Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T19:50:28.658Z Has data issue: false hasContentIssue false
  • English
  • Français

Intuitionistic fuzzy logic and intuitionistic fuzzy set theory

Published online by Cambridge University Press:  12 March 2014

Gaisi Takeuti  and
Satoko Titani
Gaisi Takeuti
Affiliation:
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Satoko Titani
Affiliation:
University of Washington, Seattle, Washington 98195
Get access Rights & Permissions [Opens in a new window]

Extract

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 p ⊃ q = 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 (P ⊃ Q). ⊃ 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 3 , September 1984 , pp. 851 - 866
Copyright
Copyright © Association for Symbolic Logic 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Gentzen, G., Untersuchungen üer das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934/1935), pp. 176–210.CrossRefGoogle Scholar
[2]Grayson, R. J., A sheaf approach to models of set theory, M.Sc. Thesis, Oxford, 1975.Google Scholar
[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. 402–414.Google Scholar
[4]Hay, L. S., Axiomatization of the infinite-valued predicate calculus, this Journal, vol. 28 (1963), pp. 77–86.Google Scholar
[5]Powell, W. C., Extending Gödel's negative interpretation to ZF, this Journal, vol. 40 (1975), pp. 221–229.Google Scholar
[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. 189–306.CrossRefGoogle Scholar