On the three-valued expansions of Kleene's logic

Abstract

The paper is devoted to one of the most famous three-valued systems – Kleene's logic. The expressive capabilities of Kleene's logic and its three-valued expansions are described. We present two results. First, all possible three-valued expansions of Kleene's logic are found up to equivalence with respect to the mutual definability of connectives. It is shown that there are only twelve such expansions. This list includes both logics already known in the literature and completely new ones. For the found expansions, we describe the structure of the lattice ordered relative to the expressive power of its elements. Secondly, for Kleene's logic and its three-valued expansions we find how many extensions each of these logics has in the same language. Kleene's logic has only two proper extensions: the classical and the trivial ones. Generally, a three-valued logic in which Kleene's matrix is definable contains no more than three proper extensions: the classical one, the trivial one, and an intermediate logic, determined by the product of the the original logic's matrix and the matrix of classical logic in the same signature. Intermediate logics exist only for two types of three-valued expansions of Kleene's logic: in expansions equivalent to Łukasiewicz's logic, and in logics whose matrices contain both a bivalent submatrix, the universe of which consists of the classical truth values, and a submatrix, the universe of which consists the intermediate value alone. All three-valued expansions of Kleene's logic that do not preserve the classical values have only one extension of their own – the trivial one.