Author Topic: I am lying!  (Read 4808 times)

Offline jknilinux

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I am lying!
« on: November 06, 2008, 02:10:17 PM »
Hi everyone,

This is a problem I've been working on a bit recently, and I'm starting to give up.
Has anyone here ever heard of the liar paradox? Basically, it states:

This sentence is false.

The problem is; is it true or is it false? And this isn't just a word problem- it's a problem with logic itself. Godel proved arithmetic must be incomplete or inconsistent using an arithmetic version of this. Unfortunately, there is a predicate-logic based version of this too:

P: "P is false"

The statement P can not fit into either the set of all true things or the set of all false things... Does this mean logic is incomplete or inconsistent as well?

Post any solutions, like modifications to logic, new truth values, or anything else you come up with to solve it here.



EDIT:
Just found another interesting paradox:

P: "P => Q"

If it's false, then the conditional evaluates to true, so what P means is true, so P is true. If P is true, then what P means must be true, which is that P implies Q, so since P is true and P => Q, then Q is true...

But what is Q? Well, Q is anything! God exists, God does not exist, you don't exist, you are an invisible pink unicorn... anything.

Anyway, if you don't understand the logic behind the liar or my new invented one, let me know.
« Last Edit: November 06, 2008, 03:29:15 PM by jknilinux »

Offline Peter

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« Reply #1 on: November 06, 2008, 04:03:26 PM »
Hmm, this reminds me of botsareus.....

Well, to keep it short what is your logic. I know the liar paradox.
As in,
The next sentance is wrong. The earlier sentance is right.

The rest just seems random-rambling. I lost you when you started at P: "P is false"
Oh my god, who the hell cares.

Offline Jez

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« Reply #2 on: November 07, 2008, 03:04:42 PM »
Isn't that like where you have the options I, O, or I and O? Quantum computers or something.

I mean you have a statement and you have an answer to a statement and because it is not working the model can be considered to be wrong or the way of looking at the model can be considered to be wrong.

And the statement exits, it makes sense, I don't see why the statement is wrong.
Its only when I try to apply the rules of logic, as I know them, to the statement that I run into problems.

So maybe the rules of logic are not sufficient to deal with the problem and they need expanding.

Or maybe the logic is ok and you just need to look at it through 4 dimensional space. Although you might consider it only a two dimensional question.

I suggest you change the rules until the statement works...

« Last Edit: November 07, 2008, 03:09:00 PM by Jez »
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Offline Peter

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« Reply #3 on: November 07, 2008, 03:16:11 PM »
Jez, are you checking out what changed. There is a new F3. A plan for a new complex behaviour league, and eric seems to be away for a few weeks(not sure why)
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Offline Numsgil

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« Reply #4 on: November 08, 2008, 03:40:27 AM »
All it basically means is that not everything that can be expressed is possible.  Language is so descriptive that it can be used to describe things which cannot be true.

Offline bacillus

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« Reply #5 on: November 08, 2008, 03:45:26 PM »
Ah, conscious deceit, the true measure of intelligence...
"They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the Clown."
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Offline jknilinux

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« Reply #6 on: November 08, 2008, 04:59:18 PM »
Peter:
Sorry, philosophy is filled with unnecessary jargon. By P:"P is false", I meant to show the formal representation of the paradox in logic. I'm attaching Tarski's proof of the liar. Let me know if you have any questions.

Jez:
1: No statement can be both True and False. That violates the principle of bivalence. If something was both true and false, then you can logically prove ANYTHING- you exist, you don't exist, god exists, etc... Let me know if you want an elaboration.

2: Believe me, we have tried expanding the rules of logic and looking at it through other ways- no matter what, the liar can still be generated. For example, we can make up a third truth value, you, for undecidable, and the liar paradox has this truth value. But what about the statement "This sentence is not true"? If it's true, then it's not true, so it must be undecidable. But if it's undecidable, then that's just what it said- that it's not true, aka false or undecidable. So, it's true, and we get a contradiction all over again.

Then, maybe we can say a statement can never refer to it's own truth value. OK, that solves the one-line liar. But what about this:

1: 2 is false.
2: 3 is false.
3: 1 is false.

