In §3, degrees are identified with sets of possible individuals. x’s degree of tallness consists of actual and possible individuals whose height is the same or less than x’s. The partial ordering of degrees is just the proper subset relation. Various properties of degrees are explained including: why degrees of tallness and weight are incommensurable, why degrees possessed in different possible worlds can be commensurable and how differentials are able to characterize a degree interval, including differentials formed with measurement vocabulary.
In §2, I offer a semantics of comparatives based on segments of a scale. The semantics is conjunctive, modeled on Davidsonian semantics for events. Scale segments provide some perspective on the use of spatial vocabulary in the formation of degree constructions across languages. Crosslinguistic variation is served by the conjunctive architecture which allow parts of the comparative to combine in different ways.
The hypothesis about degrees presupposes that gradable predicates denote relations among individuals. Degrees have to be introduced with functional vocabulary as do scalar segments. Various options are available and in §4, I briefly explore the typological landscape that results.
§5 is devoted to clausal adjectival comparatives. Adjectives denote predicates of individuals. Functional vocabulary, –er or more, quantify in and introduce segments. The Wh operator leaves an ‘adverbial trace’ – a predicate of segments. These moving parts interact in interesting ways with modals, attitude verbs and quantifiers. The standard marker than is a conjunction. It combines two clauses and it encodes the great-than-relation.