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I am lying!

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Moonfisher:
Just because you follow the rules of a laguage (in this case logic) it doesn't mean what you're saying makes sence.
I can say : "The green clouds taste sad" and I'll have followed the correct gramar to form a sentence, but it won't make any sence.

But if you realy want a rule to prevent the paradox you're mentioning, then you already stated it, just don't let the proof refer back to itself.
If a proof isn't allowed to refer to itself, then taking a "detour" isn't actualy changing the statement.

To take your example :
1: 2 is false.
2: 3 is false.
3: 1 is false.

Is the same as :
1: ((1 is false) is false) is false.

And you have a proof refering back to itself, so no matter how you try to bend it you've broken your rule. The detour isn't real, it's just an attempt to hide the fact that your proof is refering to itelf.

But in reality you're just creating an infinate loop where your state keeps changing, so the real question is if it's correct to set up a rule to prevent this or if logic can actualy have a shifting state.
In the end since we're the ones making the rules I guess it comes down to what we want

abyaly:
On the topic of Godel and incompleteness.
Godel's incompleteness theorem showed that for any axiomatic approach to number theory, there exist statements that are neither true nor false. There are many branches of mathematics that are complete (group theory is even turing complete, IIRC), but number theory isn't one of them. He did this by showing that any sufficiently strong axiomatic system for number theory will be able to operate on things that are basically equivalent to the set of axioms, and showed that a statement equivalent to "this statement cannot be proven using this set of axioms" can be generated.
Number theory is incomplete because any strong set of axioms will be able to refer to itself. It's like having a physics engine in which you can build a computer on which you can program the same physics engine. This problem is not prevalent in every field of math.

On paradoxes.
Paradox generally refers to a seeming contradiction. They are caused by a misunderstanding in how things work. Zeno's paradoxes are useful because they help people get over the idea that the sum of infinitely many distances must be infinite. Russel's paradox was useful because it illustrated that assuming you can have a set of anything is wrong.

On the liar's paradox and propositional logic.
First off, we need to agree that propositional logic only operates on propositions. I'm not going to define what a proposition is, but I think you'll agree that there are some statements that are not propositions. So propositions and their values are the only things that matter in propositional logic, and anything that is not a proposition has no weight at all.
Bivalence says that every proposition is either true or false. You tell me you have a proposition, P, which reads as follows:
P: "P is false"
We know that P cannot be true. We also know that P cannot be false. Therefore P is not a proposition, and the liar is the person who said it was one.
Let's try another twist, using the strengthened liar:
A person says they have a proposition, P, and P reads:
P: "P is not true"
We know that P cannot be true. We also know that P cannot be false. So we again deduce that P cannot be a proposition- but doesn't that make P true and give us a contradiction again? Well, no. "True" and "false" are values that can only be applied to propositions, so if we have a statement that is not a proposition, we can't talk about that any more.
So what if we have a multi-line liar?
A: "B is not true"
B: "C is not true"
C: "A is not true"
Well, we know that in order to be either true or false, something must be a proposition. So whenever we have a proposition referring to the truth values of another statement, there is certainly a problem if the other statement isn't one. Eg:
-A: "B is true" v "B is not a proposition"
-B: "C is true" v "C is not a proposition"
-C: "A is true" v "A is not a proposition"
Assuming any one of the three, it's ambiguous whether the others are propositions. We can fix this by assuming that a propositional statement can only refer to the truth value of propositional statements. We're shrinking the space they're allowed to operate on. So the negations become:
-A: "B is true"
-B: "C is true"
-C: "A is true"
And starting from any one of the three, we can determine that none of them are propositions.
This is a bit similar to how they reworked naive set theory into modern set theory.


In modern set theory (ZFC), it is not guaranteed that for every proposition, P, there exists a set of all things such that P. The axiom of replacement tells us that given any set S and proposition P, there exists a subset of S with all things that are P. The trick here is that they've loaded the word "set" with some extra baggage. Not everything is guaranteed to be a set (eg: "the set of everything"). However, a proper class (a collection which is not a set) still behaves pretty nicely, and everything holds except for the axiom of replacement (IIRC).

Moonfisher:
There's no such thing as a "multiline liar". When you say :
1 : "2 is false"
then you're saying :
1 : "(Insert proposition 2 here) is false"
It's still juts one line in the end.

It's like saying :
y = 10/x
x = 0

Splitting it up into 2 statements doesn't make it any different than saying y = 10/0.
Not saying I disagree with Abyaly, just that whenever you mention another proposition you're "suposed to" insert it... it's not realy 2 different propositions, it's just 2 parts of the same proposition wich has been split up to make it easier to read.

abyaly:

--- Quote from: Moonfisher ---just that whenever you mention another proposition you're "suposed to" insert it... it's not realy 2 different propositions, it's just 2 parts of the same proposition wich has been split up to make it easier to read.
--- End quote ---
"Supposed to" insert it? I don't think you can guarantee this will always reduce.

Moonfisher:
Wel I would never garantee anything, but my point is that saying "2 is false" means that in order to "solve" the equation you need to insert proposition 2 at that position... that's basicaly what it means.
When you say :
1: 2 is false.
2: I'm wearing a hat.

It just means : "I'm wearing a hat" is false. So I'm not ewearing a hat...

But for example you can't insert a proposition into itself, wich is also a very good argument as to why it's not a valid proposition if it refers to itself, detour or not :
So :
1: "1 is false"
Will lead to :
1: "(insert 1 here) is false" -> 1: "(1 is false) is false" -> 1: "((1 is false) is false) is false".
And so on till the end of time.

Similarly taking a detour will end up having the same result :
1: "2 is false"
2: "1 is false"
Will lead to :
1: "(1 is false) is false
Wich then takes us back to the previous issue...

Basicaly if a proposition is refering to itself you have an endless proposition.

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