Normal forms of fuzzy middle and fuzzy contradictions

Abstract

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."

Meta-linguistic axioms as a foundation for computing with words

“…However, the investigations carried out by Tü rks ßen [3][4][5][6][7][8][9][10], Resconi and Tü rks ßen [2] indicate that we ought to use both the FDCF i (AE) and FCCF i (AE), i = 1,. . .…”

“…These were investigated in Tü rks ßen et al [7] under the heading of ''Reaffirmation'' in detail. However, in that paper there is a duplication error which should be noted, and it was corrected in later papers [3][4][5].…”

“…. , 16, that have been published in various papers over about the last two decades or so [3][4][5][6][7][8]10]. If we take into the account the fact that these non-equivalences of the Fuzzy Disjunctive and Conjunctive Canonical forms, then we have to realize that the interpretation of the proposed meta-linguistic axioms must be expressed in two or four distinct forms in set symbolic notation depending on whether they are two or three element axioms, respectively, and must give two or four distinct results in computational, numeric, domain with the application of t-norms and t-conorms.…”

“…In our previous writings [3][4][5][6][7][8]10], we have explored various aspects of DNF, CNF as well as FDCF and FCCF, including their generation, their equivalence and non-equivalence, i.e., DNF i (AE) CNF i (AE) in classical theory and FDCF i (AE)TFCCF i (AE), i = 1,. . .…”

“…These meta-linguistic expressions can then be interpreted in terms of their Fuzzy Disjunctive and Conjunctive Canonical Forms, i.e., FDCF and FCCF. (Tü rks ßen [3][4][5][6][7][8]10]) which take a structure with the set theoretic symbols ''\'', ''['', and ''c(AE)'', respectively.…”