I am lying!

[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]

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.

"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.

[jknilinux]

Moonfisher-

If you disallow self-referential statements, you just disallowed a whole bunch of perfectly fine statements- "This sentence has five words." is an example. Are you saying this is wrong/not a proposition (which we'll assume for the sake of argument isn't a truth value)? Because I can safely say that that makes sense, and is a valid statement, and is true. Do you disagree?

Anyway, "the green cloud tastes sad" makes sense, but if you taste a green cloud, I think you'll find they don't taste sad. Sadness is not a known taste, so it's a self-contradictory statement (assume it's true- the cloud will therefore taste sad, but you cannot taste sadness. So it also does not taste sad. It tastes sad and doesn't taste sad, so the statement must be false, in which case it just doesn't taste sad. Same with "colorless green ideas sleep furiously", or whatever else you try.) So, it's just false.

"fbjidklgbhfil", on the other hand, is different. It has no known meaning, and so cannot be translated into logic.

"This sentence is false" has meaning, and can be translated into logic, unfortunately. The problem is that is can't be false, since you'll still get a contradiction in that case, meaning it's true.

Peter-

I know you can express false things- that's not the problem. The reason why things are false is that if they were true, there would be a true contradiction, which means everything, including all previously false things, must become true. "This is not a sentence" is false because if it were true, then it would be true that it is not a sentence. It is by definition a sentence. So, it must be both a sentence and not a sentence- a true contradiction.

If it's a true contradiction, you can represent it in logic as P and not P. "This is not a sentence" AND NOT "This is not a sentence"

If something AND something else are true, then both something is true and something else is true. So I'll just say "This is not a sentence" for right now, since it's true.

When something is true, you can add anything to it with an or- if it's raining, then you can also say "it's raining or it's not raining"- one is obviously false, but that's OK since one is still true, so the or statement is still true. For this proof, I'll use "This is not a sentence" OR "I'm a purple tomato"

When you have an OR, if one thing on one side is false, then the other thing on the other side must be true. If it's raining OR it's not raining, and it's false that it's raining, then the other one must be true, so it's true that it's not raining. So, since it's false that this is not a sentence, because it is a sentence, then the other one must be correct, which says "I'm a purple tomato", must be true.

Therefore, I'm a purple tomato.

So, "This is not a sentence" must be false, right? What happens if it's false? Well, then it's false that it's not a sentence, so it's true that it is a sentence, which is true. So, it is false.

I feel I might have missed something else you said- let me know if I did.


Anyway, the problem with the liar paradox is that if it is false, then it must be true. Hence, it is both, so it is false, so it is true, so it is neither, etc...

[Moonfisher]

Well 0 divided by any amount will give you 0, so "in theory" you could divide 0 by 0 and get zero. You could also argue that dividing by 0 would equal infinity, but this wouldn't work well with all the other rules that we apply in math. No matter how we look at it math is just a set of rules that we determine, and so is logic.
So there's no right or wrong answer... but you asked for a rule that would prevent the "paradox" that you mentioned, and not allowing a proposition to refer to itself would solve this paradox. The 3 line liar is not an exception to the rule that you yourself proposed...
You're right that it will disallow statements that are "valid" or rather statements that don't pose a paradox, since nothing is actualy valid or invalid... so it comes down to wether you feel it's more important to have access to form such a proposition or to eliminate propositions that are in theory infinate.
It's true that :
1: "1 is true"
woudl give :
1: "(1 is true) is true) is ture) asf asf..."
And an infinity of "true" statements will obviously let you know what the answer to the proposition is... but the proposition is still infinate... in theory.
I'm not saying it's wrong to have an infinate proposition... but the paradox you mention could sugest that it may help logic to disallow "infinate" propositions. In the end the rules are made by us, there's no wrong or right, just attempts to create a "world" where we can explain and justify everything.
It's sounds like you're looking for an absolute truth... but I'm not sure that can actualy be found through logic... or anywhere for that matter.

As fot the sentence I mentioned... this has nothing to do with logic, it's not a proposition, it's not logic... it's a sentence, and gramaticaly it makes sence, but from our point of view it's just nonsence.
Writing random letters does not follow the rules of gramar or spelling that we use, but the other sentence does, it just makes no sence because you can't taste a feeling. Basicaly it's applying the rules of gramar to construct a "correct" sentence that makes no sence. I'm not looking for an answer or confirmation on wether the sentence is true or false, my point is that the sentence makes no sence dispite having followed the rules of gramar... you would need a million rules to make sure you can only create meaningfull sentences (a problem when trying to create an AI capable of faking comunication). But this has nothing to do with logic or true/false statements, it was just an example from outside the world of logic. (Just because you can express logic as a sentence it does not mean that sentenses are bound by the rules of logic, the actual sentence "This sentence is false" actualy makes sence... not from a logical point of view, but the sentence itself makes sence, people can understand it)

[abyaly]

Moonfisher, you seem to be caught up in the idea that propositions that refer to each other can only be evaluated if you reduce them. This is not the case. While some propositions sets are equivalent to a single proposition, this is not always the case and is NOT a requirement.

The three line liar is not a single self-referential statement. It would be equivalent to a single self-referential statement if you allowed it, but since you don't, it isn't.

