Fuzzy logic and fuzzy set theory.

*(English)*Zbl 0786.03039On the semantic side, the authors consider the truth degree set \([0,1]\) with a constant for the degree 0.5, with min, max, Gödel implication, the negation defined by it in the intuitionistic style, but also with Łukasiewicz’s (1-..)-negation, bounded sum, and product as connectives, and with sup, inf as quantifiers. Truth degree 1 is the only designated one.

On the syntactic side they start from Gentzen’s sequent calculus LJ for intuitionistic first-order logic and add two rules (one of them infinitary) and 46 axioms. Their main result concerning this fuzzy logic is the completeness theorem.

Turning to fuzzy set theory FZF means to consider a ZF-like axiomatic theory, based on this fuzzy logic, with double complement and Zorn’s lemma as additional axioms. For FZF there is given a version of Powell’s inner model \(S\) of hereditary stable sets and proven that FZF \(\models\) “\(S\) is a model of ZFC”, and inside \(S\) Grayson’s sheaf model \(S^ l\) over \(l=[0,1]^ S\) is constructed and “\(S^ l\) is a model of FZF” proven in \(S\). Finally the authors prove that there exists a functional relation describing a bijection (via a suitable identification of elements of \(S^ l\)!) between \(S^ l\) and the universe of FZF.

{Reviewer’s remark: The paper is an interesting contribution to the theoretical foundation of fuzzy set theory. But it is more than astonishing that the authors cite besides their own papers on what they call “intuitionistic fuzzy logic” – and what is not fuzzy logic at all – no other papers concerned with foundational issues of fuzzy sets}.

On the syntactic side they start from Gentzen’s sequent calculus LJ for intuitionistic first-order logic and add two rules (one of them infinitary) and 46 axioms. Their main result concerning this fuzzy logic is the completeness theorem.

Turning to fuzzy set theory FZF means to consider a ZF-like axiomatic theory, based on this fuzzy logic, with double complement and Zorn’s lemma as additional axioms. For FZF there is given a version of Powell’s inner model \(S\) of hereditary stable sets and proven that FZF \(\models\) “\(S\) is a model of ZFC”, and inside \(S\) Grayson’s sheaf model \(S^ l\) over \(l=[0,1]^ S\) is constructed and “\(S^ l\) is a model of FZF” proven in \(S\). Finally the authors prove that there exists a functional relation describing a bijection (via a suitable identification of elements of \(S^ l\)!) between \(S^ l\) and the universe of FZF.

{Reviewer’s remark: The paper is an interesting contribution to the theoretical foundation of fuzzy set theory. But it is more than astonishing that the authors cite besides their own papers on what they call “intuitionistic fuzzy logic” – and what is not fuzzy logic at all – no other papers concerned with foundational issues of fuzzy sets}.

Reviewer: S.Gottwald (Leipzig)

##### Keywords:

truth degree set; Gentzen’s sequent calculus; fuzzy logic; completeness theorem; fuzzy set theory FZF; inner model; hereditary stable sets; sheaf model
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\textit{G. Takeuti} and \textit{S. Titani}, Arch. Math. Logic 32, No. 1, 1--32 (1992; Zbl 0786.03039)

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##### References:

[1] | Gentzen, G.: Untersuchungen über das logische Schließen. Math. Z.39, 176-210 (1935) · JFM 60.0020.02 |

[2] | Grayson, R.J.: A sheaf approach to models of set theory. Oxford: M. Sc. Thesis 1975 |

[3] | Grayson, R.J.: Heyting valued models for intuitionistic set theory. Applications of sheaves (Proceedings of the research symposius, Durham 1981). (Lect. Notes Math., vol 753, pp. 402-414. Berlin Heidelberg New York: Springer 1979 |

[4] | Powell, W.C.: Extending Gödel’s negative interpretation ofZF. J. Symb. Logic40, 221-229 (1975) · Zbl 0356.02049 |

[5] | Schütte, K.: Proof Theory. Berlin Heidelberg New York: Springer 1977 · Zbl 0367.02012 |

[6] | Takeuti, G., Titani, S.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. Symb. Logic49, 851-866 (1984) · Zbl 0575.03015 |

[7] | Takeuti, G., Titani, S.: Globalization of intuitionistic set theory. Ann. Pure Appl. Logic33, 195-211 (1987) · Zbl 0633.03050 |

[8] | Takeuti, G., Titani, S.: Global intuitionistic fuzzy set theory. The Mathematics of Fuzzy Systems, pp. 291-301 Köln: TÜV-Verlag 1986 · Zbl 0593.03031 |

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