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A WeaklyTyped Higher Order Logic with General Lambda Terms and Y Combinator

Summary: A Weakly­Typed Higher Order Logic
with General Lambda Terms and Y Combinator
James H. Andrews
Dept. of Computer Science
University of Western Ontario
London, Ontario, Canada N6A 5B7
Abstract. We define a higher order logic which has only a weak notion
of type, and which permits all terms of the untyped lambda calculus
and allows the use of the Y combinator in writing recursive predicates.
The consistency of the logic is maintained by a distinction between use
and mention, as in Gilmore's logics. We give a consistent model theory
and a proof system which is valid with respect to the model theory. We
also give examples showing what formulas can and cannot be used in the
1 Introduction
The type system of a new higher order logic must be designed with care. When­
ever we try to make the logic more expressive by permitting more well­typed
terms, we risk making the logic inconsistent; for instance, Church's higher order
logic [Chu40] cannot consistently be extended to permit the rather modestly­


Source: Andrews, Jamie - Department of Computer Science, University of Western Ontario


Collections: Computer Technologies and Information Sciences