[[!toc]] #Recursion: fixed points in the lambda calculus## Sometimes when you type in a web search, Google will suggest alternatives. For instance, if you type in "Lingusitics", it will ask you "Did you mean Linguistics?". But the engineers at Google have added some playfulness to the system. For instance, if you search for "anagram", Google asks you "Did you mean: nag a ram?" And if you search for "recursion", Google asks: "Did you mean: recursion?" ##What is the "rec" part of "letrec" doing?## How could we compute the length of a list? Without worrying yet about what lambda-calculus implementation we're using for the list, the basic idea would be to define this recursively: > the empty list has length 0 > any non-empty list has length 1 + (the length of its tail) In OCaml, you'd define that like this: let rec get_length = fun lst -> if lst == [] then 0 else 1 + get_length (tail lst) in ... (* here you go on to use the function "get_length" *) In Scheme you'd define it like this: (letrec [(get_length (lambda (lst) (if (null? lst) 0 [+ 1 (get_length (cdr lst))] )) )] ... ; here you go on to use the function "get_length" ) Some comments on this: 1. `null?` is Scheme's way of saying `isempty`. That is, `(null? lst)` returns true (which Scheme writes as `#t`) iff `lst` is the empty list (which Scheme writes as `'()` or `(list)`). 2. `cdr` is function that gets the tail of a Scheme list. (By definition, it's the function for getting the second member of an ordered pair. It just turns out to return the tail of a list because of the particular way Scheme implements lists.) 3. I use `get_length` instead of the convention we've been following so far of hyphenated names, as in `make-list`, because we're discussing OCaml code here, too, and OCaml doesn't permit the hyphenated variable names. OCaml requires variables to always start with a lower-case letter (or `_`), and then continue with only letters, numbers, `_` or `'`. Most other programming languages are similar. Scheme is very relaxed, and permits you to use `-`, `?`, `/`, and all sorts of other crazy characters in your variable names. 4. I alternate between `[ ]`s and `( )`s in the Scheme code just to make it more readable. These have no syntactic difference. The main question for us to dwell on here is: What are the `let rec` in the OCaml code and the `letrec` in the Scheme code? Answer: These work like the `let` expressions we've already seen, except that they let you use the variable `get_length` *inside* the body of the function being bound to it---with the understanding that it will there refer to the same function that you're then in the process of binding to `get_length`. So our recursively-defined function works the way we'd expect it to. In OCaml: let rec get_length = fun lst -> if lst == [] then 0 else 1 + get_length (tail lst) in get_length [20; 30] (* this evaluates to 2 *) In Scheme: (letrec [(get_length (lambda (lst) (if (null? lst) 0 [+ 1 (get_length (cdr lst))] )) )] (get_length (list 20 30))) ; this evaluates to 2 If you instead use an ordinary `let` (or `let*`), here's what would happen, in OCaml: let get_length = fun lst -> if lst == [] then 0 else 1 + get_length (tail lst) in get_length [20; 30] (* fails with error "Unbound value length" *) Here's Scheme: (let* [(get_length (lambda (lst) (if (null? lst) 0 [+ 1 (get_length (cdr lst))] )) )] (get_length (list 20 30))) ; fails with error "reference to undefined identifier: get_length" Why? Because we said that constructions of this form: let get_length = A in B really were just another way of saying: (\get_length. B) A and so the occurrences of `get_length` in A *aren't bound by the `\get_length` that wraps B*. Those occurrences are free. We can verify this by wrapping the whole expression in a more outer binding of `get_length` to some other function, say the constant function from any list to the integer 99: let get_length = fun lst -> 99 in let get_length = fun lst -> if lst == [] then 0 else 1 + get_length (tail lst) in get_length [20; 30] (* evaluates to 1 + 99 *) Here the use of `get_length` in `1 + get_length (tail lst)` can clearly be seen to be bound by the outermost `let`. And indeed, if you tried to define `get_length` in the lambda calculus, how would you do it? \lst. (isempty lst) zero (add one (get_length (extract-tail lst))) We've defined all of `isempty`, `zero`, `add`, `one`, and `extract-tail` in earlier discussion. But what about `get_length`? That's not yet defined! In fact, that's the very formula we're trying here to specify. What we really want to do is something like this: \lst. (isempty lst) zero (add one (... (extract-tail lst))) where this very same formula occupies the `...` position: \lst. (isempty lst) zero (add one ( \lst. (isempty lst) zero (add one (... (extract-tail lst))) (extract-tail lst))) but as you can see, we'd still have to plug the formula back into itself again, and again, and again... No dice. So how could we do it? And how do OCaml and Scheme manage to do it, with their `let rec` and `letrec`? 