X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week3.mdwn;h=a55659cb97b6e0f12616770115fc8680eb2ed821;hp=653b3076da7cd8a6f25afeea244bb283f63ecb3e;hb=cd20a0a226f35177c21ef48bcabfc59316e3e489;hpb=420a18f740f0c7f056c40d871595f42a0865b6c4

diff --git a/week3.mdwn b/week3.mdwn
index 653b3076..a55659cb 100644
--- a/week3.mdwn
+++ b/week3.mdwn
@@ -211,7 +211,8 @@ Instead of writing out a long formula twice, we could write:
 
 and the initial `(\x. x x)` is just what we earlier called the <code>&omega;</code> combinator (lower-case omega, not the non-terminating <code>&Omega;</code>). So the self-application of `H` can be written:
 
-<pre><code>&omega; (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))</code></pre>
+<pre><code>&omega; (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))
+</code></pre>
 
 and this will indeed implement the recursive function we couldn't earlier figure out how to define.
 
@@ -544,18 +545,17 @@ sentence in which it occurs, the sentence denotes a fixed point for
 the identity function.  Here's a fixed point for the identity
 function:
 
-<pre>
-    Y I
-    (\f. (\h. f (h h)) (\h. f (h h))) I
-    (\h. I (h h)) (\h. I (h h)))
-    (\h. (h h)) (\h. (h h)))
-    &omega; &omega;
-    &Omega
-</pre>
+<pre><code>Y I
+(\f. (\h. f (h h)) (\h. f (h h))) I
+(\h. I (h h)) (\h. I (h h)))
+(\h. (h h)) (\h. (h h)))
+&omega; &omega;
+&Omega
+</code></pre>
 
 Oh.  Well!  That feels right.  The meaning of *This sentence is true*
 in a context in which *this sentence* refers to the sentence in which
-it occurs is &Omega;, our prototypical infinite loop...
+it occurs is <code>&Omega;</code>, our prototypical infinite loop...
 
 What about the liar paradox?
 
@@ -582,3 +582,40 @@ See Barwise and Etchemendy's 1987 OUP book, [The Liar: an essay on
 truth and circularity](http://tinyurl.com/2db62bk) for an approach
 that is similar, but expressed in terms of non-well-founded sets
 rather than recursive functions.
+
+HOWEVER, you should be cautious about feeling too comfortable with
+these results.  Thinking again of the truth-teller paradox, yes,
+<code>&omega;</code> is *a* fixed point for `I`, and perhaps it has
+some a privileged status among all the fixed points for `I`, being the
+one delivered by Y and all.
+
+But one could ask: look, literally every formula is a fixed point for
+`I`, since
+
+    X <~~> I X
+
+for any choice of X whatsoever.
+
+So the Y combinator is only guaranteed to give us one fixed point out
+of infinitely many---and not always the intuitively most useful
+one. (For instance, the squaring function has zero as a fixed point,
+since 0 * 0 = 0, and 1 as a fixed point, since 1 * 1 = 1, but `Y
+(\x. mul x x)` doesn't give us 0 or 1.) So why in the reasoning we've
+just gone through should we be reaching for just this fixed point at
+just this juncture?
+
+One obstacle to thinking this through is the fact that a sentence
+normally has only two truth values.  We might consider instead a noun
+phrase such as 
+
+(3)  the entity that this noun phrase refers to
+
+The reference of (3) depends on the reference of the embedded noun
+phrase *this noun phrase*.  It's easy to see that any object is a
+fixed point for this referential function: if this pen cap is the
+referent of *this noun phrase*, then it is the referent of (3), and so
+for any object.
+
+Ultimately, in the context of this course, these paradoxes are more
+useful as a way of gaining leverage on the concepts of fixed points
+and recursion, rather than the other way around.