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Towards the Future of Fuzzy Logic pp 137–165Cite as

Fuzzy Natural Logic: Towards Mathematical Logic of Human Reasoning

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 325))

Abstract

One of the often repeated proclaims appearing in the papers on fuzzy sets and fuzzy logic is their ability to model semantics of some linguistic expressions so that the inherent vagueness of the former is also captured. Recall that this direction of research was initiated by L.A. Zadeh already in his early papers and since then, most of the applications of fuzzy sets emphasize presence of natural language, at least in hidden form. In this paper we argue that the potential of fuzzy set theory and fuzzy logic is strong enough to enable developing not only a working model of linguistic semantics but even more—to develop a model of natural human reasoning that proceeds in natural language. We bring forward the concept of fuzzy natural logic (FNL) that is a mathematical theory whose roots lay in the concept of natural logic developed by linguists and logicians. Of course, this cannot be realized without cooperation with linguists. On the other hand, it seems reasonable not to try to solve all the problems raised by the linguistic research but rather to develop a simplified model that would capture the main features of the semantics of natural language and thus made it possible to realize sophisticated technical applications. In the paper, we will show that basic formalism of FNL has already been established and has potential for further development. We also outline how model of the meaning of basic constituents of natural language (nouns, adjectives, adverbs, verbs) can be developed and the human-like reasoning can proceed.

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Notes

  1. 1.

    Cf. e.g., [60].

  2. 2.

    See, e.g., [39] where a well working mathematical model of the sorites paradox (a typical feature of vagueness) has been proposed. It is also proved in this paper that the developed model copes with the typical phenomenon of vagueness manifesting itself in the semantics of degree adjectives (adjectives such as “tall, small”, etc.).

  3. 3.

    See, e.g., [25, 58] and elsewhere.

  4. 4.

    [39]; see also [38].

  5. 5.

    In type theory, all syntactical objects including variables and connectives are taken as formulas.

  6. 6.

    By currying, we may consider only unary functions.

  7. 7.

    We speak about indefinite and specifying grammateme of a noun, respectively, cf. [33, 57].

  8. 8.

    The detailed presentation and an informal justification of \(TEV\) can be found in [39].

  9. 9.

    This is quite natural because the set of truth values is assumed to be MV-algebra, namely the Łukasiewicz one whose support set is \([0, 1]\).

  10. 10.

    This sentence is often discussed by L.A. Zadeh in his lectures.

  11. 11.

    More about this phenomenon can be found, e.g., in [14].

  12. 12.

    For simplicity, we consider only one variable in the antecedent.

  13. 13.

    See [38, 46] for the details.

  14. 14.

    Note that for specific elements assigned to \(x, y\), the intension (8.19) provides a truth value.

  15. 15.

    For the detailed linguistic analysis of these concepts, see, e.g., [14].

  16. 16.

    In fact, we only need the number of formulas to be finite and reasonably small.

  17. 17.

    For the details, see [45].

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Acknowledgments

The research was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).

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  1. Institute for Research and Applications of Fuzzy Modelling Centre of Excellence IT4Innovations, University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic

    Vilém Novák

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Editors and Affiliations

  1. Institut für Geschichte der Medizin, Naturwissenschaft und Technik, Friedrich-Schiller-Universität Jena, Berggasse, Jena, Germany

    Rudolf Seising

  2. Edificio de Investigación, 3ª Planta. C Gonzalo Gutiérrez Quirós S/N, European Centre for Soft Computing, Mieres, Asturias, Spain

    Enric Trillas

  3. Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland

    Janusz Kacprzyk

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Novák, V. (2015). Fuzzy Natural Logic: Towards Mathematical Logic of Human Reasoning. In: Seising, R., Trillas, E., Kacprzyk, J. (eds) Towards the Future of Fuzzy Logic. Studies in Fuzziness and Soft Computing, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-319-18750-1_8

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