1. OCaml and Scheme do it using a trick. Well, not a trick. Actually an impressive, conceptually deep technique, which we haven't yet developed. Since we want to build up all the techniques we're using by hand, then, we shouldn't permit ourselves to rely on `let rec` or `letrec` until we thoroughly understand what's going on under the hood. 2. If you tried this in Scheme: (define get_length (lambda (lst) (if (null? lst) 0 [+ 1 (get_length (cdr lst))] )) ) (get_length (list 20 30)) You'd find that it works! This is because `define` in Scheme is really shorthand for `letrec`, not for plain `let` or `let*`. So we should regard this as cheating, too. 3. In fact, it *is* possible to define the `get_length` function in the lambda calculus despite these obstacles. This depends on using the "version 3" implementation of lists, and exploiting its internal structure: that it takes a function and a base value and returns the result of folding that function over the list, with that base value. So we could use this as a definition of `get_length`: \lst. lst (\x sofar. successor sofar) zero What's happening here? We start with the value zero, then we apply the function `\x sofar. successor sofar` to the two arguments `xn` and `zero`, where `xn` is the last element of the list. This gives us `successor zero`, or `one`. That's the value we've accumuluted "so far." Then we go apply the function `\x sofar. successor sofar` to the two arguments `xn-1` and the value `one` that we've accumulated "so far." This gives us `two`. We continue until we get to the start of the list. The value we've then built up "so far" will be the length of the list. We can use similar techniques to define many recursive operations on lists and numbers. The reason we can do this is that our "version 3," fold-based implementation of lists, and Church's implementations of numbers, have a internal structure that *mirrors* the common recursive operations we'd use lists and numbers for. In a sense, the recursive structure of the `get_length` operation is built into the data structure we are using to represent the list. The non-recursive version of get_length exploits this embedding of the recursion into the data type. This is one of the themes of the course: using data structures to encode the state of some recursive operation. See discussions of the [[zipper]] technique, and [[defunctionalization]]. As we said before, it does take some ingenuity to define functions like `extract-tail` or `predecessor` for these implementations. However it can be done. (And it's not *that* difficult.) Given those functions, we can go on to define other functions like numeric equality, subtraction, and so on, just by exploiting the structure already present in our implementations of lists and numbers. With sufficient ingenuity, a great many functions can be defined in the same way. For example, the factorial function is straightforward. The function which returns the nth term in the Fibonacci series is a bit more difficult, but also achievable. ##Some functions require full-fledged recursive definitions## However, some computable functions are just not definable in this way. We can't, for example, define a function that tells us, for whatever function `f` we supply it, what is the smallest integer `x` where `f x` is `true`. (You may be thinking: but that smallest-integer function is not a proper algorithm, since it is not guaranteed to halt in any finite amount of time for every argument. This is the famous [[!wikipedia Halting problem]]. But the fact that an implementation may not terminate doesn't mean that such a function isn't well-defined. The point of interest here is that its definition requires recursion in the function definition.) Neither do the resources we've so far developed suffice to define the [[!wikipedia Ackermann function]]: A(m,n) = | when m == 0 -> n + 1 | else when n == 0 -> A(m-1,1) | else -> A(m-1, A(m,n-1)) A(0,y) = y+1 A(1,y) = 2+(y+3) - 3 A(2,y) = 2(y+3) - 3 A(3,y) = 2^(y+3) - 3 A(4,y) = 2^(2^(2^...2)) [where there are y+3 2s] - 3 ... Many simpler functions always *could* be defined using the resources we've so far developed, although those definitions won't always be very efficient or easily intelligible. But functions like the Ackermann function require us to develop a more general technique for doing recursion---and having developed it, it will often be easier to use it even in the cases where, in principle, we didn't have to. ##Using fixed-point combinators to define recursive functions## ###Fixed points### In general, we call a **fixed point** of a function f any value *x* such that f x is equivalent to *x*. For example, consider the squaring function `sqare` that maps natural numbers to their squares. `square 2 = 4`, so `2` is not a fixed point. But `square 1 = 1`, so `1` is a fixed point of the squaring function. There are many beautiful theorems guaranteeing the existence of a fixed point for various classes of interesting functions. For instance, imainge