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Intuitionistic modal logic and set theory

Published online by Cambridge University Press:  12 March 2014

K. Lano
K. Lano*
Affiliation:
Computing Laboratory Programming Research Group, Oxford, England
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Extract

The mathematical treatment of the concepts of vagueness and approximation is of increasing importance in artificial intelligence and related research. The theory of fuzzy sets was created by Zadeh [Z] to allow representation and mathematical manipulation of situations of partial truth, and proceeding from this a large amount of theoretical and applied development of this concept has occurred. The aim of this paper is to develop a natural logic and set theory that is a candidate for the formalisation of the theory of fuzzy sets. In these theories the underlying logic of properties and sets is intuitionistic, but there is a subset of formulae that are ‘crisp’, classical and two-valued, which represent the certain information. Quantum logic or logics weaker than intuitionistic can also be adopted as the basis, as described in [L]. The relationship of this theory to the intensional set theory MZF of [Gd] and the global intuitionistic set theory GIZF of Takeuti and Titani [TT] is also treated.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 2 , June 1991 , pp. 497 - 516
Copyright
Copyright © Association for Symbolic Logic 1991

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References

REFERENCES

[BM]Bell, J. and Machover, M., A course in mathematical logic, North-Holland, Amsterdam, 1977.Google Scholar
[Fn]Font, J. M., Modality and possibility in some intuitionistic modal logics, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 533–546.CrossRefGoogle Scholar
[Fr]Harvey Friedman's research on the foundations of mathematics (Harrington, L. A., editor), North-Holland, Amsterdam, 1985.Google Scholar
[Gd]Goodman, N. D., A genuinely intensional set theory, Intensional mathematics (Shapiro, S., editor), North-Holland, Amsterdam, 1985, pp. 63–79.CrossRefGoogle Scholar
[Gr]Greyson, R., Heyting valued models for intuitionistic set theory, Applications of sheaves, Lecture Notes in Mathematics, vol. 753, Springer-Verlag, Berlin, 1979, pp. 402–414.CrossRefGoogle Scholar
[L]Lano, K., Mathematical frameworks for vagueness and approximate reasoning, Ph.D. thesis, Bristol University, Bristol, 1988.Google Scholar
[Pw]Powell, W. C., Extending Gödel's negative interpretation to ZF, this Journal, vol. 40 (1975), pp. 221–229.Google Scholar
[RS]Rasiowa, H. and Sikorski, R., The mathematics of metamathematics, PWN, Warsaw, 1963.Google Scholar
[Sce]Scedrov, A., Extending Gödel's negative interpretation to type theory and set theory, Intensional mathematics (Shapiro, S., editor), North-Holland, Amsterdam, 1985, pp. 81–119.CrossRefGoogle Scholar
[TT]Takeuti, G. and Titani, S., Globalization of intuitionistic set theory, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 195–211.CrossRefGoogle Scholar
[Z]Zadeh, L. A., Fuzzy sets, Information and Control, vol. 8 (1965), pp. 338–353.CrossRefGoogle Scholar
[Zh]Jinwen, Zhang, Fuzzy set structure with strong implication, Advances in fuzzy sets, possibility theory and applications (Wang, H., editor), Plenum Press, New York, 1983, pp. 107–136.CrossRefGoogle Scholar