First Steps Toward Fuzzy Number-valued Propositional Logic
Jorma K. Mattila
Suitable sets of fuzzy numbers are described and examined for presenting the special conditions for those fuzzy numbers which can form a fuzzy number-valued Zadeh algebra. Zadeh algebra is a Kleene algebra of fuzzy sets. A set of triangular fuzzy numbers, called lambda-standard fuzzy numbers are considered. Some sets of these fuzzy numbers satisfy the conditions of Kleene algebra. These sets of fuzzy numbers can be used as sets of truth values in Kleene-like fuzzy-valued logics, called fuzzy number-valued many-valued logics. Some preliminary examples for this kind of logics are sketched. The connection of Łukasiewicz’ implication to these logics is shown. This connection gives a possibility to construct fuzzy number-valued Łukasiewicz’ logics. Any fuzzy number-valued logics are not defined yet. Only some possible sets of fuzzy truth values are considered and tested with many-valued connectives creating by means of the operations of fuzzy-valued Zadeh algebra in a propositional language.
Keywords: Ordering of fuzzy numbers, fuzzy number-valued Zadeh algebra, fuzzy number-valued truth values, fuzzy number-valued propositional logic, fuzzy-valued Kleene type logic, fuzzy-valued Łukasiewicz type logic, lambda-standard fuzzy number