the Sorites Paradox. Springer (to appear)

    Google Scholar 

  27. Paoli, F.: Comparative logic as an approach to comparison in natural language. J. Semant. 16, 67–96 (1999)

    Article  Google Scholar 

  28. Raffman, D.: Vagueness without paradox. Philos. Rev. 103, 41–74 (1994)

    Article  Google Scholar 

  29. Shapiro, S.: Vagueness in Context. Oxford University Press, Oxford (2006)

    Book  Google Scholar 

  30. Smith, N.J.J.: Consonance and dissonance in solutions to the sorites. In: Bueno, O., Abasnezhad, A. (eds.) On the Sorites Paradox. Springer (to appear)

    Google Scholar 

  31. Smith, N.J.J.: Undead argument: the truth-functionality objection to fuzzy theories of vagueness. To appear in Synthese. doi:10.1007/s11229-014-0651-7

  32. Smith, N.J.J.: Vagueness as closeness. Australas. J. Philos. 83, 157–183 (2005)

    Article  Google Scholar 

  33. Smith, N.J.J.: Vagueness and Degrees of Truth. Oxford University Press, Oxford (2008). Paperback 2013

    Book  Google Scholar 

  34. Sorensen, R.: Vagueness and Contradiction. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  35. Stenning, K., van Lambalgen, M.: Human Reasoning and Cognitive Science. MIT Press, Cambridge (2008)

    Google Scholar 

  36. Williamson, T.: Vagueness. Routledge, London (1994)

    Google Scholar 

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Acknowledgments

Petr Cintula and Carles Noguera are supported by the project GA17-04630S of the Czech Science Foundation (GAČR); both authors have also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (SYSMICS project). Petr Cintula also acknowledges the support of RVO 67985807. Nicholas Smith is supported by the SOPHI Research Support Scheme at the University of Sydney.

Author information

Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 271/2, 182 07, Prague, Czech Republic

    Petr Cintula

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08, Prague, Czech Republic

    Carles Noguera

  3. Department of Philosophy, University of Sydney, Main Quadrangle A14, Sydney, NSW, 2006, Australia

    Nicholas J. J. Smith

Authors
  1. Petr Cintula
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  2. Carles Noguera
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  3. Nicholas J. J. Smith
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Correspondence to Petr Cintula .

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Editors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam , Amsterdam, Noord-Holland, The Netherlands

    Alexandru Baltag

  2. Department of Philosophy, The University of Auckland, Auckland, Auckland, New Zealand

    Jeremy Seligman

  3. Graduate School of Letters, Hokkaido University , Sapporo, Japan

    Tomoyuki Yamada

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Cintula, P., Noguera, C., Smith, N.J.J. (2017). A Logical Framework for Graded Predicates. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_1

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  • DOI: https://doi.org/10.1007/978-3-662-55665-8_1

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