We argue that the epistemic modal ‘might’ is a generator of alternatives in the sense of Hamblin semantics (Kratzer & Shimoyama 2002) or inquisitive semantics (Ciardelli, Groenendijk & Roelofsen 2009). Building on methodologies from the psychology of reasoning, we show that ‘might’ patterns with disjunctions and with indefinites in giving rise to a particular kind of illusory inference. The best extant accounts of these illusory inferences crucially involve alternatives, paired with matching strategies (Walsh & Johnson-Laird 2004) or with question-answer dynamics (Koralus & Mascarenhas 2013). Our results constitute further steps toward convergence between theories and methodologies in natural language semantics and the psychology of reasoning.