that you are looking at a map of Manhattan, and you are standing somewhere in Manhattan. The the [[!wikipedia Brouwer fixed point]] guarantees that there is a spot on the map that is directly above the corresponding spot in Manhattan. It's the spot where the blue you-are-here dot should be. Whether a function has a fixed point depends on the set of arguments it is defined for. For instance, consider the successor function `succ` that maps each natural number to its successor. If we limit our attention to the natural numbers, then this function has no fixed point. (See the discussion below concerning a way of understanding the successor function on which it does have a fixed point.) In the lambda calculus, we say a fixed point of an expression `f` is any formula `X` such that: X <~~> f X You should be able to immediately provide a fixed point of the identity combinator I. In fact, you should be able to provide a whole bunch of distinct fixed points. With a little thought, you should be able to provide a fixed point of the false combinator, KI. Here's how to find it: recall that KI throws away its first argument, and always returns I. Therefore, if we give it I as an argument, it will throw away the argument, and return I. So KII ~~> I, which is all it takes for I to qualify as a fixed point of KI. What about K? Does it have a fixed point? You might not think so, after trying on paper for a while. However, it's a theorem of the lambda calculus that every formula has a fixed point. In fact, it will have infinitely many, non-equivalent fixed points. And we don't just know that they exist: for any given formula, we can explicit define many of them. Yes, even the formula that you're using the define the successor function will have a fixed point. Isn't that weird? Think about how it might be true. We'll return to this point below. ###How fixed points help definie recursive functions### Recall our initial, abortive attempt above to define the `get_length` function in the lambda calculus. We said "What we really want to do is something like this: \list. if empty list then zero else add one (... (tail lst)) where this very same formula occupies the `...` position." Imagine replacing the `...` with some function that computes the length function. Call that function `length`. Then we have \list. if empty list then zero else add one (length (tail lst)) At this point, we have a definition of the length function, though it's not complete, since we don't know what value to use for the symbol `length`. Technically, it has the status of an unbound variable. Imagine now binding the mysterious variable: h := \length \list . if empty list then zero else add one (length (tail list)) Now we have no unbound variables, and we have complete non-recursive definitions of each of the other symbols. Let's call this function `h`. Then `h` takes an argument, and returns a function that accurately computes the length of a list---as long as the argument we supply is already the length function we are trying to define. (Dehydrated water: to reconstitute, just add water!) But this is just another way of saying that we are looking for a fixed point. Assume that `h` has a fixed point, call it `LEN`. To say that `LEN` is a fixed point means that h LEN <~~> LEN But this means that (\list . if empty list then zero else add one (LEN (tail list))) <~~> LEN So at this point, we are going to search for fixed point. The strategy we will present will turn out to be a general way of finding a fixed point for any lambda term. ##Deriving Y, a fixed point combinator## How shall we begin? Well, we need to find an argument to supply to `h`. The argument has to be a function that computes the length of a list. The function `h` is *almost* a function that computes the length of a list. Let's try applying `h` to itself. It won't quite work, but examining the way in which it fails will lead to a solution. h h <~~> \list . if empty list then zero else 1 + h (tail list) There's a problem. The diagnosis is that in the subexpression `h (tail list)`, we've applied `h` to a list, but `h` expects as its first argument the length function. So let's adjust h, calling the adjusted function H: H = \h \list . if empty list then zero else one plus ((h h) (tail list)) This is the key creative step. Since `h` is expecting a length-computing function as its first argument, the adjustment tries supplying the closest candidate avaiable, namely, `h` itself. We now reason about `H`. What exactly is H expecting as its first argument? Based on the excerpt `(h h) (tail l)`, it appears that `H`'s argument, `h`, should be a function that is ready to take itself as an argument, and that returns a function that takes a list as an argument. `H` itself fits the bill: H H <~~> (\h \list . if empty list then zero else 1 + ((h h) (tail list))) H <~~> \list . if empty list then zero else 1 + ((H H) (tail list)) == \list . if empty list then zero else 1 + ((\list . if empty list then zero else 1 + ((H H) (tail list))) (tail