[a1369209993]

The axiom of excluded middle is that for every proposition, either that proposion is true or it is false. That's a universal quantification, so any counterexample makes it false. "This proposition is false." is such a counterexample (it can't be true, because than it's false and your logic is inconsistent, and it can't be false, because than it's true and ditto). This reasoning applies to pretty much any recognizable system of propositional logic (including, per Godel nineteen-thirty-something, systems that explicitly prohibit the basic self-referential counter example, as long as they're expressive enough to describe arithmetic), so it's at least colloquially correct to simply dismiss it as false, even if the pedantic version is that it's a axiom that cannot be part of a consistent set of axioms.

The theorem of included middle doesn't prevent specific classes of propositions from being exclusively divided into true and false, though. As a simple example, any proposition of the form "natural number SSS...SS0 is prime" will be either true or false, and there's nothing inconsistent about (say) "for every proposion P matching regex /(S)*0 is prime/, either P is true or P is false". Just like there's nothing weird about a linter that can reliably tell you that "while(1) {...}" doesn't terminate, halting problem or not.

[ithinkso]

I understand what you're saying (in the first paragraph). I really do because I also went down this path. Self-reference is very tricky and it's way easier to trick yourself into proving things than to prove things.

You are referencing Godel and there is a reason why he only proved 'non-provability', namely Tarski's undefinability theorem[1]. In short, you cannot express the truthiness of a statement in the system within the system itself. This prohibits you to draw conclusions in a way you did above.

I'm of course no expert on the matter, and I might be wrong just as likely, but I'll encourage you not to make such strong, definitive statements. Maybe the reason for it is only that what you stated above would be a huge result in mathematics so one would think there is a mistake somewhere (especially since Liar's paradox is nothing new). And one would explore the subject further to find it rather than 'I post it on online forums because no one wants to accept my theory'

[1] https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theo...

[a1369209993]

> you cannot express the truthiness of a statement in the system within the system itself.

Of course not; I'm not talking about truthiness; I'm talking about truth.

> that what you stated above would be a huge result in mathematics

Not really; it's the same sort of trivially-blatantly-obvious thing as "two-boxing on Newcomb's problem is irrational" or "quantum superpositions never actually collapse".

[ithinkso]

> it's the same sort of trivially-blatantly-obvious thing as

> "quantum superpositions never actually collapse"

Popsci is fun,

Never stop learning, even if it's obvious.

[perl4ever]

>"This proposition is false."

What about "this proposition is true"? Does it have a truth value?

If "this proposition is true" is true, then it's true. But if "this proposition is true" is false, then it's false.

It's self consistent whether it's true or false, but there's no way to determine if it is true or false.

I've been wondering what the formal name is for this kind of statement; surely it's not a paradox. Searching for "the opposite of a paradox" is apparently not the right query.

I feel like there's something questionable going on with the word "is" in either statement.

[a1369209993]

> What about "this proposition is true"? Does it have a truth value?

I think it'll probably end up being neither true nor false too in most 'nice' systems (since general rules that could be used to prove or disprove it tend to set off things like Curry's paradox[0]), but unlike the other one it's (I think) dependent on the details of how you're defining things.

0: https://en.wikipedia.org/wiki/Curry's_paradox