What is the truth value of 1? If it's false, then 2 is true, then 3 is false, so 1 is true. If 1 is true, then 2 is false, then 3 is true, so 1 is false. Again, a paradox, and it doesn't refer to it's self.

No matter what you do, it just doesn't go away.



Numsgil:

Are you saying that it's false? If it's false, then it's true.
If you're saying it's instead undecidable/meaningless, then that counts as a third truth value, which brings you back to "This sentence is not true", discussed above.


So, any way to save logic? Philetas of Cos was an ancient philosopher. His tombstone reads:

Greetings, stranger. Philetas of Cos am I.
      'Twas the Liar that made me die
     And the bad nights caused thereby.

Kinda creepy.

Offline Moonfisher

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« Reply #7 on: November 08, 2008, 08:38:55 PM »
I think what numsgil was saying is that the sentense is just jiberish, and just because you can say giberish in a languare it doesn't make the langueage invalid, same would go for logic...
Sure you can express something that makes no sence through logic... you can also try to eat soup with a hammer and fail, but the hammer will still be good when used right.
You're more or less just creating an infinite loop... just like garbage collection won't catch 3 pointers that point at eachother with no entry... doesn't mean that garbage collection doesn't work, just means it isn't fool proof. Sure you can say that it's a flaw... but everything we do is flawed... you can't divide by 0 either, then you can set up a rule, but then you can say that X = X + 1... then X will never stop growing and always have a new value... doesn't mean that math is flawed... just means that once you get to a certain point rules aren't enough, you need to think about how you're applying the tools you have.
If what I'm saying seems like jibberish it's just because I'm fairly drunk... again IMHO this doesn't make english flawed, just means I'm drunk.

You can probably find a way around any rules concerning this, so in the end you would need a rules that states that your logic need to reach a finite answer or the entire statement is invalid.
« Last Edit: November 08, 2008, 08:40:37 PM by Moonfisher »

Offline Jez

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« Reply #8 on: November 09, 2008, 04:27:54 PM »
Cheers Peter,

I've only been back a couple of times but had noticed the new plans and am interested in how that pans out.

jk;

Had to look at this bivalence thingy but pretty quickly "Lukasiewicz introduced a third value" jumps out.
While I won't pretend to understand the depth to which logic can descend a 'third value' is what I had imagined the statement needed.

When it comes to logically proving anything: All the philosophers I have known have been pretty happy at their ability to argue that they can prove anything, the argument for a horse having an infinite number of legs jumps to mind.

While it was harder for me to picture the 1,2,3 statement I eventually figured that it was no different to saying [1: 1 is false], it's just a trick to take your attention away from the important bit, the distraction technique that illusionists use if you like.
If you try and take a cat apart to see how it works, the first thing you have in your hands is a non-working cat.
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Offline peterb

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« Reply #9 on: November 09, 2008, 07:08:20 PM »
hmmm

in drwinbots it means simply
cond
x x !=
start
stop

or simpley x isnt true, so whatever x is doesnt matter. it might be some wisdom but from the perspeciteve of its carbon god it isnt true whatever it is.
Oke well that is fun, a type of fun that only gods can have. knowing of some genes that just don't fire. as the condintion would never happen. Dough on the other side, if a carbon based god would put such a logic in a silicon based heaven..
Then perhaps that silicon world woud behave different. And wel so it does.
Even the carbon gods rulers of darwinbots, note that over many generations you get problems with reproduction and DNA..
Hack that even happens in the carbon world..
Wel anyway in the end it means that after many cycles (and just a few errors in DNA copying) you'll see that there is answer to the problem rule.
Simply it was a condtition which never could be solved... until the condtition changed.. just by random DNA copy errors. suddenly x x != became x Y !=  or   x x =
welll ofcourse you dont have to believe me, nor the rules of genetics, there are no rules for it..
DNA wouldnt allow me describe a rule, knowing that there are no absolutes, in the game of DNA some strings play a false game.
is that a trouble?? ... no rather it is a solution just chang the rules to get solved

No its rather a way of peeling around the main problem to survive and to solve a problems wich couldnt get solved before.
Unless you took different math..

