Login Register Cart 0
Generic placeholder image

Journal of Fuzzy Logic and Modeling in Engineering

Editor-in-Chief

ISSN (Print): 2666-2949
ISSN (Online): 2666-2957

Back
Journal Home About Journal Editorial Board Current Issue Volumes/Issues
Subscribe
Research Article

Partial Fuzzy Quantifiers and their Computation

Author(s): Vilém Novák* and Michal Burda

Volume 1, Issue 1, 2022

Published on: 05 January, 2021

Article ID: e010621189940 Pages: 10

DOI: 10.2174/2666294901666210105141618

Abstract

Background: In computer science, one often needs to deal with undefined values. For example, they naturally increase when a mistake such as the square root of a negative number or division by zero occurs. A similar problem occurs in the logical analysis of natural language. For example, the expression “Czech president in the 18th century” has no denotation because there was no Czech president before 1918. Such a situation in mathematics is characterized by partial functions, i.e., functions that may be undefined for specific arguments.

Methods: In this paper, we will extend the theory of intermediate quantifiers (i.e., expressions such as “most, almost all, many, a few,” etc.) to deal with partially defined fuzzy sets. First, we will extend algebraic operations that are used in fuzzy logic by the additional value “undefined.” Then we will introduce intermediate quantifiers using the former. The theory of intermediate quantifiers has been developed as a special theory of higher-order fuzzy logic.

Results: In this paper, we introduce the quantifiers semantically and show how they can be computed. The latter is also demonstrated in three illustrative examples.

Conclusion: The paper contributes to the development of fuzzy quantifier theory and its extension by undefined values and suggests methods for computation of truth values.

Keywords: Higher-order fuzzy logic, partial fuzzy type theory, fuzzy generalized quantifiers, intermediate quantifiers, Bochvar connectives, Sobocinski connectives, partial fuzzy sets.

