Does projection strength vary by context or by word type?

Standard accounts of presupposition treat projection as categorical: presuppositions project (survive embedding under negation, questions, modals) and non-presuppositions do not. The Gradient Projection Principle (Tonhauser et al.) challenges this with robust empirical evidence.

Across 19 American English expressions, projectivity varies continuously, not discretely. The strength of projection is determined by a single organizing principle: content projects to the extent it is not at-issue — to the extent it does not address the Question Under Discussion (QUD) in that context.

This has a consequence for how we think about presupposition triggers. The standard typology — factive predicates (know) vs. semi-factive (discover) vs. non-factive (believe) — partially tracks projection but misses the contextual sensitivity. The same predicate can generate stronger or weaker projection depending on whether its complement addresses the current QUD. "Obama improved the economy" under "knows" projects strongly when the QUD is about someone's mental state, but less strongly when the QUD is about the economy itself.

For understanding LLM presupposition handling, this matters because: LLMs learn categorical trigger patterns from training data (factive = presupposition trigger = projects). But the actual projection strength is context-sensitive and gradient. A model that learned only the categorical pattern will systematically miscalibrate when context shifts at-issueness — which is precisely the kind of context-sensitivity that Why do embedding contexts confuse LLM entailment predictions? shows LLMs lack.

This also connects to the broader principle that language is gradient and functional, not categorical and defective. Like Why do speakers deliberately use ambiguous language?, gradient projection is a feature, not a failure — it allows presuppositions to project more or less based on discourse context.