Uniform Weak Kleene Logics

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 (K3w). The aim of this work is to define two new substructural systems called uniform Weak Kleene logics (uWK logics), uPWK and uK3w, 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 K3w and show that, with minor adjustments, we can achieve sound and complete sequent calculi for uPWK and uK3w. Finally, we generalize our results for WK logics and their uniform counterparts to other infectious Tarskian logics that meet specific criteria.