Changing math always expanded science like SQR (-1) = ....

or like f(n) = 1  2 3 5 7 11 13 17 19 13 17 19 23
uh oh not invented yet...(or didnt it be not invented... [I lwill not say]   .)        )
In the end only real math brakes you trough the limits, and so it can set you free.


ehhmmm ok I admit just a few beeers......
still doupts did he have a prime function or not?, would he tell ?
« Last Edit: November 09, 2008, 07:16:26 PM by peterb »

Offline bacillus

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« Reply #10 on: November 09, 2008, 11:18:13 PM »
Quote
cond
x x !=
start
stop
The problem with this, or any paradox for that matter, is that it has no useful applications.
"They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the Clown."
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Offline jknilinux

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« Reply #11 on: November 10, 2008, 01:18:41 PM »
Moonfisher- If by meaningless or invalid you mean something that cannot be true or false, then meaninglessness or invalidity is a new truth-value, by definition, since that means that "if a sentence is meaningless, then it's not true or false. If a sentence is not true or false, then it's meaningless."

So, something must be true, false, or meaningless- what about "This statement is not true"? If it's true, then it must not be true, so it's true and not true, so it's false or meaningless. If it's false, then what it's saying is false, so it's false that it's not true, so it's true or meaningless. So, it can't be true, it can't be false, so it's just meaningless... yay! And everything works, right?

Well, no. As it turns out, if it's meaningless, then what it's saying is true- it says it's not true, so it says "I'm false or meaningless". Since it's right, it's true. So, if it's meaningless, it's true, and we get a paradox all over again.



Jez- Just like I showed above, you can have 3 truth values, or 4 truth values, or infinitely many truth values, and you'll solve the original paradox, but you can't solve what's known as the strenghtened liar- "This sentence is not true." Because, if it's truth value Q, or whatever else you make up, then it's true, so it's false, etc...

Also, although the 1-2-3 statement is weirder, it's important because many solutions to the original liar or the strengthened liar fail in front of paradox 1-2-3  - aka what I call the "ultimate liar". For example, we could say a statement cannot refer to itself. That solves the original AND the strengthened liar, but the ultimate liar is not self-referential, so that doesn't work. It's just an acid test for any solution that people come up with.



Peterb- Actually, no. A contradiction is something of the form "x and ~x", while a paradox is a proof of a contradiction from acceptable premises. Just writing down "X and ~X" AKA "X = ~X" is not a paradox, since it must be false according to the rules of logic. Think about it- can x ever possibly equal ~x? Of course not, so it's just false.

However, this isn't the case with the liar- if it's false, then it must be true. It is not saying it isn't itself. It's a paradox because I can prove a contradiction based off of it, but it itself isn't a contradiction. To see the proof, you'll need to look at the "tarski's proof" document I included.



Bacillus-
I'm glad no logicians heard you. Zeno's paradoxes are helping guide current quantum physics, and have even helped spawn a new area of physics- digital physics. Russel's paradox showed that the original set theory was wrong, and Godel's incompleteness theorem, which is basically a version of the liar paradox that's been proven in math, showed that math must be incomplete or inconsistent. Once (if ever) the liar paradox is solved, it will reveal new fundamental truths about truth itself. So far, it seems that logic itself must simply be incomplete or inconsistent to solve the liar, which itself would be earth-shattering in all fields dealing with logic (aka all fields) if proved.
« Last Edit: November 10, 2008, 01:58:56 PM by jknilinux »

Offline Peter

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« Reply #12 on: November 10, 2008, 02:42:51 PM »
Quote from: jknilinux
Moonfisher- If by meaningless or invalid you mean something that cannot be true or false, then meaninglessness or invalidity is a new truth-value, by definition, since that means that "if a sentence is meaningless, then it's not true or false. If a sentence is not true or false, then it's meaningless."

So, something must be true, false, or meaningless- what about "This statement is not true"? If it's true, then it must not be true, so it's true and not true, so it's false or meaningless. If it's false, then what it's saying is false, so it's false that it's not true, so it's true or meaningless. So, it can't be true, it can't be false, so it's just meaningless... yay! And everything works, right?

Well, no. As it turns out, if it's meaningless, then what it's saying is true- it says it's not true, so it says "I'm false or meaningless". Since it's right, it's true. So, if it's meaningless, it's true, and we get a paradox all over again.
I think if something is meaningless, then it is. Then it doesn't say anything at all. And the paradox stops.

This statement is not true. Meaningless, the statement isn't true or false, it could be both, maybe neather.

