Non-denoting Terms in Fuzzy Logic: An Initial Exploration

  • Libor Běhounek
  • Antonín DvořákEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 641)


We introduce two variants of first-order fuzzy logic that can deal with non-denoting terms, or terms that lack existing referents, e.g., Pegasus, the current king of France, the largest number, or 0/0. Logics designed for this purpose in the classical setting are known as free logics. In this paper we discuss the features of free logics and select the options best suited for fuzzification, deciding on the so-called dual-domain semantics for positive free logic with truth-value gaps and outer quantifiers. We fuzzify the latter semantics in two levels of generality, first with a crisp and subsequently with a fuzzy predicate of existence. To accommodate truth-valueless statements about nonexistent objects, we employ a recently proposed first-order partial fuzzy logic with a single undefined truth value. Combining the dual-domain semantics with partial fuzzy logic, we define several kinds of ‘inner-domain’ quantifiers, relativized by the predicate of existence. Finally, we make a few observations on some of the resulting rules of free fuzzy quantification that illustrate the differences between the two proposed systems of free fuzzy logic and their well known non-free or non-fuzzy variants.


Quantifier Free logic Existence Referent Partial fuzzy logic 



The work was supported by grant No. 16–19170S “Fuzzy partial logic” of the Czech Science Foundation.


  1. 1.
    Běhounek, L., Cintula, P., Hájek, P.: Introduction to mathematical fuzzy logic. In: Cintula, P., Hájek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic, Chap. 1, vol. I, pp. 1–101. College Publications (2011)Google Scholar
  2. 2.
    Běhounek, L., Daňková, M.: Towards fuzzy partial set theory. In: Carvalho, J., et al. (eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016), Part II. Communications in Computer and Information Science, vol. 611, pp. 482–494. Springer, Cham (2016)Google Scholar
  3. 3.
    Běhounek, L., Novák, V.: Towards fuzzy partial logic. In: Proceedings of the IEEE 45th International Symposium on Multiple-Valued Logics (ISMVL 2015), pp. 139–144 (2015)Google Scholar
  4. 4.
    Bencivenga, E.: Free logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 5, 2nd edn., pp. 147–196. Kluwer, Dordrecht (2002)Google Scholar
  5. 5.
    Lehmann, S.: More free logic. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 5, 2nd edn., pp. 197–259. Kluwer, Dordrecht (2002)Google Scholar
  6. 6.
    Nolt, J.: Free logics. In: Jacquette, D. (ed.) Philosophy of Logic, Handbook of the Philosophy of Science, pp. 1023–1060. North-Holland, Amsterdam (2007)Google Scholar
  7. 7.
    Nolt, J.: Free logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Winter 2014 edn. (2014). URL \(\langle \) \(\rangle \)

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute for Research and Applications of Fuzzy ModelingUniversity of Ostrava, NSC IT4InnovationsOstravaCzech Republic

Personalised recommendations