A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms

  • Authors:

  • Siegfried Weber

  • Affiliations:

  • -

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 1983

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Abstract

All known connectives 'and'/'or' for fuzzy sets or some classes can be introduced as t-norms/t-conorms, where Ling's representation theorem is used as a basic tool, and which is illustrated by various known and new examples (Section 2). Given a strict negation function and one connective, the other can be constructed, so that the corresponding De Morgan law is valid. In case of given Archimedean connectives, there can be constructed negation functions (Section 3). Given a non-strict Archimedean connective, a negation function and the other connective can be constructed, so that in addition to the De Morgan laws, the excluded middle law and the law of non-contradiction are valid, i.e. the negation function results to be complementary (Section 4). In function of connectives and negation three types of fuzzy implication operators are introduced, which include almost all known implications, and that of type I using 'and' only; of type II using 'or'/'non'; of type III using 'and'/'non'. In the non-strict Archimedean cases the formulas become particularly lucid (Section 5). Finally the different types are compared with respect to some logical properties (Section 6).