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Intuitionistic fuzzy logic and intuitionistic fuzzy set theory

  • Gaisi Takeuti (a1) and Satoko Titani (a2)

In 1965 Zadeh introduced the concept of fuzzy sets. The characteristic of fuzzy sets is that the range of truth value of the membership relation is the closed interval [0, 1] of real numbers. The logical operations ⊃, ∼ on [0, 1] which are used for Zadeh's fuzzy sets seem to be Łukasiewciz's logic, where pq = min(1, 1 − p + q), ∼ p = 1 − p. L. S. Hay extended in [4] Łukasiewicz's logic to a predicate logic and proved its weak completeness theorem: if P is valid then P + Pn is provable for each positive integer n. She also showed that one can without losing consistency obtain completeness of the system by use of additional infinitary rule.

Now, from a logical standpoint, each logic has its corresponding set theory in which each logical operation is translated into a basic operation for set theory; namely, the relation ⊆ and = on sets are translation of the logical operations → and ↔. For Łukasiewicz's logic, P Λ (PQ). ⊃ Q is not valid. Translating it to the set version, it follows that the axiom of extensionality does not hold. Thus this very basic principle of set theory is not valid in the corresponding set theory.

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[1]Gentzen, G., Untersuchungen üer das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934/1935), pp. 176210.
[2]Grayson, R. J., A sheaf approach to models of set theory, M.Sc. Thesis, Oxford, 1975.
[3]Grayson, R. J., Heyting valued models for intuitionistic set theory, Applications of sheaves (Proceedings of the research symposium, Durham, 1981), Lecture Notes in Mathematics, vol. 753, Springer-Verlag, Berlin, 1979, pp. 402414.
[4]Hay, L. S., Axiomatization of the infinite-valued predicate calculus, this Journal, vol. 28 (1963), pp. 7786.
[5]Powell, W. C., Extending Gödel's negative interpretation to ZF, this Journal, vol. 40 (1975), pp. 221229.
[6]Takeuti, G. and Titani, S., Heyting valued universes of intuitionistic set theory, Logic symposium, Hakone, 1979, 1980, Lecture Notes in Mathematics, vol. 891, Springer-Verlag, Berlin, 1981, pp. 189306.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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