So first of all, a purely syntactic remark: this involves a rather unnatural construction, taking “X’s quotation” to mean what you get by putting X in quotation marks. So far as I know, no one ever uses the word “quotation” in this way except when talking about Quine’s construction (i.e., the thing we’re talking about now). OK, let’s proceed.
The version of this I’ve seen is slightly different (and avoids Creutzer’s complaint):
“yields a falsehood when preceded by its quotation” yields a falsehood when preceded by its quotation.
So, let Q be the sentence-fragment “yields a falsehood when preceded by its quotation”. Then for any sentence-fragment R we can construct a sentence [“R” Q], and what it says that [“R” R] is false.
E.g., if you say [“foo” yields a falsehood when preceded by its quotation”] you’re saying that [“foo” foo] is a falsehood. Now, of course in this particular case that’s wrong because [“foo” foo] is just nonsense, not a falsehood. So [“foo” yields a falsehood when preceded by its quotation] is wrong.
Or consider [“is a sentence fragment” is a sentence fragment]. That’s true, so [“is a sentence fragment” yields a falsehood when preceded by its quotation] is wrong.
So, in general, [“R” Q] is a sentence—call it S—saying what you get when you construct the possibly-a-sentence [“R” R] -- call that T—and in particular claiming that it’s a falsehood. That is: S says that T is false.
Now, what happens when you let R=Q? Well, in that case S and T are both [“Q” Q] or, writing it out in full, [“yields a falsehood when preceded by its quotation” yields a falsehood when preceded by its quotation]. S says that T is false, but in this case S=T, so S says that S is false.
In other words, it’s an Epimenides-style paradox but without explicit self-reference. Instead we have a sort of self-reference by construction: we’ve got a sentence that says “Build a sentence according to such-and-such a recipe; the result is false”, and it just so happens that when you do what it says the sentence you get is that sentence.
This is quite closely analogous to how some metamathematical proofs work—e.g., Goedel’s theorem is proved by constructing a mathematical proposition S that in some sense says “S is not provable”, and Tarski’s theorem by constructing (assuming you have a way of expressing “is true”, which is what we’re trying to prove is impossible) an S that says “S is not true”. (Note: Everything in this paragraph is at best an approximation to the truth, with the possible exception of this parenthetical remark.)
So first of all, a purely syntactic remark: this involves a rather unnatural construction, taking “X’s quotation” to mean what you get by putting X in quotation marks. So far as I know, no one ever uses the word “quotation” in this way except when talking about Quine’s construction (i.e., the thing we’re talking about now). OK, let’s proceed.
The version of this I’ve seen is slightly different (and avoids Creutzer’s complaint):
So, let Q be the sentence-fragment “yields a falsehood when preceded by its quotation”. Then for any sentence-fragment R we can construct a sentence [“R” Q], and what it says that [“R” R] is false.
E.g., if you say [“foo” yields a falsehood when preceded by its quotation”] you’re saying that [“foo” foo] is a falsehood. Now, of course in this particular case that’s wrong because [“foo” foo] is just nonsense, not a falsehood. So [“foo” yields a falsehood when preceded by its quotation] is wrong.
Or consider [“is a sentence fragment” is a sentence fragment]. That’s true, so [“is a sentence fragment” yields a falsehood when preceded by its quotation] is wrong.
So, in general, [“R” Q] is a sentence—call it S—saying what you get when you construct the possibly-a-sentence [“R” R] -- call that T—and in particular claiming that it’s a falsehood. That is: S says that T is false.
Now, what happens when you let R=Q? Well, in that case S and T are both [“Q” Q] or, writing it out in full, [“yields a falsehood when preceded by its quotation” yields a falsehood when preceded by its quotation]. S says that T is false, but in this case S=T, so S says that S is false.
In other words, it’s an Epimenides-style paradox but without explicit self-reference. Instead we have a sort of self-reference by construction: we’ve got a sentence that says “Build a sentence according to such-and-such a recipe; the result is false”, and it just so happens that when you do what it says the sentence you get is that sentence.
This is quite closely analogous to how some metamathematical proofs work—e.g., Goedel’s theorem is proved by constructing a mathematical proposition S that in some sense says “S is not provable”, and Tarski’s theorem by constructing (assuming you have a way of expressing “is true”, which is what we’re trying to prove is impossible) an S that says “S is not true”. (Note: Everything in this paragraph is at best an approximation to the truth, with the possible exception of this parenthetical remark.)