Simultaneous Abstraction and Semantic Theories

Abstract

I present a simple Simultaneous Abstraction Calculus, where the familiar lambda-abstraction over single variables is replaced by abstraction over whole sets of them. Terms are applied to partial assignments of objects to variables. Variants of the system are investigated and compared, with respect to their semantic and proof theoretic properties. The system overcomes the strict ordering requirements of the standard lambda-calculus,and is shown to provide the kind of "non-selective" binding needed for Dynamic Montague Grammar and Discourse Representation Theory. It is closely related to a more complex system, due to Peter Aczel and Rachel Lunon, and can be used for Situation Theory in a similar way. I present versions of these theories within an axiomatic, property-theoretic framework, based on Aczels Frege Structures. The aim of this work is to provide the means for integrating various semantic theories within a formal framework,so that they can share what is common between them, and adopt from each other what is compatible with them.