Presupposition Projection and Repair Strategies in Trivalent Semantics

In binary propositional constructions S1 con S2, the Strong Kleene connectives explain filtering of S1’s and S2’s presuppositions depending on their logical relations with their non-presuppositional content. However, the presuppositions derived by the Strong Kleene connectives are weak conditional presuppositions, which raise the “proviso problem” in cases where no filtering is motivated. Weak Kleene connectives do not face this problem, but only because their presuppositions are often too strong, and hence do not account for filtering phenomena altogether. While various mechanisms have been proposed to allow filtering without the proviso problem, their relations with the standard trivalent Kleene systems have remained unclear. This paper shows that by sacrificing truth-functionality, we uncover a rich domain of possibilities in trivalent semantics in between the Weak Kleene and Strong Kleene connectives. These systems derive presupposition filtering while avoiding the proviso problem. The Kleene-style operators studied are generalized to arbitrary binary functions, which further clarifies the connection between their different “repair” strategies and presupposition projection.