I Am a Strange Loop - Douglas R. Hofstadter [89]
Gödel’s Elephant-in-Matchbox Trick via Quine’s Analogy
Suppose you wanted to write a sentence in English that talks about itself without using the phrase “this sentence”. You would probably find the challenge pretty tricky, because you’d have to actually describe the sentence inside itself, using quoted words and phrases. For example, consider this first (somewhat feeble) attempt:
The sentence “This sentence has five words” has five words.
Now what I’ve just written (and you’ve just read) is a sentence that is true, but unfortunately it’s not about itself. After all, the full thing contains ten words, as well as some quotation marks. This 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 Gödel 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.) Gödel 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 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