Abduction with Estimates for Statements in Fuzzy Propositional Logic

Abstract

Estimates are expressions of the form $\upvarphi \ge r, \upvarphi > r$, $\upvarphi \le r, \upvarphi < r$, $\upvarphi {\rm{ }} \le \uppsi$ or $\upvarphi < \uppsi$ where $\upvarphi$ and $\uppsi$ are propositional formulas and r is a real number from the unit interval [0, 1]. We consider the classic fuzzy interpretations of formulas $\upvarphi$, i.e., those based on the t-norm min{x, y} and negation 1 − x. Such interpretations is naturally extended to estimates. Logic of estimates LE is the set of all Boolean compositions of estimates that are interpreted with the usual sense of the propositional connectives. We have developed, for the logic LE, a complete system of inference rules in the style of analytic tableaux. It is shown how to apply the rules for abduction in the logic LE.