X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week2.mdwn;h=bd86cbbf3a0cadaadc2f259b3602b5c04ff878c9;hp=ce2df76df252e30b20e142599925ed771e6097a2;hb=c595c61a3fc33a2343e3ef906a382652c655d518;hpb=6b354c56b73998dfcdd5be0f148ad24f1b9fed1d
diff --git a/week2.mdwn b/week2.mdwn
index ce2df76d..bd86cbbf 100644
--- a/week2.mdwn
+++ b/week2.mdwn
@@ -241,7 +241,7 @@ 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 Processs). Steedman attempts to build
-a syntax/semantics interface using a small number of combinators, including T = `\xy.yx`, B = `\fxy.f(xy)`,
+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.
@@ -318,6 +318,8 @@ will eta-reduce by n steps to:
\x. M
+When we add eta-reduction to our proof system, we end up reconstruing the meaning of `~~>` and `<~~>` and "normal form", all in terms that permit eta-reduction as well. Sometimes these expressions will be annotated to indicate whether only beta-reduction is allowed (~~>β
) or whether both beta- and eta-reduction is allowed (~~>βη
).
+
The logical system you get when eta-reduction is added to the proof system has the following property:
> if `M`, `N` are normal forms with no free variables, then M ≡ N
iff `M` and `N` behave the same with respect to every possible sequence of arguments.
@@ -328,7 +330,7 @@ That is, when `M` and `N` are (closed normal forms that are) syntactically disti
N L1 ... Ln x y ~~> y
-That is, closed normal forms that are not just beta-reduced but also fully eta-reduced, will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
+That is, closed beta-plus-eta-normal forms will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
So the proof theory with eta-reduction added is called "extensional," because its notion of normal form makes syntactic identity of closed normal forms coincide with extensional equivalence.