The paper has two principal aims. First, Pietroski's sparse account of semantic composition will be favourably compared to the familiar full Frege hierarchy of composed semantic types. Second, I shall introduce the idea of a head as employed in linguistics and pose the problem of how to square Pietroski's compositional principles with the asymmetry entailed by headedness. The paper will leave the problem of headedness as an outstanding issue to be resolved.

Proportional quantification and progressive aspect interact in English in revealing ways. This paper investigates these interactions and draws conclusions about the semantics of the progressive and telicity. In the scope of the progressive, the proportion named by a proportionality quantifier (e.g. most in The software was detecting most errors) must hold in every subevent of the event so described, indicating that a predicate in the scope of the progressive is interpreted as an internally homogeneous activity. Such an activity interpretation is (...) argued to be available for telic predicates (e.g. cross the street) because these receive a partitive interpretation except in combination with a completive operator, which asserts that the event so described has culminated. The obligatoriness of the completive operator in the preterit is shown to parametrically distinguish those languages that show completion entailments in the preterit, e.g. English, from those that do not, e.g. Malagasy, Hindi, and Japanese. (shrink)