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 ≡ `\xy.yx`, B ≡ `\fxy.f(xy)`,
and our friend S. Steedman used Smullyan's fanciful bird
names for the combinators, Thrush, Bluebird, and Starling.
\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>β</sub></code>) or whether both beta- and eta-reduction is allowed (<code>~~><sub>βη</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 ≡ N</code> iff `M` and `N` behave the same with respect to every possible sequence of arguments.
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.