Approximate reasoning based on the preference implication

Abstract

In fuzzy logic, most of the implication operators are based on generalizations of the classical, material implication. That is, these implications are defined as the disjunction of the negated value of the first argument and the value of the second argument, while the underlying disjunction operators are associative triangular conorms. In our study, we concentrate on how a class of implication operators, called the preference implication operators, can be used in approximate reasoning. Using this implication operator family, we present a novel, Modus Ponens-like approximate reasoning method, in which we have two premises: (1) a statement and (2) a preference implication with an antecedent of this statement. Here, we show how the continuous logical value of the consequent of the preference implication can be derived from the continuous logical values of the premises. We point out that this novel approximate reasoning method is strongly connected with the so-called aggregative operator, which is a representable uninorm. Next, we also present a threshold value-based generalization of the Modus Ponens syllogism and demonstrate that the Modus Tollens syllogism can be generalized in the same way. Lastly, we provide an illustrative example.