A unified approach to type theory through a refined λ-calculus

Abstract

In the area of foundations of mathematics and computer science, three related topics dominate. These are λ-calculus, type theory and logic. There are moreover, many versions of λ-calculi and type theories. In these versions, the presence of logic ranges from the non-existent to the dominant. In fact, the three subjects of λ-calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube, for example, is an attempt to unify various type systems and his associated logic cube is an attempt to find connections between type theories and logic. We devise a new λ-notation which enables categorising most of the known systems in a unified way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automath-like systems to singly typed pure type systems. The system and the notation that we propose have far reaching advantages than just being used as a generalisation formalism. These advantages range from generalising reduction and substitution to representing Mathematics and are investigated in detail in various articles cited in the bibliography.