A formulation of the simple theory of types

Abstract

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.

On the Specification of Sequent Systems

“…As in Church's Simple Theory of Types [Chu40], types for both terms and formulas are built using a simply typed λ-calculus. Variables are simply typed that do not contain the type o, which is reserved for the type of formulas.…”

Variants of Gödel’s Ontological Proof in a Natural Deduction Calculus

“…The language of higher-order modal logic used here is inspired by that of Church's simple type theory (Church, 1940).…”

Horn clause programs with polymorphic types: Semantics and resolution

“…According to [Chu40], we embed types in teiTas~ i.e., each symbol in a term is annotated with a type expression: Let V C_ Var~,x. A (~, X , V ) -t e r m of t y p e rE TH(X) is either a v a r i a b l e x:~" E V, a c o n s t a n t c:v with c:--+ vc E Func so that there exists a a E T S ( H , X ) with or(re) = r, or a c o m p o s i t e t e r m of the form f ( t V r l , .…”