On three-valued logics with the variable sharing property

Abstract

The class of the logics free from paradoxes of relevance determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with two designated values is defined. These logics are free from paradoxes of relevance in the sense that they have the "variable sharing property". They have "natural conditionals" in the sense that the function defining them coincides
with the classical function when restricted to the "classical values", satisfies Modus Ponens and, finally, assigns a designated value to a conditional whenever its antecedent and its consequence are assigned the same value. These logics are defined by using a "two-valued" overdetermined Belnap-Dunn semantics. Thus, the interpretation of the three values is crystalline. The logics here introduced enjoy the properties customarily demanded of many-valued implicative logics except, of course, the satisfaction of the rule "Verum ex quodlibet".