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 kn ow that a statement A is true "to some extent", i.e., if we know that the s tatement m(A) is true, for some logical modifier m ("approximately", "sligh tly", 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 . This mapping describes the meaning of the fuzzy implication; so, it is de sirable 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 t o modifiers that satisfy some reasonable properties. For an important parti cular case of invertible mapping, we get a complete description of all modi fier-to-modifier mapping that characterize fuzzy implication. Our main mathematical result can be also used in the foundations of knowled ge elicitation. (C) 1999 Published by Elsevier Science Inc. All rights rese rved.