ON THREE-VALUED PRESENTATIONS OF CLASSICAL LOGIC

Abstract

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 $st$ , $ss$ , $tt$ , $ss\cap tt$ , and $ts$ , when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ 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.