week3 tweak

[lambda.git] / week2.mdwn

diff --git a/week2.mdwn b/week2.mdwn
index ce2df76..bd86cbb 100644 (file)
--- 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, <cite>The Syntactic Processs</cite>).  Steedman attempts to build
 
 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)`,
+a syntax/semantics interface using a small number of combinators, including T &equiv; `\xy.yx`, B &equiv; `\fxy.f(xy)`,

 and our friend S.  Steedman used Smullyan's fanciful bird 
 names for the combinators, Thrush, Bluebird, and Starling.
 
 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
 
 
        \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 (<code>~~><sub>&beta;</sub></code>) or whether both beta- and eta-reduction is allowed (<code>~~><sub>&beta;&eta;</sub></code>).
+

 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 <code>M &equiv; N</code> iff `M` and `N` behave the same with respect to every possible sequence of arguments.
 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 <code>M &equiv; N</code> 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 L<sub>1</sub> ... L<sub>n</sub> x y ~~> y
 </code></pre>
 
 N L<sub>1</sub> ... L<sub>n</sub> x y ~~> y
 </code></pre>
 

-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.
 
 
 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.