This article presents a method to construct event semantic representations incrementally from left to right. The proposed theory focusses on the interaction between events and other scope taking expressions, in particular quantificational DPs and adverbials that are sensitive to aspect. Psycholinguistic experiments have revealed that semantic mismatches can be detected as early as their lexical triggers are encountered, without waiting for a complete semantic or syntactic representation. Our theory thus provides for ‘incomplete’ representations that nonetheless allow for an immediate explanation of such mismatches, right at the point where left to right parsing detects the offending triggers. Viewed from a broader perspective, our framework can be seen as a logical background for theories of semantic parsing in general. In contrast to existing implementations of incremental interpretation, we only employ a single compositional processing rule, namely functional composition of only one logical type of expressions. On the other hand, for this (otherwise unmatched) uniformity to work, we rely on advanced semantic techniques like continuations, dynamic binding and unrestricted β-reduction (cf. Barker 2002; Gronendijk & Stokhof 1991; Klein & Sternefeld 2013, respectively). In the context of adverbials, these methods can be shown to interact in a non-trivial way with the mereology of events as proposed in Krifka (1992).