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. As we said, the most common system uses S, K, and I as defined here.
###The equivalence of the untyped lambda calculus and combinatory logic###
IY (KX)Y =
Y X
-Viola: the combinator takes any X and Y as arguments, and returns Y applied to 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,
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').
+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
+express their 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.
+A philosophical connection: 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
+ 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 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'.
+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 *From Frege to Goedel,
- a source book in mathematical logic, 1879--1931*.
+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, *The Syntactic Process*). Steedman attempts to build
-a syntax/semantics interface using a small number of combinators, including T = \xy.yx, B = \fxy.f(xy),
+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.