list)) <~~> \list . if empty list then zero else 1 + (if empty (tail list) then zero else 1 + ((H H) (tail (tail list)))) We're in business! How does the recursion work? We've defined `H` in such a way that `H H` turns out to be the length function. In order to evaluate `H H`, we substitute `H` into the body of the lambda term. Inside the lambda term, once the substitution has occurred, we are once again faced with evaluating `H H`. And so on. We've got the infinite regress we desired, defined in terms of a finite lambda term with no undefined symbols. Since `H H` turns out to be the length function, we can think of `H` by itself as half of the length function (which is why we called it `H`, of course). Given the implementation of addition as function application for Church numerals, this (H H) is quite literally H + H. Can you think up a recursion strategy that involves "dividing" the recursive function into equal thirds `T`, such that the length function <~~> T T T? We've starting with a particular recursive definition, and arrived at a fixed point for that definition. What's the general recipe? 1. Start with any recursive definition `h` that takes itself as an arg: `h := \fn ... fn ...` 2. Next, define `H := \f . h (f f)` 3. Then compute `H H = ((\f . h (f f)) (\f . h (f f)))` 4. That's the fixed point, the recursive function we're trying to define So here is a general method for taking an arbitrary h-style recursive function and returning a fixed point for that function: Y := \h. ((\f.h(ff))(\f.h(ff))) Test: Yh == ((\f.h(ff))(\f.h(ff))) <~~> h((\f.h(ff))(\f.h(ff))) == h(Yh) That is, Yh is a fixed point for h. Works! ##What is a fixed point for the successor function?## Well, you might think, only some of the formulas that we might give to the `successor` as arguments would really represent numbers. If we said something like: successor make-pair who knows what we'd get back? Perhaps there's some non-number-representing formula such that when we feed it to `successor` as an argument, we get the same formula back. Yes! That's exactly right. And which formula this is will depend on the particular way you've implemented the successor function. Moreover, the recipes that enable us to name fixed points for any given formula aren't *guaranteed* to give us *terminating* fixed points. They might give us formulas X such that neither `X` nor `f X` have normal forms. (Indeed, what they give us for the square function isn't any of the Church numerals, but is rather an expression with no normal form.) However, if we take care we can ensure that we *do* get terminating fixed points. And this gives us a principled, fully general strategy for doing recursion. It lets us define even functions like the Ackermann function, which were until now out of our reach. It would also let us define arithmetic and list functions on the "version 1" and "version 2" implementations, where it wasn't always clear how to force the computation to "keep going." OK, so how do we make use of this? Many fixed-point combinators have been discovered. (And some fixed-point combinators give us models for building infinitely many more, non-equivalent fixed-point combinators.) Two of the simplest:
``````Θ′ ≡ (\u f. f (\n. u u f n)) (\u f. f (\n. u u f n))
Y′ ≡ \f. (\u. f (\n. u u n)) (\u. f (\n. u u n))``````
`Θ′` has the advantage that `f (Θ′ f)` really *reduces to* `Θ′ f`. Whereas `f (Y′ f)` is only *convertible with* `Y′ f`; that is, there's a common formula they both reduce to. For most purposes, though, either will do. You may notice that both of these formulas have eta-redexes inside them: why can't we simplify the two `\n. u u f n` inside `Θ′` to just `u u f`? And similarly for `Y′`? Indeed you can, getting the simpler:
``````Θ ≡ (\u f. f (u u f)) (\u f. f (u u f))
Y ≡ \f. (\u. f (u u)) (\u. f (u u))``````
I stated the more complex formulas for the following reason: in a language whose evaluation order is *call-by-value*, the evaluation of `Θ (\self. BODY)` and `Y (\self. BODY)` will in general not terminate. But evaluation of the eta-unreduced primed versions will. Of course, if you define your `\self. BODY` stupidly, your formula will never terminate. For example, it doesn't matter what fixed point combinator you use for `Ψ` in:
``Ψ (\self. \n. self n)``
When you try to evaluate the application of that to some argument `M`, it's going to try to give you back: (\n. self n) M where `self` is equivalent to the very formula `\n. self n` that contains it. So the evaluation will proceed: (\n. self n) M ~~> self M ~~> (\n. self n) M ~~> self M ~~> ... You've written an infinite loop! However, when we evaluate the application of our:
``Ψ (\self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) ))``
