Using the programming-language concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of in-situ quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, type-logical way to model evaluation order and side effects in natural language. We illustrate with an improved account of quantificational binding, weak crossover, wh-questions, superiority, and polarity licensing.

This paper and our conference paper (Abbott, Altenkirch, Ghani, and McBride, 2003b) explain and analyse the notion of the derivative of a data structure as the type of its one-hole contexts based on the central observation made by McBride (2001). To make the idea precise we need a generic notion of a data type, which leads to the notion of a container, introduced in (Abbott, Altenkirch, and Ghani, 2003a) and investigated extensively in (Abbott, 2003). Using containers we can provide a notion of linear map which is the concept missing from McBride’s first analysis. We verify the usual laws of differential calculus including the chain rule and establish laws for initial algebras and terminal coalgebras.

Abstract. A theory of traces of computations has emerged within the field of coalgebra, via finality in Kleisli categories. In concurrency theory, traces are traditionally obtained from executions, by projecting away states. These traces and executions are sequences and will be called “thin”. The coalgebraic approach gives rise to both “thin ” and “fat” traces/executions, where in the “fat ” case the structure of computations is preserved. This distinction between thin and fat will be introduced first. It is needed for a theory of schedulers in a coalgebraic setting, of which we only present the very basic definitions and results. 1