Three-valued generalizations of classical logic in weak languages: the degree of maximality

This paper is deals with the degrees of maximality of consequence relations within the class of C-extending three-valued logics with languages of minimal expressive power. A logic is defined as C-extending if its operations coincide with those of classical logic when their domain is restricted to classical truth values. The degree of maximality of a logic is understood as the set of all its extensions in the same language. Every three-valued C-extending logic can be seen as a linguistic expansion of one of ten three-valued logics. We provide an assesment of the degree of maximality for each of these ten logics. In cases where the degree of maximality is finite, we have obtained exact values. In instances where it proves to be infinite, a lower bound for the cardinality of the lattice of extensions of the corresponding logic is provided. The exception is one system, for which it is demonstrated that the degree of maximality of its consequence relation has the cardinality of the continuum. The paper is devoted to the investigation of proof-theoretic properties of three-valued logics based on the expressive capabilities of their languages. The obtained results raise several questions, outlining new directions for research in this field.