Keith Devens .com

Ha! I'd be interested to know what he says, but I think it will be too long and weird for you to really want to type about later.

Remembering back to my Philosophy of Logic class, my prof. DID say that there's only a paradox if you consider the sentence to have a truth value (ie, that it is a proposition and not just a random sentence.)

my prof. DID say that there's only a paradox if you consider the sentence to have a truth value

Well, that's good to know. This problem seems to be a particularly easy one. There are much harder problems, such as the "The king of France is bald" problem. Is that statement false? If it's false you should be able to get a true statement by negating it ("The king of France is not bald"), but that's not true either because there's no king of France. In fact, there's a whole question of what it means when you use a definite article when referring to something that doesn't exist.

I'm not quite sure what to make of all that. But at least I have the "This sentence is false" stuff figured out.

On the King of France, you get into problems with "equivalent rewrites":

"The king of France is bald." -> "There exists a king of France, and he is bald."

Are the sentences equivalent? Maybe! What about:

-> "If there exists a king of France, he is bald."

Another possible rewrite, but with a different truth table.

This sentence is about truth.

"""Truth and falsehood simply don't apply to sentences, they apply to statements or propositions."""

But what if you say "this statement is false"? :-)

Hans, what statement? "This statement" is not a statement. Above, I used "statement" and "proposition" interchangeably. Maybe it would have been clearer if I had merely used "proposition". There is no proposition contained within "This statement".

ok, how about this: "today is opposite day"

Crawford, you rock.

Y'know, the one interesting book I read of this kind of thing is Hofstadter's Goedel, Escher, Bach. You read it yet? It's a whole heap of fun.

Hmm, so what about... "the statement made in this sentence is false"?

Haven't read Gödel, Escher, Bach yet. I started to, and I got a very mystic, Kabbalah-ish feel from it. Maybe I'll finish it someday.

Now, let's analyze the "today is opposite day" statement. There are four cases:

So, that has no paradox. It just turns out that whenever you say "It's opposite day" you must be lying, and whenever you say "It's not opposite day" you must be telling the truth. Unfortunately, that makes it impossible to tell someone whether or not it's opposite day. It would be interesting to see if anyone could come up with a way to reliably indicate whether it's opposite day or not, though that's a separate question.

What this problem is really trying to get at is that you're trying to come up with a proposition whose falsehood implies its truth and whose truth inplies its falsehood. That's where the contradiction comes from.

So, first I'd like to revise myself a bit. What I was getting at with the "but it has no content" complaints is really the following: What you're really doing is making endlessly recursive sentences that have no base case. That's why they have a problem. So if you say "the statement made in this sentence is false" you get in infinite recursion, such as "Ok, what statement?", "The statement made in this sentence", "What statement", "The statement!". Ultimately, the sentence makes no statement and has no proposition. The fact that you can make a recursive sentence like that I'm not sure is an interesting problem, or a problem at all.

The original goal of this type of paradox was to create a sentence whose truth implied its falsehood and whose falsehood implied its truth, but which does contain a proposition and therefore seems like it should have some defined logical value. That is interesting. My professor replied to me by e-mail and gave me this sentence: "Either this sentence expresses no proposition, or the proposition it expresses is false". If it has no proposition like the other sentences we've been looking at then it's true. But clearly, since it's true, it does contain a proposition, but then the sentence is false because it says that the proposition it expresses is false. But then if the proposition is false, then it's true because the second half of the disjunction states "the proposition it expresses is false". So, you get a contradiction.

I'm honestly not sure what to think about this. Is this some deep epistemelogical problem that has some relevance beyond telling us that we can construct this type of strange construct, or is it just a neat trick of language or logic? This is why Keith's mention of Gödel, Escher, Bach is probably extremely relevant, because it seems like this runs us right up against the wall of undecidability that Gödel explored. I'll have to read more about his discoveries before I can go any further with this, I think