Partial Fuzzy Quantifiers and their Computation

Vilém       Novák; Michal       Burda

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.

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DOI https://dx.doi.org/10.2174/2666294901666210105141618	Print ISSN 2666-2949
Publisher Name Bentham Science Publisher	Online ISSN 2666-2957

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