Linguistic expressions
of a natural language are trans
formed at least three times before they are expressed in computational
formulas. Between the linguistic and the computational expressions, there are the meta-linguistic and propositional expressions under various interpretations and assumptions. Fuzzy
normal forms are canonical expressions derived at the propositional level with particular assumptions. The fuzzy
normal forms can be generated from fuzzy truth tables with a
normal form derivation algorithm and an orthodox interpretation
of conjunction, disjunction and complementation operations. It is shown that these fuzzy
normal forms display a unique structure for certain general classes
of crisp operators over fuzzy sets. The fuzzy
normal forms, generated by crisp operators over fuzzy sets that correspond to max-min-standard-complement De
Morgan triple, are shown to be equivalent to the classical
normal forms
of two-valued logic, i.e.,
disjunctive and
conjunctive normal forms known as DNF and CNF, with a particular property, i.e., the crisp containment FDNF
(3) FCNF
(3). Whereas the fuzzy
normal forms, generated by crisp operators that correspond to non-idempotent t-norms, t-conorms and standard complement De Morgan triples do not have, in general, a similar crisp containment property, i.e., FDNF
(2) FCNF
(2). Neither do the fuzzy
normal forms, generated by crisp operators over fuzzy sets that correspond to non-idempotent and non-commutative and non-associative conjunction, disjunction and standard complement De Morgan triples have a crisp containment property, i.e., FDNF
(1) FCNF
(1). However, they still provide bounds on the combination
of concepts.