`ω`

combinator (lower-case omega, not the non-terminating `Ω`

). So the self-application of `H` can be written:
-`ω (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))`

+```
ω (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))
+
```

and this will indeed implement the recursive function we couldn't earlier figure out how to define.
@@ -414,7 +424,7 @@ to *the tail* of the list we were evaluating its application to at the previous
##Fixed-point Combinators Are a Bit Intoxicating##
-![tatoo](/y-combinator.jpg)
+![tatoo](/y-combinator-fixed.jpg)
There's a tendency for people to say "Y-combinator" to refer to fixed-point combinators generally. We'll probably fall into that usage ourselves. Speaking correctly, though, the Y-combinator is only one of many fixed-point combinators.
@@ -544,18 +554,17 @@ sentence in which it occurs, the sentence denotes a fixed point for
the identity function. Here's a fixed point for the identity
function:
-- 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))) - ω ω - &Omega -+

```
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)))
+ω ω
+&Omega
+
```

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 Ω, our prototypical infinite loop...
+it occurs is `Ω`

, our prototypical infinite loop...
What about the liar paradox?
@@ -582,3 +591,50 @@ 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,
+`Ω`

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 (though it is not obvious why Y should have
+any special status).
+
+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 with respect to the
+truth-teller paradox, 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.
+
+The chameleon nature of (3), by the way (a description that is equally
+good at describing any object), makes it particularly well suited as a
+gloss on pronouns such as *it*. In the system of
+[Jacobson 1999](http://www.springerlink.com/content/j706674r4w217jj5/),
+pronouns denote (you guessed it!) identity functions...
+
+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.