to some list `L`, we're not going to go into an infinite evaluation loop of that sort. At each cycle, we're going to be evaluating the application of: \lst. (isempty lst) zero (add one (self (extract-tail lst))) to *the tail* of the list we were evaluating its application to at the previous stage. Assuming our lists are finite (and the implementations we're using don't permit otherwise), at some point one will get a list whose tail is empty, and then the evaluation of that formula to that tail will return `zero`. So the recursion eventually bottoms out in a base value. ##Fixed-point Combinators Are a Bit Intoxicating## ![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. I used `Ψ` above to stand in for an arbitrary fixed-point combinator. I don't know of any broad conventions for this. But this seems a useful one. As we said, there are many other fixed-point combinators as well. For example, Jan Willem Klop pointed out that if we define `L` to be: \a b c d e f g h i j k l m n o p q s t u v w x y z r. (r (t h i s i s a f i x e d p o i n t c o m b i n a t o r)) then this is a fixed-point combinator: L L L L L L L L L L L L L L L L L L L L L L L L L L ##Watching Y in action## For those of you who like to watch ultra slow-mo movies of bullets piercing apples, here's a stepwise computation of the application of a recursive function. We'll use a function `sink`, which takes one argument. If the argument is boolean true (i.e., `\x y.x`), it returns itself (a copy of `sink`); if the argument is boolean false (`\x y. y`), it returns `I`. That is, we want the following behavior: sink false ~~> I sink true false ~~> I sink true true false ~~> I sink true true true false ~~> I So we make `sink = Y (\f b. b f I)`: 1. sink false 2. Y (\fb.bfI) false 3. (\f. (\h. f (h h)) (\h. f (h h))) (\fb.bfI) false 4. (\h. [\fb.bfI] (h h)) (\h. [\fb.bfI] (h h)) false 5. [\fb.bfI] ((\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))) false 6. (\b.b[(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))]I) false 7. false [(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))] I -------------------------------------------- 8. I So far so good. The crucial thing to note is that as long as we always reduce the outermost redex first, we never have to get around to computing the underlined redex: because `false` ignores its first argument, we can throw it away unreduced. Now we try the next most complex example: 1. sink true false 2. Y (\fb.bfI) true false 3. (\f. (\h. f (h h)) (\h. f (h h))) (\fb.bfI) true false 4. (\h. [\fb.bfI] (h h)) (\h. [\fb.bfI] (h h)) true false 5. [\fb.bfI] ((\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))) true false 6. (\b.b[(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))]I) true false 7. true [(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))] I false 8. [(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))] false We've now arrived at line (4) of the first computation, so the result is again I. You should be able to see that `sink` will consume as many `true`s as we throw at it, then turn into the identity function after it encounters the first `false`. The key to the recursion is that, thanks to Y, the definition of `sink` contains within it the ability to fully regenerate itself as many times as is necessary. The key to *ending* the recursion is that the behavior of `sink` is sensitive to the nature of the input: if the input is the magic function `false`, the self-regeneration machinery will be discarded, and the recursion will stop. That's about as simple as recursion gets. ##Application to the truth teller/liar paradoxes## ###Base cases, and their lack### As any functional programmer quickly learns, writing a recursive function divides into two tasks: figuring out how to handle the recursive case, and remembering to insert a base case. The interesting and enjoyable part is figuring out the recursive pattern, but the base case cannot be ignored, since leaving out the base case creates a program that runs forever. For instance, consider computing a factorial: `n!` is `n * (n-1) * (n-2) * ... * 1`. The recursive case says that the factorial of a number `n` is `n` times the factorial of `n-1`. But if we leave out the base case, we get 3! = 3 * 2! = 3 * 2 * 1! = 3 * 2 * 1 * 0! = 3 * 2 * 1 * 0 * -1! ... That's why it's crucial to declare that 0! = 1, in which case the recursive rule does not apply. In our terms, fac = Y (\fac n. iszero n 1 (fac (predecessor n))) If `n` is 0, `fac` reduces to 1, without computing the recursive case. 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: the truth-teller paradox (and related paradoxes). (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`. 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
(\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... 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 it occurs, (2) will denote a fixed point for `\f.neg f`, or `\f l r. f r l`, which is the `C` combinator. So in such a context, (2) might denote Y C (\f. (\h. f (h h)) (\h. f (h h))) I (\h. C (h h)) (\h. C (h h))) C ((\h. C (h h)) (\h. C (h h))) C (C ((\h. C (h h))(\h. C (h h)))) C (C (C ((\h. C (h h))(\h. C (h h))))) ... 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 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.