Questions about proof theory vis-à-vis natural language semantics

Abstract
Semantics plays a role in grammar in at least three guises. (A) Linguists seek to account for speakers‘ knowledge of what linguistic expressions mean. This goal is typically achieved by assigning a model theoretic interpretation2 in a compositional fashion. For example, No whale flies is true if and only if the intersection of the sets of whales and fliers is empty in the model. (B) Linguists seek to account for the ability of speakers to make various inferences based on semantic knowledge. For example, No whale flies entails No blue whale flies and No whale flies high. (C) The wellformedness of a variety of syntactic constructions depends on morpho-syntactic features with a semantic flavor. For example, Under no circumstances would a whale fly is grammatical, whereas Under some circumstances would a whale fly is not, corresponding to the downward vs. upward monotonic features of the preposed phrases
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