On All Strong Kleene Generalizations of Classical Logic

Abstract

By using the notions of exact truth (`true and not false') and exact falsity (`false and not true'), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the $${{\bf SK}^\mathcal{P}}$$SKP and $${\bf SK}^{\mathcal{N}}$$SKN calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the $${\bf SK}^{\mathcal{P}}$$SKP and the $${\bf SK}^{\mathcal{N}}$$SKN calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the $${\bf SK}^{\mathcal{P}}$$SKP and the $${\bf SK}^{\mathcal{N}}$$SKN calculus, we also hint at its philosophical significance.