+Here's an example of the translation:
+
+ [\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.
+
+ Quine, Willard. 1960. Variables explained away. {\it Proceedings of
+ the American Philosophical Society}. Volume 104: 343--347. Also in
+ W.~V.~Quine. 1960. {\it Selected Logical Papers}. Random House: New
+ York. 227--235.
+
+The reason this was important to Quine is similar to the worries that Jim was talking about
+in the first class in which using non-referring expressions such as Santa Clause might commit
+one to believing in non-existant things. Quine's slogan was that `to be is to be the value of a variable'.
+What this was supposed to mean is that if and only if an object could serve as the value of some variable, we
+are committed to recognizing the existence of that object in our ontology.
+Obviously, if there ARE no variables, this slogan has to be rethought.
+
+Quine did not appear to appreciate that Shoenfinkel had already invented combinatory logic, though
+he later wrote an introduction to Shoenfinkel's key paper reprinted in Jean
+van Heijenoort (ed) 1967 *From Frege to Goedel,
+ a source book in mathematical logic, 1879--1931*.
+Cresswell has also developed a variable-free approach of some philosophical and linguistic interest
+in two books in the 1990's.