“Is false when preceded by its quotation” is false when preceded by its quotation.
I feel stupid for this, but I can’t quite wrap my head around it. Can somebody please ELI5? (I’m asking LW because it seems to have more than its fair share of math & logic whizzes.)
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.)
The grammar of the sentence is a bit hard to follow. When I am presenting this paradox to friends (I have interesting friends), I hand them a piece of paper with the following words on it:
Take another piece of paper and copy these words:
“Take another piece of paper and copy these words: “QQQ” Then replace the three consecutive capital letters with another copy of those words. The resulting paragraph will make a false claim.”
Then replace the three consecutive capital letters with another copy of those words. The resulting paragraph will make a false claim.
I urge you to carry out the task. You should wind up with a paper that has the exact same words on it as the paper I gave you.
If you believe that the statement on my paper is true, then you should believe that the statement on your paper is false, and vice versa. Yet they are the same statement! Assuming that you think truth or falsehood is a property of grammatical sentences, independent of where they are written, this should bother you. Moreover, unlike the standard liar paradox, the paper I gave never talks about itself, it only talks about a message you will write on some other piece of paper (which does not, in turn, talk about the original message) when you perform some simple typographical operations.
Quine constructed this example to demonstrate the sort of subtleties that come up in order to invent a mathematical formalism that can talk about truth, and can talk about manipulating symbols, without bringing in the liar paradox. (To learn how this problem is solved, take a course on mathematical logic and Goedel’s theorem.)
I suppse the intention is for this to be another version of This sentence is false, but it fails.
The sentence
(1) “Is false when preceded by its quotation” is false when preceded by its quotation.
says that the string “Is false when preceded by its quotation” has the property of being false when preceded by its quotation. The sentence is, in fact, that string preceded by its own quotation. However, there is no paradox. The whole sentence (1) can be true, and then the italicised occurrence of the string “Is false when preceded by its quotation” is false. No problem there. Incidentally, it’s dubious to ascribe “is false when preceded by its quotation” a truth-value anyway, since it’s not a sentence, but merely a verb-phrase.
One could change it to “yields falsehood when predicated of its own quotation”. Then you get the sentence “Yields falsehood when predicated of its quotation” yields falsehood when predicated of its quotation, which looks more paradoxical. However, I’m not entirely clear that it really is a paradox, either. There might be some confusion there between predicates and strings that refer to predicates. It depends on how quotation in natural language really works...
“Is false when preceded by its quotation” is false when preceded by its quotation.
I feel stupid for this, but I can’t quite wrap my head around it. Can somebody please ELI5? (I’m asking LW because it seems to have more than its fair share of math & logic whizzes.)
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.)
The grammar of the sentence is a bit hard to follow. When I am presenting this paradox to friends (I have interesting friends), I hand them a piece of paper with the following words on it:
I urge you to carry out the task. You should wind up with a paper that has the exact same words on it as the paper I gave you.
If you believe that the statement on my paper is true, then you should believe that the statement on your paper is false, and vice versa. Yet they are the same statement! Assuming that you think truth or falsehood is a property of grammatical sentences, independent of where they are written, this should bother you. Moreover, unlike the standard liar paradox, the paper I gave never talks about itself, it only talks about a message you will write on some other piece of paper (which does not, in turn, talk about the original message) when you perform some simple typographical operations.
Quine constructed this example to demonstrate the sort of subtleties that come up in order to invent a mathematical formalism that can talk about truth, and can talk about manipulating symbols, without bringing in the liar paradox. (To learn how this problem is solved, take a course on mathematical logic and Goedel’s theorem.)
Have you read the Wikipedia article?
Yes, and I’m still not getting it. Hence the stupid questions thread.
I suppse the intention is for this to be another version of This sentence is false, but it fails.
The sentence
(1) “Is false when preceded by its quotation” is false when preceded by its quotation.
says that the string “Is false when preceded by its quotation” has the property of being false when preceded by its quotation. The sentence is, in fact, that string preceded by its own quotation. However, there is no paradox. The whole sentence (1) can be true, and then the italicised occurrence of the string “Is false when preceded by its quotation” is false. No problem there. Incidentally, it’s dubious to ascribe “is false when preceded by its quotation” a truth-value anyway, since it’s not a sentence, but merely a verb-phrase.
One could change it to “yields falsehood when predicated of its own quotation”. Then you get the sentence “Yields falsehood when predicated of its quotation” yields falsehood when predicated of its quotation, which looks more paradoxical. However, I’m not entirely clear that it really is a paradox, either. There might be some confusion there between predicates and strings that refer to predicates. It depends on how quotation in natural language really works...