A New Perspective on Reasoning with Fuzzy Rules

This paper puts forward the idea that information encoded on a computer may have a negative or a positive emphasis. Negative information corresponds to the statement that some situations are impossible. It is often the case for pieces of background knowledge expressed in a logical format. Positive information corresponds to observed cases. It is often encountered in data-driven mathematical models, learning, etc. The notion of an “if. . . , then. . . ” rule is examined in the context of positive and negative information. It is shown that it leads to the three-valued representation of a rule, after De Finetti, according to which a given state of the world is an example of the rule, a counterexample to the rule, or is irrelevant for the rule. This view also sheds light on the typology of fuzzy rules. It explains the difference between a fuzzy rule modeled by a many-valued implication, and expressing negative information, and a fuzzy rule modeled by a conjunction (a la Mamdani) and expressing positive information. A new compositional rule of inference adapted to conjunctive rules, specific to positive information, is proposed. Consequences of this framework on interpolation between sparse rules are also presented.