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+++ b/week3.mdwn
@@ -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.