How do I solve this paradox?

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The statement "this statement is false" is self referential, and it is only self referential, and nothing but self referential. It doesn't describe anything other than its own content, which is solely content about its own content, which is only about its own content, and so on.

Consider the opposite statement "this statement is true".

Is that statement "true", or "false"?

You could declare "this statement is true" to be true, that it is true, that it is true, that..... . Or you could declare it to be false...that its true. Either way there would be no contradiction. No seeming "paradox".

So "this statement is true" would seem to be unlike its negative near twin, and to be like most statements (like "all crows are black"), in that in principle it could be judged as being one or the other (as being true, or as being false). So which thing is it?

Is "this statement is true" true, or false?

The fact is that the statement is meaningless. And stating that is true, or stating that is false, are both also equally meaningless.

Is it "true" that the content of the statement "this statement is true" is true? Or is it false? Its meaningless either way.

So the negative version of the statement is also meaningless. So its not a paradox because there is no contradiction because you have to have meaning in order to have a contradiction in meaning.

"This statement is false" is not a contradiction because it doesn't mean anything anyway.

Utter nonsense.

You picked the statement for the same reason that everyone else in the last 3000 years picked it, because it SEEMS to be a contradiction. You dont need to tell us that water is wet. We know that.

What I am saying is the the way out of this paradox that bothers you so much is to do something different than what has been done in the last 3000 years, and that's to consider the statement's opposite.

The statement "this statement is true" is self referential, referential only to its own empty content. Empty content is meaningless, and thus cant be classified as either true or false. Its like trying to determine whether or not a belch is true or false. True and false don't apply to a belch.

Saying "it is meaningful insofar that it shows how you cant prove a negative" is nonsense. It has nothing to do with the adage that "you cant prove a negative" because that adage only applies to statements that refer to things other the statement itself. Statements like "Trump is whatever" or "the earth is being visited by alien space craft" refer to things outside the statement (Trump and the world or UFOs etc).

IF you find one real UFO crashed in the swamp then you have proven that alien space craft really are visiting us. But you cant really disprove the assertion because you cant show that there wasn't some one real UFO that visited and got away without being detected.

Saying "this statement is true" is "true" because "it says so" is nonsense. True about what? About the fact that it is true? True about the fact that it is true, about the fact that is true...about what? True about the fact that it is true about the fact that it is true, about the fact......ad infintum. Its a meaningless statement.

Further you could just as easily declare the statement "this statement is true" to be FALSE. Like the first statement there is no self-contradiction. It makes just as much sense to say "this statement is true" is false, as it does to say that this statement is true is true. Further where is the evidence either way? Both are self referential statements. Neither statement refers to any reality in which can gather evidence.

Likewise the third statement "this statement is false" is also nothing but self referential, and therefore meaningless. Unlike the first two statements it seems to be a contradiction. But since like the first two statements it is meaningless it cannot be a contradiction because you have to have meaning before you can have either contradiction or consistency.

It isn't a valid statement, or a statement at all. Not everything that can be said or written is a valid statement, even when it's grammatically in the form of a statement.

congratulations, you broke language.

also: Read Goedel, Escher, Bach to understand how a formal system (in this case language) can be used to break it, given that it's strong enough to be self-referential.
works also with math.

does it mean anything, outisde of logic and AI? meh.

http://aynrandlexicon.com/lexicon/contradictions.html

also: don't take dogma from ayn rand. she's really not that great a thinker. she just got popular with mediocre thinkers who love to think they're misunderstood geniuses, itself a romantic era concept we owe to Kant and Hegel.

Since long this paradox has been analyzed in mathematical logic, namely as a basis of the proof of the incompleteness theorem, and what was discovered, as surprising as it may sound, is the following.
In the sentence, "this sentence is false", the problem is neither about the concept of "sentence" (as formulas can be described as a kind of mathematical systems), nor about self-reference (it is actually possible for a formula to quote itself, this quotation included !), but about the truth predicate over the class of formulas (the qualification for a formula as true or false) as soon as the formula in question is not explicitly written in full as a sub-formula in the right place of the main formula, but only given as an object (mathematical system).
The proof of the incompleteness is based on the established possibility for a formula to quote itself, so we can form a formula which says "This formula is unprovable in System X" where X is some given axiomatic system of arithmetic, or of set theory for its ability to prove formulas of arithmetic as particular cases. Then after some work this self-referencing formula (but we only qualify it as self-referencing when looking at it on a high level, as it is but a super abbreviation of some lengthy formula of pure arithmetic) turns out to be logically equivalent to the formula "System X does not lead to contradiction"
If you wonder the status of another formula saying "This formula is provable in System X" the answer is known !
Namely such a formula is actually provable, and its proof comes as a corollary of the Incompleteness theorem, called Löb's theorem.
I wrote a page about it : http://settheory.net/model-theory/incompleteness

I've run across this one before, mainly in how it supposedly destroyed Logicism (Gottlob Frege's the idea that math is a subset of logic). It's one of those problems I find really tedious and haven't examined much on a formal or academic level because it involves extended research of definitions to figure out what ideas are actually destroyed by this sentence, or two sentences, being writable or utterable.

I remember in a theism debate someone, I think AG, brought up the concept of a square circle (ie. in reference to questions like 'Could God draw a square circle') as a good example of a linguistic absurdity which was grammatically possible but held no valid meaning. It reminds me of something which I think is intuitively obvious, ie. that typical written and spoken language was never intended to be a perfectly consistent set of logical operators and it would be very strange if it did pass any sort of test for perfect logical consistency being that's somewhat orthogonal to the original goal of language which is meant to carry truth as much as its meant to carry BS, poetry, propaganda, and anything else you can imagine one person saying to another. Past that I don't really know. It might very well be that there's plenty of math that gives untestable answers but I'd also think there's plenty of logic that can create absurdities, even properly applied, if it doesn't doesn't check in with physical reality on a routine basis.

Could God build a boulder so big that even he (God) couldn't lift it?

First thank you for the well-built synopsis.