Abstract
Kleene first investigated a three-valued system which follows the evaluations of the Lukasiewicz infinite-valued logic ${\L}C$ with respect to negation, conjunction, and disjunction, and treats $\rightarrow$ as material-like implication in the sense that A $\rightarrow$ B is defined as ${\sim}A{\vee}B$ in its evaluation. Diense and Rescher extended it to many-valued logic and infinite-valued logic, respectively. This paper investigates a variant of the infinite-valued Kleene-Diense logic KD, which we shall call strong Kleene-Diense logic (sKD): sKD has the same evaluations as KD except that sKD takes a variant of Kleene-Diense implication. Following the idea of Dunn [2], we provide algebraic completeness for sKD together with its deduction theorem.