lambda-Terms, Logic, Determiners and Quantifiers

In this paper, a theory T H based on combining type freeness with logic is introduced and is then used to build a theory of properties which is applied to determiners and quantiiers. It is well known that mixing type freeness and logic leads to contradictions. For example, by taking the following syntax of terms: t := xjx:tjt 1 t 2 j:t and applying the term x::xx to itself one gets a contradiction (known as Russell's paradox). Church was aware of the problem when he started the-calculus which he intended to be a theory of functions and logic. But his rst theory of the-calculus was type free and so was inconsistent. The paradox could be described as follows: take a to be x:(xx ! ?). Then from Modus Ponens (MP), the Deduction Theorem (DT), and-conversion, we could derive Curry's paradox: 1: