On three-valued presentations of Classical Logic

Abstract

Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations st, ss, tt, ss ∩ tt, and ts, when the connectives are negation, conjunction, and disjunction. For ts and ss ∩ tt the answer is trivial (no scheme works), and for ss and tt it is straightforward (they are the collapsible schemes, in which the middle value acts like one of the classical values). For st, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible.