sentence is about a shorter sentence embedded inside it, in quote marks. And changing “five” to “ten” still won’t make it refer to itself; all that this simple act does is to turn my sentence, which was true, into a false one. Take a look:
The sentence “This sentence has ten words” has ten words.
This sentence is false. And more importantly, it’s still merely about a shorter sentence embedded inside itself. As you see, so far we are not yet very close to having devised a sentence that talks about itself.
The problem is that anything I put inside quote marks will necessarily be shorter than the entire sentence of which it is a part. This is trivially obvious, and in fact it is an exact linguistic analogue to the stumbling block of trying to stick a formula’s own Godel number directly inside the formula itself. An elephant will not fit inside a matchbox! On the other hand, an elephant’s DNA will easily fit inside a matchbox…
And indeed, just as DNA is a description of an elephant rather than the elephant itself, so there is a way of getting around the obstacle by using a description of the huge number rather than the huge number itself. (To be slightly more precise, we can use a concise symbolic description instead of using a huge numeral.) Godel discovered this trick, and although it is quite subtle, Quine’s analogy makes it fairly easy to understand. Look at the following sentence fragment, which I’ll call “Quine’s Quasi-Quip”:
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 Godel 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 Godel’s PM formula that talked about itself. The point is that Godel 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).
Godel, 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 Godel 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 Godel 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 Godel’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 Godel number yields a non-prim number”
when fed its own Godel number yields a non-prim number.
The above sentence is neither fish nor fowl, for it is not a formula of Principia
