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Implicit definability of truth constants in Łukasiewicz logic

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Abstract

In the framework of propositional Łukasiewicz logic, a suitable notion of implicit definability, tailored to the intended real-valued semantics and referring to the elements of its domain, is introduced. Several variants of implicitly defining each of the rational elements in the standard semantics are explored, and based on that, a faithful interpretation of theories in Rational Pavelka logic in theories in Łukasiewicz logic is obtained. Some of these results were already presented in Hájek (Metamathematics of fuzzy logic, 1998) as technical statements. A connection to the lack of (deductive) Beth property in Łukasiewicz logic is drawn. Moreover, while irrational elements of the standard semantics are not implicitly definable by finitary means, a parallel development is possible for them in infinitary Łukasiewicz logic. As an application of definability of the rationals, it is shown how computational complexity results for Rational Pavelka logic can be obtained from analogous results for Łukasiewicz logic. The complexity of the definability notion itself is studied as well. Finally, we review the import of these results for the precision/vagueness discussion for fuzzy logic, and for the general standing of truth constants in Łukasiewicz logic.

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Notes

  1. We also use a horizontal bar to denote complement of a set. A vector of variables is denoted with a tilde (such as \({\tilde{x}}\)).

  2. In Łukasiewicz logic, the canonical interpretation of rational constants is the only one that consistently expands \({[0,\!1]_{\mathchoice{\mathrm{\L }}{\mathrm{\L }}{\mathrm{\L }}{\mathrm{\tiny \L }}}}\) under the usual bookkeeping axioms; see Sect. 3.

  3. The discourse in mathematical fuzzy logic so far confirms that fuzzy logics are semantics-based; accordingly, constants are tied to the intended (real-valued) semantics, standing for the rational or the real numbers thereof. Indeed, when all rational constants from the interval [0, 1] are present, one may claim that semantics has leaked into syntax in a substantial way, in particular, such a process narrows down the range of algebraic interpretations of the calculus noticeably.

  4. Cf. also Hájek et al. (2000), Theorem 3.1 and above.

  5. This is also reflected in (Hájek et al. 2000, §3):“Lemma 2.3 shows that \(\mathrm {RPL}\forall \) is a very conservative extension indeed of \(\mathrm {{\mathchoice{\mathrm{\L }}{\mathrm{\L }}{\mathrm{\L }}{\mathrm{\tiny \L }}}}\forall \). There is a sense in which even its new formulae don’t express anything which can’t be expressed by old formulae”.

  6. Admittedly, \(T_Q\) imposes its own semantics: it does not have a model over algebras that do not contain a copy of the rationals.

References

  • Běhounek L (2014) In which sense is fuzzy logic a logic for vagueness? In: T Łukasiewicz, R Peñaloza, and A-Y Turhan (eds) Proceedings of the first workshop on logics for reasoning about preferences, uncertainty, and vagueness, pp 26–39. Vienna

  • Caicedo X (2004) Implicit connectives of algebraizable logics. Stud Log 78:155–170

    Article  MathSciNet  MATH  Google Scholar 

  • Caicedo X (2007) Implicit operations in MV-algebras and the connectives of Łukasiewicz logic. In: Algebraic and proof-theoretic aspects of many-valued logics, volume 4460 of Lecture Notes in Computer Science, pp 50–68. Springer

  • Cignoli R, D’Ottaviano IML, Mundici D (1999) Algebraic foundations of many-valued reasoning, volume 7 of Trends in Logic. Kluwer, Dordrecht

    Google Scholar 

  • Cintula P (2016) A note on axiomatizations of Pavelka-style complete fuzzy logics. Fuzzy Sets Syst 292:160–174

    Article  MathSciNet  MATH  Google Scholar 

  • Cintula P, Esteva F, Gispert J, Godo L, Montagna F, Noguera C (2009) Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann Pure Appl Logic 160(1):53–81

    Article  MathSciNet  MATH  Google Scholar 

  • Di Nola A, Leuştean I (2011) Łukasiewicz logic and MV-algebras. In: Cintula P, Hájek P, Noguera C (eds) Handbook of mathematical fuzzy logic, vol 2. College Publications, New York, pp 469–583

    Google Scholar 

  • Esteva F, Gispert J, Godo L, Noguera C (2007) Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results. Fuzzy Sets Syst 158(6):597–618

    Article  MathSciNet  MATH  Google Scholar 

  • Gaitán H (1993) The number of simple one-generated bounded commutative BCK-chains. Math Japonica 38(3):483–486

    MathSciNet  MATH  Google Scholar 

  • Gispert J (2002) Universal classes of MV-chains with applications to many-valued logic. Math Logic Q 42:581–601

    Article  MathSciNet  MATH  Google Scholar 

  • Goguen JA, Joseph Amadee Goguen (1969) The logic of inexact concepts. Synthese 19(3–4):325–373

    Article  MATH  Google Scholar 

  • Hájek P (1998) Metamathematics of fuzzy logic, volume 4 of Trends in Logic. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Hájek P (2006) Computational complexity of t-norm based propositional fuzzy logics with rational truth constants. Fuzzy Sets Syst 157(5):677–682

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek P, Paris J, Shepherdson JC (2000) Rational Pavelka logic is a conservative extension of Łukasiewicz logic. J Symb Logic 65(2):669–682

    Article  MATH  Google Scholar 

  • Haniková Z (2011) Computational complexity of propositional fuzzy logics. In: Cintula P, Hájek P, Noguera C (eds) Handbook of mathematical fuzzy logic, vol 2. College Publications, New York, pp 793–851

    Google Scholar 

  • Haniková Z, Savický P (2016) Term satisfiability in \(\text{ FL }_{ew}\)-algebras. Theor Comput Sci 631:1–15

    Article  MATH  Google Scholar 

  • Hoogland E (2001) Definability and interpolation: model-theoretic investigations. Ph.D. thesis, Institute for Logic, Language and Computation

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

    MATH  Google Scholar 

  • Marra V (2014) The problem of artificial precision in theories of vagueness: a note on the rôle of maximal consistency. Erkenntnis 79(5):1015–1026

    Article  MathSciNet  MATH  Google Scholar 

  • McNaughton R (1951) A theorem about infinite-valued sentential logic. J Symb Logic 16(1):1–13

    Article  MathSciNet  MATH  Google Scholar 

  • Montagna F (2006) Interpolation and Beth’s property in propositional many-valued logics: a semantic investigation. Ann Pure Appl Logic 141(1–2):148–179

    Article  MathSciNet  MATH  Google Scholar 

  • Mundici D (2011) Advanced Łukasiewicz calculus and MV-algebras, volume 35 of Trends in Logics. Springer, Berlin

    Book  MATH  Google Scholar 

  • Mundici D (2014) Universal properties of Łukasiewicz consequence. Log Univ 8(1):17–24

    Article  MATH  Google Scholar 

  • Pavelka J (1979) On fuzzy logic I, II, III. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 25:45–52, 119–134, 447–464

  • Torrens A (1994) Cyclic elements in MV-algebras and Post algebras. Math Log Q 40(4):431–444

    Article  MathSciNet  MATH  Google Scholar 

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Funding

The author was supported by CE-ITI and GAČR under grant GBP202/12/G061 and by long-term strategic development financing of the Institute of Computer Science RVO:67985807.

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  1. Institute of Computer Science Czech Academy of Sciences, 182 07, Prague, Czech Republic

    Zuzana Haniková

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Haniková, Z. Implicit definability of truth constants in Łukasiewicz logic. Soft Comput 23, 2279–2287 (2019). https://doi.org/10.1007/s00500-018-3461-x

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