> **K** is defined to be `\x y. x`. That is, it throws away its
second argument. So `K x` is a constant function from any
(further) argument to `x`. ("K" for "constant".) Compare K
- to our definition of **true**.
+ to our definition of `true`.
-> **get-first** was our function for extracting the first element of an ordered pair: `\fst snd. fst`. Compare this to **K** and **true** as well.
+> **get-first** was our function for extracting the first element of an ordered pair: `\fst snd. fst`. Compare this to K and `true` as well.
-> **get-second** was our function for extracting the second element of an ordered pair: `\fst snd. snd`. Compare this to our definition of **false**.
+> **get-second** was our function for extracting the second element of an ordered pair: `\fst snd. snd`. Compare this to our definition of `false`.
> **B** is defined to be: `\f g x. f (g x)`. (So `B f g` is the composition `\x. f (g x)` of `f` and `g`.)
> **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.
-Notice that the semantic value of *himself* is exactly W.
+Notice that the semantic value of *himself* is exactly `W`.
The reflexive pronoun in direct object position combines first with the transitive verb (through compositional magic we won't go into here). The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.
Note that `W <~~> S(CI)`:
SFGX ~~> FX(GX)
If the meaning of a function is nothing more than how it behaves with respect to its arguments,
-these reduction rules capture the behavior of the combinators S,K, and I completely.
-We can use these rules to compute without resorting to beta reduction. For instance, we can show how the I combinator is equivalent to a certain crafty combination of S's and K's:
+these reduction rules capture the behavior of the combinators S, K, and I completely.
+We can use these rules to compute without resorting to beta reduction. For instance, we can show how the I combinator is equivalent to a certain crafty combination of Ss and Ks:
SKKX ~~> KX(KX) ~~> X
-So the combinator SKK is equivalent to the combinator I.
+So the combinator `SKK` is equivalent to the combinator I.
-Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. The most common system uses S,K, and I as defined here.
+Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. The most common system uses S, K, and I as defined here.
###The equivalence of the untyped lambda calculus and combinatory logic###