Strict-tolerant conditional logics

Abstract We construct a conditional logic using selection models in a three-valued setting. When the selection function selects an empty set, the corresponding conditional is neither true nor false. The semantic consequence is defined as in strict-tolerant logic. The resulting logic validates conditional excluded middle without assuming that a selection function always chooses a single world, reconciling Stalnaker and Lewis. We give a labelled sequent calculus for the logic and consider three extensions of it. With the sequent calculus in hand, we show the decidability of two systems. It turns out that two extensions are strongly connexive, hyperconnexive, superconnexive and almost totally connexive. We compare our logics with other connexive conditional logics and show the philosophical consequence concerning paraconsistency and überconsistency.