Classical Implication for Three-Valued Logic

Abstract The article proposes a new implication for three-valued logical systems. The tautologies of this implication are exactly the same as for the classical implication in the two-valued Boolean logic. In the setting of this new implication, the conjunction and disjunction can be viewed as binary modalities. The article studies the definability and complete axiomatisation of the three versions of these modalities capturing Kleene’s, weak Kleene’s, and short-circuit versions of the connectives.