Semantical analysis of weak Kleene logics

This paper presents a semantical analysis of the Weak Kleene Logics K3W and PWK from the tradition of Bochvar and Halldén. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical consequence in PWK–that is, we individuate necessary and sufficient conditions for a set Δ of formulas to follow from a set Γ in PWK–and a characterisation result for logical consequence in K3W. The paper also investigates two subsystems of K3W and PWK and discusses the relevance of the results against existing background. Finally, the paper discusses some issues related to Weak Kleene Logics–in particular, their philosophical interpretation and the reading of conjunction and disjunction–and points at some open issues.