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Semisimple Algebras of Infinite Valued Logic and Bold Fuzzy Set Theory

Published online by Cambridge University Press:  20 November 2018

L. P. Belluce
L. P. Belluce*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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In classical two-valued logic there is a three way relationship among formal systems, Boolean algebras and set theory. In the case of infinite-valued logic we have a similar relationship among formal systems, MV-algebras and what is called Bold fuzzy set theory. The relationship, in the latter case, between formal systems and MV-algebras has been known for many years while the relationship between MV-algebras and fuzzy set theory has hardly been studied. This is not surprising. MV-algebras were invented by C. C. Chang [1] in order to provide an algebraic proof of the completeness theorem of the infinitevalued logic of Lukasiewicz and Tarski. Having served this purpose (see [2]), the study of these algebras has been minimal, see for example [6], [7]. Fuzzy set theory was also being born around the same time and only in recent years has its connection with infinite-valued logic been made, see e.g. [3], [4], [5]. It seems appropriate then, to take a further look at the structure of MV-algebras and their relation to fuzzy set theory.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 6 , 01 December 1986 , pp. 1356 - 1379
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Chang, C. C., Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467–490.Google Scholar
2. Chang, C. C., A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc. 95 (1959), 74–80.Google Scholar
3. Dubois, D. and Prade, H., Fuzzy sets and systems: theory and applications (Academic Press, 1980).Google Scholar
4. Giles, R., Lukasiewicz logic and fuzzy set theory, Proc. of the 1975 International Symposium on Multiple-valued Logic (Indiana Univ., Bloomington, Ind., 1975), 197–211. (IEEE Comp. Soc, Long Beach, Calif., 1975).Google Scholar
5. Skala, H. J., On many-valued logics, fuzzy sets, fuzzy logics and their applications, Fuzzy sets and systems 1 (1978), 141–148.Google Scholar
6. Schwartz, D., Arithmetishe theory der MV-algebren endlicher ordnung, Math. Nachr. 77 (1977), 65–73.Google Scholar
7. Schwartz, D., Das homomorphietheorem fur MV-algebren endlicher ordnung, Z. Math Logik Grundlagen Math. 22 (1976), 141–148.Google Scholar