Quote
Also, although the 1-2-3 statement is weirder, it's important because many solutions to the original liar or the strengthened liar fail in front of paradox 1-2-3  - aka what I call the "ultimate liar". For example, we could say a statement cannot refer to itself. That solves the original AND the strengthened liar, but the ultimate liar is not self-referential, so that doesn't work. It's just an acid test for any solution that people come up with.
Try to put something like it in excel if excel says there is a loop coming back to itself, it refers to itself in the end. Excel is a clear number program, try it.

A=B+1
B=C+1
C=A+1

(with numbers and letters you can do everything, what is the point anyway?)

Quote
Peterb- Actually, no. A contradiction is something of the form "x and ~x", while a paradox is a proof of a contradiction from acceptable premises. Just writing down "X and ~X" AKA "X = ~X" is not a paradox, since it must be false according to the rules of logic. Think about it- can x ever possibly equal ~x? Of course not, so it's just false.

However, this isn't the case with the liar- if it's false, then it must be true. It is not saying it isn't itself.
That apple is the same as that apple. It says is it the same to itself. Not something like the apple is the same to a apple.
If I look at it,
maybe you try the condition ''X X != '' in other ''strangely'' chosen symbols. But then how is ''~'' false?

Quote
Bacillus-
I'm glad no logicians heard you. Zeno's paradoxes are helping guide current quantum physics, and have even helped spawn a new area of physics- digital physics. Russel's paradox showed that the original set theory was wrong, and Godel's incompleteness theorem, which is basically a version of the liar paradox that's been proven in math, showed that math must be incomplete or inconsistent. Once (if ever) the liar paradox is solved, it will reveal new fundamental truths about truth itself. So far, it seems that logic itself must simply be incomplete or inconsistent to solve the liar, which itself would be earth-shattering in all fields dealing with logic (aka all fields) if proved.
Well now is the time to show me the use for this. It sounds a little like a made-up kind op phycics.(not that you made it up)
But more on the stage of, what is the use for this?
What do true and false have to do with math?
Oh my god, who the hell cares.

Offline jknilinux

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« Reply #13 on: November 10, 2008, 03:52:22 PM »
Peter-
Math is based on ZFC set theory, which is based on logic. You don't know it, but you use formal logic every day.
~x is the same as not x, and !x, and "x is false". In logic, we use ~ most.

By the way, are you saying that even though it cannot be meaningless if it is meaningless, that it should still be meaningless? "This is a sentence" leads to a contradiction if you try to make it meaningless, and "this is not true" also leads to a contradiction if you try to make it meaningless. Why does one remain meaningless while the other remains true?

A statement can never be both true and false. It is impossible by definition, and if even if it was, then I could prove p*~p (p and not p) from it. So, it must be false that something is both true and false.

And what does excel have to do with this? How does excel solve the paradox?

Offline Peter

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« Reply #14 on: November 10, 2008, 05:01:09 PM »
Quote from: jknilinux
Peter-
Math is based on ZFC set theory, which is based on logic. You don't know it, but you use formal logic every day.
~x is the same as not x, and !x, and "x is false". In logic, we use ~ most.
~X sounds to like is close to X. Not ''not X''. Doesn´t sound logic.

Quote
By the way, are you saying that even though it cannot be meaningless if it is meaningless, that it should still be meaningless? "This is a sentence" leads to a contradiction if you try to make it meaningless, and "this is not true" also leads to a contradiction if you try to make it meaningless. Why does one remain meaningless while the other remains true?
How, what contradiction. Explain further. I´m not that fast.

Quote
A statement can never be both true and false. It is impossible by definition, and if even if it was, then I could prove p*~p (p and not p) from it. So, it must be false that something is both true and false.
It is inpossible by definition and just you said the definition could be wrong becouse the statement are wrong.

You can express things that aren't true. That doesn't make the paper it is written on wrong. It is like saying the paper is wrong becouse you can write something wrong on it.
AB=AB2
1+1=3
I'm smart and I rule the world and 1+1=2, and you gimme now a cup of coffee.

Most of the sentense is rambling, but 1+1=2, it is,is it?
Anyway does this make the sentance wrong or right

Again, give a link. How did you come at this, anyway.
Oh my god, who the hell cares.