The Class of All Natural Implicative Expansions of Kleene’s Strong Logic Functionally Equivalent to Łkasiewicz’s 3-Valued Logic Ł3

Abstract

We consider the logics determined by the set of all natural implicative expansions of Kleene's strong 3-valued matrix (with both only one and two designated values) and select the class of all logics functionally equivalent to Łukasiewicz's 3-valued logic Ł3. The concept of a "natural implicative matrix" is based upon the notion of a "natural conditional" defined in Tomova (2012), "A lattice of implicative extensions of regular Kleene's logics", Reports on Mathematical Logic 47, pp. 173-182.

The class of all 3-valued natural conditional variants of RM3 that are Plumwood Algebras

“…: the classical truth-values, the conditional is the one of classical logic; (II) the conditional verifies Modus Ponens; and (III) whenever the antecedent of the conditional is assigned a lower value than the consequent, the interpretation of the conditional is a designated truth-value; that is to say that for any propositional variables p and q, if p ≤ q, then I(p → q) ∈ D 12 . While there are certain weakenings of Tomova's definition [17] we prefer to focus on the original iteration of the idea.…”