> **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###
[\x\y.yx] = [\x[\y.yx]] = [\x.S[\y.y][\y.x]] = [\x.(SI)(Kx)] = S[\x.SI][\x.Kx] = S(K(SI))(S[\x.K][\x.x]) = S(K(SI))(S(KK)I)
+We can test this translation by seeing if it behaves like the original lambda term does.
+The orginal lambda term lifts its first argument (think of it as reversing the order of its two arguments):
+
+ S(K(SI))(S(KK)I) X Y =
+ (K(SI))X ((S(KK)I) X) Y =
+ SI ((KK)X (IX)) Y =
+ SI (KX) Y =
+ IY (KX)Y =
+ Y X
+
+Viola: the combinator takes any X and Y as arguments, and returns Y applied to X.
+
+Back to linguistic applications: one consequence of the equivalence between the lambda calculus and combinatory
+logic is that anything that can be done by binding variables can just as well be done with combinators.
+This has given rise to a style of semantic analysis called Variable Free Semantics (in addition to
+Szabolcsi's papers, see, for instance,
+Pauline Jacobson's 1999 *Linguistics and Philosophy* paper, `Towards a variable-free Semantics').
+Somewhat ironically, reading strings of combinators is so difficult that most practitioners of variable-free semantics
+express there meanings using the lambda-calculus rather than combinatory logic; perhaps they should call their
+enterprise Free Variable Free Semantics.
+
+A philosophical application: Quine went through a phase in which he developed a variable free logic.
+
+
+
+
+
+
+
+
[Fussy notes: if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of S, K, and I (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In that intermediate stage, we have `\x.I`. It's possible to avoid this, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.]