Fuzzy Modus Ponens as a Calculus of Logical Modifiers: Towards Zadeh's Vision of Implication Calculus

If we know that an implication A ! B is to some extent true, and if we know that a statement A is true \to some extent", i.e., if we know that the statement m(A) is true, for some logical modiier m (\approximately", \slightly", etc.), then we can conclude that B is also true to some extent, i.e., that m 0 (B) is true for some logical modiier m 0. Thus, every fuzzy implication deenes a mapping from modiiers to modiiers. This mapping describes the meaning of the fuzzy implication, so, it is desirable to nd out what mapping 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 deene implication as a mapping from mod-iiers to modiiers that satisfy some reasonable properties. For an important particular case of invertible mappings, we get a complete description of all modiier-to-modiier mappings that characterize fuzzy implication. Our main mathematical result can be also used in the foundations of knowledge elicitation.