A well-known challenge for compositionality is the phenomenon of so-called scope splitting, which is observed with non-monotone-increasing quantifiers in the scope of modal verbs. 'You can have at most two bites' has a salient reading which is neither 'there are at most two bites which you can have' nor 'I don't mind if you have at most two', but rather 'you cannot have more than two.' In this reading, it appears as if the quantifier 'at most two' has been split into a negation and a part meaning 'more than two', with one part taking scope above the modal 'can' and the other one below.

A less known challenge is an ambiguity in less-comparatives (first noted by Seuren). 'She drove less fast than was allowed' can mean either that she drove below the speed limit, or else that she drove below the minimum speed. Rullmann has argued that this ambiguity must be attributed either to a lexical ambiguity in 'less', or else to a decomposition of 'less' into two parts, one of which has no meaning of its own, but combines with 'fast' into a non-compositional unit meaning 'slow'.

I will argue that the two problems are related and have a common solution. The solution, a close but improved relative of Rullmann's proposal, relies on two assumptions. One concerns lexical semantics and amounts to the thesis that antonymy (the relation between pairs of adjectives like 'fast'-'slow') is negation. The other assumption is about the logical syntax of comparatives and other degree and measure constructions: these constructions contain a constituent which is a generalized quantifier over degrees and can scopally interact with other elements.