<p>This paper examines two approaches to presuppositions: one viewing them as inferences projecting from sentences under negation and other logical operators, and another defining them as admittance conditions of utterances. Neither approach fully accounts for the “proviso problem”, which arises when a sentence’s presuppositional inferences are logically stronger than its necessary admittance conditions. To address this challenge, we propose a calculus of a trivalent logic that formally distinguishes between admittance and projection, extending Karttunen’s dynamic, logical form-based analysis. The resulting framework enables a simple pragmatic strategy: presuppositional conclusions are accommodated unless overridden by a contextually likelier admittance condition. We provide evidence that this approach is empirically superior to methods that address the proviso problem using pragmatic strengthening.</p>