[1]
J. Kacprzyk, A. Wilbik, and S. Zadrozny, "Linguistic summarization of time series using a fuzzy quantifier driven aggregation", Fuzzy Sets Syst., vol. 159, pp. 1485-1499, 2008.
[http://dx.doi.org/10.1016/j.fss.2008.01.025]
[2]
R. Gilsing, A. Wilbik, P. Grefen, O. Turetken, and B. Ozkan, A formal basis for business model evaluation with linguistic summaries. S. Nurcan, I. Reinhartz-Berger, P. Soffer, and J. Zdravkovic, Eds.25th Int. Conf. EMMSAD 2020, vol. 387. 2020, pp. 428-442. Grenoble, France
[3]
A. Wilbik, D. Barreto, and G. Backus, On relevance of linguistic summaries - a case study from the agro-food domain. M. Lesot, S. Vieira, M. Reformat, J. Carvalho, A. Wilbik, B. Bouchon-Meunier, and R. Yager, Eds.,Information Processing and Management of Uncertainty in Knowledge-Based Systems - 18th International Conference, IPMU, vol. 1237. 2020, pp. 289-300. Lisbon, Portugal Part I
[4]
P. Peterson, “Intermediate quantifiers - Logic, linguistics, and Aristotelian semantics., Ashgate: Aldershot, 2000.
[5]
A. Mostowski, "On a generalization of quantifiers", Fundam. Math., vol. 44, pp. 12-36, 1957.
[http://dx.doi.org/10.4064/fm-44-1-12-36]
[6]
R.C.J. Barwise, "Generalized quantifiers and natural language", Linguist. Philos., vol. 4, pp. 159-219, 1981.
[http://dx.doi.org/10.1007/BF00350139]
[7]
M. Brown, "Generalized quantifiers and the square of opposition", Notre Dame J. Form. Log., vol. 25, pp. 303-322, 1984.
[http://dx.doi.org/10.1305/ndjfl/1093870683]
[8]
S. Peters, and D. Westerstahl, “Quantifiers in Language and Logic.”, Clarendon Press: Oxford, 2006.
[9]
D. Westerstahl, “Quantifiers in formal and natural languages.”Handbook of philosophical logic., vol. IV. D. Reidel: Dordrecht, 1989, pp. 1-131.
[http://dx.doi.org/10.1007/978-94-009-1171-0_1]
[10]
P. Lindstrom, "First order predicate logic with generalized quantifiers", Theoria, vol. 32, pp. 186-195, 1966.
[11]
M. Holcapek, "Monadic L-fuzzy quantifiers of the type <1n,1>", Fuzzy Sets Syst., vol. 159, pp. 1811-1835, 2008.
[http://dx.doi.org/10.1016/j.fss.2008.03.028]
[12]
P. Murinova, and V. Nov´ak, "A formal theory of generalized´ intermediate syllogisms", Fuzzy Sets Syst., vol. 186, pp. 47-80, 2012.
[http://dx.doi.org/10.1016/j.fss.2011.07.004]
[13]
P. Murinova, and V. Nov´ak, "Analysis of generalized square of´ opposition with intermediate quantifiers", Fuzzy Sets Syst., vol. 242, pp. 89-113, 2014.
[http://dx.doi.org/10.1016/j.fss.2013.05.006]
[14]
V. Novak, "A formal theory of intermediate quantifiers", Fuzzy Sets Syst., vol. 159, no. 10, pp. 1229-1246, 2008.
[http://dx.doi.org/10.1016/j.fss.2007.12.008]
[15]
L.A. Zadeh, "A computational approach to fuzzy quantifiers in natural languages", Comput. Math. Appl., vol. 9, pp. 149-184, 1983.
[http://dx.doi.org/10.1016/0898-1221(83)90013-5]
[16]
L.A. Zadeh, "Syllogistic reasoning in fuzzy logic and its applications to usuality and reasoning with dispositions", IEEE Trans. Syst. Man Cybern., vol. 15, pp. 754-765, 1985.
[http://dx.doi.org/10.1109/TSMC.1985.6313459]
[17]
M. Delgado, M. Ruiz, D. Sanchez, and M. Vila, "Fuzzy quantification: a state of the art", Fuzzy Sets Syst., vol. 242, pp. 1-30, 2014.
[http://dx.doi.org/10.1016/j.fss.2013.10.012]
[18]
J. Galindo, R. Carrasco, and P. del R’ıo, "Fuzzy quantifiers with and without arguments for databases: Definition, implementation and application to fuzzy dependencies", In: Proceedings of IPMU, Torremolinos, Malaga, 2008, pp. 227-234.
[19]
I. Glockner, “Fuzzy Quantifiers: A Computational Theory”, Springer: Berlin, 2006.
[20]
M. Ying, and B. Bouchon-Meunier, "Quantifiers, modifiers and qualifiers in fuzzy logic", J. App. Non Class. Logics, vol. 7, pp. 335-342, 1997.
[http://dx.doi.org/10.1080/11663081.1997.10510918]
[21]
L. Behounek, and V. Novˇak, "Towards fuzzy partial logic", 45th Intl. Symposium on Multiple-Valued Logics, 2015pp. 139-144
[22]
L. Behounek, and M. Daˇnkovˇa, "Variable-domain fuzzy sets - part I: Representation", Fuzzy Sets Syst., vol. 380, pp. 1-18, 2020.
[http://dx.doi.org/10.1016/j.fss.2018.11.002]
[23]
P. Murinova, and V. Nov´ak, "The theory of intermediate quantifiers in fuzzy natural logic revisited and the model of “Many”", Fuzzy Sets Syst., vol. 388, pp. 56-89, 2020.
[http://dx.doi.org/10.1016/j.fss.2019.12.010]
[24]
V. Novak, I. Perfilieva, and A. Dvoˇrak, “Insight into Fuzzy´ Modeling,”, In Wiley & Sons: Hoboken, New Jersey, 2016, pp. 1-12.
[http://dx.doi.org/10.1002/9781119193210]
[25]
V. Novak, "Intermediate quantifiers in presence of partial fuzzy sets", S. Sundaram, Ed., Proc. 2018 IEEE Symposium Series on Computational Intelligence (SSCI 2018), 2018pp. 427-433
[http://dx.doi.org/10.1109/SSCI.2018.8628922]
[26]
L. Behounek, and M. Daˇnkovˇa, "Variable-domain fuzzy sets-part II: Apparatus", Fuzzy Sets Syst., vol. 380, pp. 19-43, 2020.
[http://dx.doi.org/10.1016/j.fss.2019.04.026]

Rights & Permissions Print Cite
Article Metrics
37
1
© 2024 Bentham Science Publishers | Privacy Policy