Douglas Hofstadter - I Am a Strange Loop

preceded by itself in quote marks yields a full sentence.

As you will note, Quine’s Quasi-Quip is certainly not a full sentence, for it has no grammatical subject (that is, “yields” has no subject); that’s why I gave it the prefix “Quasi”. But what if we were to put a noun at the head of the Quasi-Quip — say, the title “Professor Quine”? Then Quine’s Quasi-Quip will turn into a full sentence, so I’ll call it “Quine’s Quip”:

“Professor Quine” preceded by itself in quote marks yields a full sentence.

Here, the verb “yields” does have a subject — namely, Professor Quine’s title, modified by a trailing adjectival phrase that is six words long.

But what does Quine’s Quip mean ? In order to figure this out, we have to actually construct the entity that it’s talking about, which means we have to precede Professor Quine’s title by itself in quote marks. This gives us:

“Professor Quine” Professor Quine

The Quine’s Quip that we created a moment ago merely asserts (or rather, claims) that this somewhat silly phrase is a full sentence. Well, that claim is obviously false. The above phrase is not a full sentence; it doesn’t even contain a verb.

However, we arbitrarily used Professor Quine’s title when we could have used a million different things. Is there some other noun that we might place at the head of Quine’s Quasi-Quip that will make Quine’s Quip come out true ? What Gödel realized, and what Quine’s analogy helps to make clear, is that for this to happen, you have to use, as your subject of the verb “yields”, a subjectless sentence fragment.

What is an example of a subjectless sentence fragment? Well, just take any old sentence such as “Snow is white”, and cut off its subject. What you get is a subjectless sentence fragment: “is white”. So let’s use this as the noun to place in front of Quine’s Quasi-Quip:

“is white” preceded by itself in quote marks yields a full sentence.

This medium-sized mouthful makes a claim about a construction that we have yet to exhibit, and so let’s do so without further ado:

“is white” is white.

(I threw in the period for good measure, but let’s not quibble.)

Now what we have just produced certainly is a full sentence, because it has a verb (“is”), and that verb has a subject (the quoted phrase), and the whole thing makes sense. I’m not saying that it is true, mind you, for indeed it is blatantly false: “is white” is in fact black (although, to be fair, letters and words do contain some white space along with their black ink, otherwise we couldn’t read them). In any case, Quine’s Quasi-Quip when fed “is white” as its input yielded a full sentence, and that’s exactly what Quine’s Quip claimed. We’re definitely making headway.

The Trickiest Step

Our last devilish trick will be to use Quine’s Quasi-Quip itself as the noun to place at its head. Here, then, is Quine’s Quasi-Quip with a quoted copy of itself installed in front:

“preceded by itself in quote marks yields a full sentence”

preceded by itself in quote marks yields a full sentence.

What does this Quip claim? Well, first we have to determine what entity it is talking about, and that means we have to construct the analogue to “ ‘is white’ is white”. Well, in this case, the analogue is the following:

“preceded by itself in quote marks yields a full sentence”

preceded by itself in quote marks yields a full sentence.

I hope you are not lost at this point, for we really have hit the crux of the matter. Quine’s Quip turns out to be talking about a phrase that is identical to the Quip itself! It is claiming that something is a full sentence, and when you go about constructing that thing, it turns out to be Quine’s Quip itself. So Quine’s Quip talks about itself, claiming of itself that it is a full sentence (which it surely is, even though it is built out of two subjectless sentence fragments, one in quote marks and one not).

While you are pondering this, I will jump back to the source of it all, which was Gödel’s PM formula that talked about itself. The point is that Gödel numbers, since they can be used as names for formulas and can be inserted into formulas, are precisely analogous to quoted phrases. Now we have just seen that there is a way to use quotation marks and sentence fragments to make a full sentence that talks about itself (or if you prefer, a sentence that talks about another sentence, but one that is a clone to it, so that whatever is true of the one is true of the other).

Gödel, analogously, created a “subjectless formula fragment” (by which I mean a PM formula that is not about any specific integer, but just about some unspecified variable number x ). And then, making a move analogous to that of feeding Quine’s Quasi-Quip into itself (but in quotes), he took that formula fragment’s Gödel number k (which is a specific number, not a variable) and replaced the variable x by it, thus producing a formula (not just a fragment) that made a claim about a much larger integer, g. And g is the Gödel number of that very claim. And last but not least, the claim was not about whether the entity in question was a full sentence or not, but about whether the entity in question was a provable formula or not.

An Elephant in a Matchbox is Neither Fish Nor Fowl

I know this is a lot to swallow in one gulp, and so if it takes you several gulps (careful rereadings), please don’t feel discouraged. I’ve met quite a few sophisticated mathematicians who admit that they never understood this argument totally!

I think it would be helpful at this juncture to exhibit a kind of hybrid sentence that gets across the essential flavor of Gödel’s self-referential construction but that does so in Quinean terms — that is, using the ideas we’ve just been discussing. The hybrid sentence looks like this:

“when fed its own Gödel number yields a non-prim number”

when fed its own Gödel number yields a non-prim number.

The above sentence is neither fish nor fowl, for it is not a formula of Principia Mathematica but an English sentence, so of course it doesn’t have a Gödel number and it couldn’t possibly be a theorem (or a nontheorem) of PM. What a mixed metaphor!

And yet, mixed metaphor though it is, it still does a pretty decent job of getting across the flavor of the PM formula that Gödel actually concocted. You just have to keep in mind that using quote marks is a metaphor for taking Gödel numbers, so the upper line should be thought of as being a Gödel number ( k ) rather than as being a sentence fragment in quote marks. This means that metaphorically, the lower line (an English sentence fragment) has been fed its own Gödel number as its subject. Very cute!

I know that this is very tricky, so let me state it once again, slightly differently. Gödel asks you to imagine the formula that k stands for (that formula happens to contain the variable x ), and then to feed k into it (this means to replace the single letter x by the extremely long numeral k, thus giving you a much bigger formula than you started with), and to take the Gödel number of the result. That will be the number g, huger far than k — and lastly, Gödel asserts that this walloping number is not a prim number. If you’ve followed my hand-waving argument, you will agree that the full formula’s Gödel number ( g ) is not found explicitly inside the formula, but instead is very subtly described by the formula. The elephant’s DNA has been used to get a description of the entire elephant into the matchbox.