-
-Curry originally called `Y` the "paradoxical" combinator, and discussed
-it in connection with certain well-known paradoxes from the philosophy
-literature. The truth-teller paradox has the flavor of a recursive
-function without a base case:
-
-(1) This sentence is true.
-
-If we assume that the complex demonstrative "this sentence" can refer
-to (1), then the proposition expressed by (1) will be true just in
-case the thing referred to by *this sentence* is true. Thus (1) will
-be true just in case (1) is true, and (1) is true just in case (1) is
-true, and so on. If (1) is true, then (1) is true; but if (1) is not
-true, then (1) is not true.
-
-Without pretending to give a serious analysis of the paradox, let's
-assume that sentences can have for their meaning boolean functions
-like the ones we have been working with here. Then the sentence *John
-is John* might denote the function `\x y. x`, our `true`.
-
-<!-- Jim says: I haven't yet followed the next chunk to my satisfaction -->
-
-Then (1) denotes a function from whatever the referent of *this
-sentence* is to a boolean. So (1) denotes `\f. f true false`, where
-the argument `f` is the referent of *this sentence*. Of course, if
-`f` is a boolean, `f true false <~~> f`, so for our purposes, we can
-assume that (1) denotes the identity function `I`.
-
-If we use (1) in a context in which *this sentence* refers to the
-sentence in which the demonstrative occurs, then we must find a
-meaning `m` such that `I m = I`. But since in this context `m` is the
-same as the meaning `I`, so we have `m = I m`. In other words, `m` is
-a fixed point for the denotation of the sentence (when used in the
-appropriate context).
-
-That means that in a context in which *this sentence* refers to the
-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 ≡
- (\h. (\u. h (u u)) (\u. h (u u))) I ~~>
- (\u. I (u u)) (\u. I (u u))) ~~>
- (\u. (u u)) (\u. (u u))) ≡
- ω ω
- Ω
-
-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...
-
-What about the liar paradox?
-
-(2) This sentence is false.
-
-Used in a context in which *this sentence* refers to the utterance of
-(2) in which that noun phrase occurs, (2) will denote a fixed point for `\f. neg f`,
-or `\f y n. f n y`, which is the `C` combinator. So in such a
-context, (2) might denote
-
- Y C
- (\h. (\u. h (u u)) (\u. h (u u))) C
- (\u. C (u u)) (\u. C (u u)))
- C ((\u. C (u u)) (\u. C (u u)))
- C (C ((\u. C (u u)) (\u. C (u u))))
- C (C (C ((\u. C (u u)) (\u. C (u u)))))
- ...
-
-And infinite sequence of `C`s, each one negating the remainder of the
-sequence. Yep, that feels like a reasonable representation of the
-liar paradox.
-
-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 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, versus other fixed point combinators).
-
-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 `\x. mul x x` has `0` as a fixed point,
-since `square 0 <~~> 0`, and `1` as a fixed point, since `square 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...
-
-<!-- Jim says: haven't made clear how we got from the self-referential (3) to I. -->
-
-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.