A two-valued logic for reasoning about different types of consequence in Kleene's three-valued logic

A formal language of two-valued logic is developed, whose terms are formulas of the language of Kleene's three-valued logic. The atomic formulas of the former language are pairs of formulas of the latter language joined by consequence operators. These operators correspond to the three sensible types of consequence (strong-strong, strong-weak and weak-weak) in Kleene's logic in analogous way as the implication connective in the classical logic corresponds to the classical consequence relation. The composed formulas of the considered language are built from the atomic ones by means of the classical connectives and quantifiers.A deduction system for the developed language is given, consisting of a set of decomposition rules for sequences of formulas. It is shown that the deduction system is sound and complete.