Uniform Weak Kleene Logics

Abstract

In a multiple-premise and multiple-conclusion setting, logicians typically use the metalinguistic expression "," to gather premises and conclusions. In most logical systems, this comma can be freely interchanged with the object-language conjunction on the premise side and with the object-language disjunction on the conclusion side. However, this is not the case for Weak Kleene Logics (WK logics), which include Paraconsistent Weak Kleene (PWK) and its paracomplete counterpart (K₃^w).

The aim of this work is to define two new substructural systems called uniform Weak Kleene logics (uWK logics), uPWK and uK₃^w, which re-establish what we call "Uniformity": the correspondence between the structural comma and the object-language WK operators. We begin by providing some philosophical arguments advocating for it as a desirable logical feature. We then introduce a novel two-sided sequent calculus free of linguistic restrictions both for PWK and K₃^w and show that, with minor adjustments, we can achieve sound and complete sequent calculi for uPWK and uK₃^w. Finally, we generalize our results for WK logics and their uniform counterparts to other infectious Tarskian logics that meet specific criteria.