 
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
andrews@csd.uwo.ca
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
logic.
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