[Moonfisher]

You're right, it's not a requirement... never said it was, just saying, in THEORY you would enter the value of the proposition you're refering to in that position, just like you would with math.
The statement X = X+1 in math would in theory create an infinate loop... but it all depends on the rules you want to apply.
In the end, you have an artifitial world wich obeys the rules that we define ourselves... so it may not be a requirement, but it could be, and if it was it would make some sence in some ways...

[abyaly]

It might make sense to you to create such a requirement, but it would cut away a lot of good stuff. You would pretty much disallow all recursion, which is a really useful way to describe things.
The programming language Haskell is a good example of the benefit. Self-referential objects that can be expanded indefinitely are core to the language. You would have a hard time getting anything done without them. The language itself includes very powerful abstraction methods, so destroying the entire language just to get an easy restriction on logic would create a lot of extra work for some people.

When coming up rules of logic, you need to realize that your scope includes a lot.

[Moonfisher]

I'm not saying it's THE thing to do, I'm saying the rule he proposed would work for eliminating the paradox he mentioned... that's all.
And the rule would make sense in a lot of ways. As you say yourself "Self-referential objects that can be expanded indefinitely"... this is a gray area that allows infinate loops. So you can either accept that this also allows a paradox or proposition with a shifting state that will never setle (However you wish to look at it) or have a rule that prevents this "paradox" but also eliminates a lot of usefull propositions.

I keep saying, there's no right or wrong, and I'm not advocating any of these ideas... in the end I don't realy care where this is going, don't know much about logic and I rarely use it in this way. I just found the whole conversation interesting.
I don't know Haskell so it's hard for me to say what it allows or doesn't allow. I have no idea how it works and how bound it is to the rules of logic, so I can't say what would work well for this laguage. If I understand correctly the laguage is strictly bound to the rules of logic and uses self referential statements to create loops? It seems like most statements who end up refering to themselves would create an infinate loop... but is that the whole idea?

Anyway as said, I don't know much about this stuff, on nothing about Haskell, I just know what makes sence. And if I had to pick then I'm not sure if I would find either rule or no rule more apealing for it's uses, but I do think logic with shifting states seems to make less sence to me.
You said a valid proposition was either true or false and could not be both or it would not be a valid proposition, so it seems to me that a "Self-referential objects that can be expanded indefinitely" wouldn't be able to garantee a value of true or false and therefor wouldn't be a valid proposition...
Or maybe I misunderstood something... as mentioned, don't know all that much about logic, I just know what makes sence

[abyaly]

Banning self-reference is not necessary to solve the paradox. I don't think any new rules are needed at all.

Logic doesn't have shifting states. Every statement is completely stable. I think you are making the mistake of thinking about the logical statement in steps. The trick is that no matter what angle you think of it from, the original statement (as it was written) isn't changing, so it's value can't be changing either.

Also, Haskell is really cool. It's a pure, lazy, functional language inspired by category theory. It's worth taking a look if you're interested in programming. Functional languages in general will warp your mindset a bit the first time you encounter them.

[jknilinux]

Well, aby, your idea just doesn't "feel" right to me- if you think something's false, but you get a contradiction when it's false, then it must be true. But, if something's not-a-proposition, even if we get a contradiction from this, we still leave it in the not-a-proposition set, because things in that set we just ignore.
The cover story of the new york times today is "everything we'll say tomorrow is true"... this seems fine. The next day, the cover story is "everything we said yesterday was false"... Uhoh, contradiction. So, let's just leave it in the not-a-proposition set, and ignore it.


Anyway, there's one final paradox that deserves some attention here. It's closely related to the liar, but not quite...

Take the statement x:"x => y", (aka x, defined as "x implies y", aka the statement "if me, then y")
The only way implication, aka if...then, can be false, is if the "if" part is true and the "then" part is false. For example:

If it's raining, then the street is wet (AKA R => W). This can only be false if it's raining and the street is not wet. So, it can only be false if R is true and W is false.

So, back to x. If x is false, then what it means must be false, so it is false that "x => y". This can only be false if x is true and y is false. So x is true. However, we just assumed x is false! So, if x is false, then x is true and false, so x cannot be false. So, x must be true...

If x is true, then what x means must be true. So, x is true, if x then y is true, so y must be true.

Uh, one problem... What was y? Here's the awful truth: y was anything. You can put anything in y: "I exist", "I don't exist", "I both exist AND don't exist"... I can prove anything with this!

Hence, a paradox is born. Plus, you can't just cheat and call it "not a proposition" either...

BWAHaaHaaHaaaaaaa! (sry- I just had to do that...)

[Peter]

Uh, one problem... What was y? Here's the awful truth: y was anything. You can put anything in y: "I exist", "I don't exist", "I both exist AND don't exist"... I can prove anything with this!

Hence, a paradox is born. Plus, you can't just cheat and call it "not a proposition" either...

BWAHaaHaaHaaaaaaa! (sry- I just had to do that...)

Why should it mean anything at all. What is wrong with the sentance at all.

If it is raining then the earth is round.(yes it is, the fact that it's too if it isn't raining is a detail)

[abyaly]

Take the statement x:"x => y", (aka x, defined as "x implies y", aka the statement "if me, then y")
The only way implication, aka if...then, can be false, is if the "if" part is true and the "then" part is false. For example:

Curry's paradox is a good one, and I'll admit that logical systems that allow its construction are broken.

PS - The programming language Haskell was named after Haskell Curry, which is the same person this paradox is attributed to.