Nieuwe inzichten over het voorzetselvoorwerp

New insights on the prepositional objectAn influential article by Hans Broekhuis in Nederlandse Taalkunde 9 (2004) treated PP complements (type He thought of her all day) in Dutch. It contained sharp observations, intriguing hypotheses and a somewhat puzzling classification of the verbs allowing these complements. It provoked a lively discussion, on matters such as the number of possible PP complements in one sentence, the value of distinctive criteria such as an isolation test with en doet dat, the functional status of the preposition. Broekhuis’ generative-style syntactic reasoning at times clashed with approaches inspired by form/content analysis, valency theory and constructional grammar. In all, the original contribution to this journal and the subsequent discussion brought new and interesting insights in this traditionally hard-to-deal-with syntactic category.

Big-step normalisation

Traditionally, decidability of conversion for typed λ-calculi is established by showing that small-step reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflects and preserves conversion. We apply our approach to the simply typed λ-calculus with explicit substitutions and βη-equality, a system which is not strongly normalising. We also show how the construction can be extended to system T with the usual β-rules for the recursion combinator. Our approach is practical, since it does verify an actual implementation of normalisation which, unlike normalisation by evaluation, is first order. An important feature of our approach is that we are using logical relations to establish equational soundness (identity of normal forms reflects the equational theory), instead of the usual syntactic reasoning using the Church–Rosser property of a term rewriting system.