Explanatory Model for the Break of Logic Equivalence by Irrational Agents in Elkan’s Paradox

Abstract

Fuzzy logic breaks logic equivalence of statements such as (A∧B)∨(¬A∧B)∨(A∧¬B) and A∨B. It breaks the symmetry of use of such logically equivalent statements. There is a controversy about this property. It is called a paradox (Elkan's paradox) and interpreted as a logical weakness of fuzzy logic. In the opposite view, it is not a paradox but a fundamental postulate of fuzzy logic and one of the sources of its success in applications. There is no explanatory model to resolve this controversy. This paper provides such a model using a vector/matrix logic of rational and irrational agents that covers scalar classical and fuzzy logics. It is shown that the classical logic models rational agents, while fuzzy logic can model irrational agents. Rational agents do not break logic equivalence in contrast with irrational agents. We resolve the paradox by showing that the classical and fuzzy logics have different domains of rational and irrational agents.