In Chapter 2 some care was taken to distinguish the Curry and Church approaches to type-theory from each other. Curry's approach involved assigning types to preexisting untyped terms with each term receiving either an infinite set of types or none at all, whereas in Church's the terms were defined with built-in types with each term having a single type (see 2A3). In Curry's approach the types contained variables, in Church's they contained only constants.

This book focuses on the Curry approach. However, even in this approach it turns out to be very useful to introduce a typed-term language as an alternative notation for TAλ-deductions. Although the tree-notation introduced in Chapter 2 shows very clearly what assumptions are needed in deducing what conclusions, it takes up a lot of space and is hard to visualise when the deduction is in any way complicated. And when manipulations and reductions of deductions are under discussion it is almost unmanageable. A much more compact alternative notation is needed, and this is what the typed terms in the present chapter will give.

We shall also define reduction of typed terms; typed terms will be shown in the next chapter to encode deductions in propositional logic as well as in TAλ, and their reduction will be essentially the same as the reduction of deductions that is a standard tool in proof theory.