Fuzzy modus ponens as a calculus of logical modifiers : towards Zadeh's vision of implication calculus
Résumé
If we know that an implication A → B is to some extent true, and if we know that astatement A is true “to some extent”, i.e., if we know that the statement m(A) is true, for some logical modifier m (“approximately”, “slightly”, etc.), then we can conclude that B is also true to some extent, i.e., that m′(B) is true for some logical modifier m′.Thus, every fuzzy implication defines a mapping from modifiers to modifiers. Thismapping describes the meaning of the fuzzy implication; so, it is desirable to find out what mappings are possible. This desire is in line with Zadeh's suggestion that the future of fuzzy logic lies with treating fuzzy implications as a calculus, rather than as analytical formulas.In this paper, we formally define implication as a mapping from modifiers to modifiers that satisfy some reasonable properties. For an important particular case of invertible mapping, we get a complete description of all modifier-to-modifier mapping that characterize fuzzy implication.Our main mathematical result can be also used in the foundations of knowledgeelicitation.