The trivalent theory of indicative conditionals -- on which ‘if A, B’ is true if A and B are true, false if A is true and B is false, and neither true nor false when A is false -- has a lot going for it. Yet even the most sympathetic theorists have thought the theory comes at a high price, given how it appears to interact with the trivalent theory of presupposition. In this paper, I argue that by distinguishing between the ways in which conditional and categorical assertions affect a conversational context, we can avoid paying this price. I then develop a bilateral update semantics for a language that includes presuppositions, indicative conditionals, negation, and modals. The semantics makes a number of desirable compositional predictions, and on three different ways of reducing update rules to truth and falsity conditions, both conditionals and presupposition-carrying sentences give rise to truth value gaps.

According to a simple theory of the relationship between ‘want’ ascriptions and the desires they ascribe, when I learn that 'A wants p' is true, I learn that the truth of p is necessary and sufficient for satisfying one of A’s desires. I argue that this simple theory is false: 'A wants p' can be true and underspecific, that is, there can be cases in which 'A wants p' is true and p is necessary but not sufficient for the satisfaction of A’s desire. I show that existing semantics for ‘want’ cannot account for this kind of underspecificity, and I propose a desire-based semantics for ‘want’ that can. I go on to argue that my semantics has empirical and methodological advantages over existing theories of ‘want’ that give truth conditions in terms of an agent’s preferences rather than their desires.

Orthodox semantics for natural language modals give rise to two puzzles for their interactions with disjunction: Ross’s puzzle and the puzzle of free choice permission. It is widely assumed that each puzzle can be explained in terms of the licensing of 'Diversity' inferences: from the truth of a possibility or necessity modal with an embedded disjunction, hearers infer that each disjunct is compatible with the relevant set of worlds. I argue that Diversity inferences are too weak to explain the full range of data. Instead, modals with embedded disjunctions license 'Independence' inferences: from the truth of a modal with an embedded disjunction, hearers infer that each disjunct is an independent alternative among the relevant set of worlds. I then develop a bilateral inquisitive semantics for modals that predicts the validity of these Independence inferences. My account vindicates common intuitions about both Ross’s puzzle and the puzzle of free choice permission, and explains the full range of data.

Orthodox theories of necessity modals make the counterintuitive prediction that 'Must(p)' entails `Must(p or q)`. Ross's Puzzle is the challenge of reconciling this prediction with the common, unorthodox intuition that it is false. It is widely assumed that the unorthodox intuition can be explained by the licensing of 'Diversity' or 'free choice' inferences: from the truth of `Must(p or q)`, hearers infer that p and q are both compatible with the set of worlds 'Must' quantifies over in a context. I argue that Diversity inferences are too weak to explain the full range of data. Instead, necessity modals with embedded disjunctions license 'Independence' inferences: from the truth of `Must(p or q)`, hearers infer that p and q are both independent alternatives among the relevant set of worlds. I give a bilateral inquisitive semantics for necessity modals that predicts the validity of these Independence inferences. Then, I argue that my semantics offers an assimilation of the unorthodox logic of necessity modals with embedded disjunctions to the logic of collective predicates (e.g., `performed Happy Days') in the nominal domain.