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Normal forms of fuzzy middle and fuzzy contradictions

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3 Author(s)
Turksen, I.B. ; Dept. of Mech. & Ind. Eng., Toronto Univ., Ont., Canada ; Kandel, A. ; Yan-Qing Zhang

The expressions of “excluded middle” and “crisp contradiction” are reexamined starting with their original linguistic expressions which are first restated in propositional and then predicate forms. It is shown that, in order to generalize the truth tables and hence the normal forms, the membership assignments in predicate expressions must be separated from their truth qualification. In two-valued logic, there is no need to separate them from each other due to reductionist Aristotalean dichotomy. Whereas, in infinite (fuzzy) valued set and logic, the separation of membership assignments from their truth qualification forms the bases of a new reconstruction of the truth tables. The results obtained from these extended truth tables are reducible to their Boolean equivalents under the axioms of Boolean theory. Whereas, in fuzzy set and logic theory, we obtain a richer and more complex interpretations of the “fuzzy middle” and “fuzzy contradiction.”

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Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on  (Volume:29 ,  Issue: 2 )