Abstract
This publication builds on previous publications in which we constructed syntactic proofs of fuzzy Peterson’s syllogisms related to the graded square of opposition. The aim of the publication is to be formally able to find syntactic proofs of fuzzy Peterson’s logical syllogisms with forms of fuzzy intermediate quantifiers that design the graded Peterson’s cube of opposition.
Highlights
MethodsThe main approach of this section is to recall the theory of fuzzy natural logic (FNL)
In [18], we focused on the syntactic construction of proofs of all 105 basic fuzzy logical syllogisms that relate to the graded Peterson’s square of opposition
Fuzzy syllogisms are obtained by replacing the classical quantifier with the fuzzy quantifier
Summary
The main approach of this section is to recall the theory of fuzzy natural logic (FNL). That was designed based on the fuzzy type theory (FTT). FNL is a formal mathematical theory that includes three theories: . Theory of evaluative linguistic expressions (see [17]); Theory of fuzzy IF– rules and approximate reasoning (see [31]); Theory of intermediate quantifiers, generalized syllogisms, and graded structures of opposition (see [16,18]). This section focuses on the reminder of the main symbols and of the fuzzy type theory. Let us recall, at this point, that the mathematical theory of fuzzy quantifiers was proposed over the Łukasiewicz fuzzy type theory (Ł-FTT). The structure of truth values is represented by a linearly ordered MV∆ -algebra that is extended by the delta operation (see [33,34]). A particular case is the standard Łukasiewicz MV∆ -algebra:
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