> **W** is defined to be: `\f x . f x x`. (So `W f` accepts one argument and gives it to `f` twice. What is the meaning of `W multiply`?)
-> **ω** is defined to be: `\x. x x`
+> **ω** (that is, lower-case omega) is defined to be: `\x. x x`
It's possible to build a logical system equally powerful as the lambda calculus (and readily intertranslatable with it) using just combinators, considered as atomic operations. Such a language doesn't have any variables in it: not just no free variables, but no variables at all.
-One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: K and I from above, and also one more, **S**, which behaves the same as the lambda expression `\f g x. f x (g x)`. behaves. But it's possible to be even more minimalistic, and get by with only a single combinator. (And there are different single-combinator bases you can choose.)
+One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: **K** and **I** from above, and also one more, **S**, which behaves the same as the lambda expression `\f g x. f x (g x)`. behaves. But it's possible to be even more minimalistic, and get by with only a single combinator. (And there are different single-combinator bases you can choose.)
There are some well-known linguistic applications of Combinatory
Logic, due to Anna Szabolcsi, Mark Steedman, and Pauline Jacobson.