Paradox

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Paradox. One of those metaphysical insights that strikes a chord of amusement into the darkest of my hours.

Here is a paradoxical challenge that has, to-date, apparently been unresolved. Curry’s paradox, stated in natural language (not mathematics) could be expressed thus:

"If this sentence is true, then Santa Claus exists."

In my own opinion, as of today, it seems clear to me that the paradox can be resolved by qualifying that the configuration of letters that make up the words “Santa Claus” exists. This would be true of any word we can (or can’t) imagine. The word need not even be meaningful. We could as easily say “If this sentence is true, then peanuts exist” or “If this sentence is true, then schplieterinkerdink exists”. It proves only that the configuration of letters exist, not whether the configuration of letters bear any relevance to an entity or known form of something that it represents in the real world. Not so? Ah, now there’s the rub…

'cos if we think Aspies have problems with non-verbal communication, what does this say about verbal communication?

If X is true Then Y is not Null.

Simple enough statement when you break it down mathematically.

The key point is the word "exists" actually. If you use the meaning of exist of "have an existence, be extant" and extant being "Publicly known; conspicuous" then the original statement of "Santa Claus" would be a true statement as Y (Santa Claus) is not Null. That particular being may be fictional but satisfys the condition because it is publicly known and conspicous.

Precisely, my dear Grimfaire. Mathematics tells us it is true. It exists, regardless of what words we use. That's what tickles my brain, 'cos there are clearly examples of little-known, publically unknown, mythical creatures that we could insert into that sentence. And by doing so, call them into "being"? Don't take me too seriously, as I said, paradox is my friend, most of the time. An amusing companion.

It's not a paradox because it's not actually stating anything that isn't already stated by filling in the blank for what's assumed to exist. If the "foo exists" clause is false, then the statement is false. If the "foo exists" clause is true, then the statement is true. If Santa did in fact exist, then it would of course be a true statement. But he doesn't, so the clause "if this statement is true..." cannot be true or else it would contradict itself.

The problem with this sentence is that it is self-referencing, thus a pointless statement. It is like saying : "If Santa Claus exists, then Santa Claus exists."

But if you take it a step further, you actually find out why it's an unsolved paradox...

Basically :

Is it true that : "If this sentence is true, then Santa Claus exists."
Answer : Yes it is true on its own, just as "If Santa Claus exists, then Santa Claus exists." is a true statement. Pointless, but true.

From this point, you just established that "If this sentence is true, then Santa Claus exists." is a true sentence. So the sentence was true after all and therefore Santa Claus exists.

Confused?

No.

Firstly, because you're making the assumption that the statement is true, when there is no reason to assume that. It is neither true nor false until you input a truth value for Santa's existence.

Examine the two following statements which do not rely on outside truth values and see if you can catch my meaning.

"If this statement is true, then this statement is true."
"If this statement is true, then this statement is false."

Neither of those create a paradox; the first is obviously true, and the second is obviously false.

Or, take it this way - the same form of statement, but with the first clause known to be true and the second known to be false:

"If it's true that 1=1, then it must also be true that 1=2."

This is an absurd statement. But now, if we put a variable in the first clause...

"If it's true that x=1, then it must also be true that 1=2."

Thus, x is not equal to 1.

If you lived in a world where 1 and 2 were symbols representing the same mathematical quantity, things would be different, but that's not the case here. The "paradox" only comes into play when you assume that a falsity - Santa's existence- is true.



In sum: Perfectly composed logical statements, given false assumptions, are still failures and prove nothing at all.

Coyote27,

It seems like we agree to disagree. At least I do.

I too think it is pretty much pointless, and rest assured that I'll never use such a sentence to prove the existence of Santa Claus (or God for that matter).

But it is a paradox, as it leads to a logical conclusion that defies the intuitive answer everyone expects. Such a paradox bothers logical fields such as science, philosophy and mathematics, as the naive or intuitive answer prevails over its logical counterpart, something that cannot be accepted in such fields. Even when processed mathematically by pure logic, one end up with the conclusion that Santa Claus does exist. So the problem really is about finding out why pure logic does not work in this case, something that might be relevant to fields that use logic to provide truths.

"If this statement is true, then this statement is false." as you mentionned is also a paradox that could be presented otherwise in the sentence : "I'm always lying."

If this statement is true, it actually has to be false. So you end up with an affirmative statement that is both true and false, thus a paradox. I'm not sure if this one remains unresolved thought.