+ [\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 (KXY) ~~>
+ Y X
+
+Voilà: the combinator takes any X and Y as arguments, and returns Y applied to X.
+
+One very nice property of combinatory logic is that there is no need to worry about alphabetic variance, or
+variable collision---since there are no (bound) variables, there is no possibility of accidental variable capture,
+and so reduction can be performed without any fear of variable collision. We haven't mentioned the intricacies of
+alpha equivalence or safe variable substitution, but they are in fact quite intricate. (The best way to gain
+an appreciation of that intricacy is to write a program that performs lambda reduction.)
+
+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 their meanings using the lambda-calculus rather than combinatory logic; perhaps they should call their
+enterprise Free Variable Free Semantics.
+
+A philosophical connection: Quine went through a phase in which he developed a variable free logic.
+
+ Quine, Willard. 1960. "Variables explained away" <cite>Proceedings of the American Philosophical Society</cite>. Volume 104: 343--347. Also in W. V. Quine. 1960. <cite>Selected Logical Papers</cite>. 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 Claus 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 <cite>From Frege to Goedel, a source book in mathematical logic, 1879--1931</cite>.
+
+Cresswell has also developed a variable-free approach of some philosophical and linguistic interest
+in two books in the 1990's.
+
+A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is
+from combinatory logic (see especially his 2000 book, <cite>The Syntactic Processs</cite>). Steedman attempts to build
+a syntax/semantics interface using a small number of combinators, including T ≡ `\xy.yx`, B ≡ `\fxy.f(xy)`,
+and our friend S. Steedman used Smullyan's fanciful bird
+names for the combinators, Thrush, Bluebird, and Starling.
+
+Many of these combinatory logics, in particular, the SKI system,
+are Turing complete. In other words: every computation we know how to describe can be represented in a logical system consisting of only a single primitive operation!