The intended meaning of these constants can be described simply by showing a corresponding transformation or reduction which the constant may be thought of as effecting.
The transformations are as follows:
|K x y||=||x|
|S f g x||=||(f x)(g x)|
There exists an expression E in S,K and I such that:
This property establishes the close relationship between Pure Combinatory Logic and the Pure lambda Calculus, in which a special notation for functional abstraction is available. It shows that the notation for functional abstraction, though a great convenience, adds no expressiveness to the language.
As well as having combinatorial completeness, Pure Combinatory Logic is able to express all effectively computable functions over natural numbers appropriately represented as combinators.
Pure Combinatory Logic can be extended by moving along any of The Axes of The Cube.