Author Topic: I am lying!  (Read 15807 times)

Offline jknilinux

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I am lying!
« Reply #30 on: November 16, 2008, 11:59:12 AM »
So, how would you solve it? It seems to imply an innate problem with modus ponens* itself.

(modus ponens is a fancy way of saying "if p then q", "p", therefore "q".)

Offline abyaly

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« Reply #31 on: November 17, 2008, 08:04:53 PM »
The paradox only arises in logical systems that allow the construction of that type of statement. So if you're working in a system that doesn't, then there is no problem.
Moonfisher suggested an example of one such: a language with no words referring to elements of the language itself. So it wouldn't have the word "sentence", "proposition" or anything along those lines, preventing the construction of self-referential statements.

The problem isn't with modus ponens, but rather that casual languages are not designed to be logically consistent. Their purpose is strictly communication.
This is why more logic reliant issues like mathematical proofs are generally not delivered in casual language.
Lancre operated on the feudal system, which was to say, everyone feuded all
the time and handed on the fight to their descendants.
        -- (Terry Pratchett, Carpe Jugulum)

Offline d-EVO

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« Reply #32 on: November 23, 2008, 09:38:38 PM »
does my sig have anything to do with the beginning of this thread

easiest way to solve is to say:
 (this sentence is false) is false and forget about it.

Quote
the green cloud tastes sad is ilogical

did you know that some people can taste sound and hear colour.
no jokes. the visual , taste and orditary parts of the brain are all  wired together in those people
« Last Edit: November 24, 2008, 08:38:57 AM by d-EVO »
1:      2 is true
2:      1 is false

Offline jknilinux

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« Reply #33 on: November 23, 2008, 09:54:40 PM »
It can't be false, by definition of falsehood

btw, logic allows curry's paradox to be constructed.
« Last Edit: November 23, 2008, 09:55:39 PM by jknilinux »

Offline Numsgil

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« Reply #34 on: November 23, 2008, 10:48:43 PM »
Quote from: jknilinux
It can't be false, by definition of falsehood

btw, logic allows curry's paradox to be constructed.

Think more that it's nonsense.  Not every proof possibly constructed using logic is a valid proof.  That's all it means.  It doesn't mean that all proofs are suspect, just that a proof is only as good as the set of axioms upon which it's built.  To build a self contradicting proof, you have to start with an invalid axiom, which nullifies your whole proof before you get started.

"This sentence is false" is an invalid axiom.  If you expand it out to something like:

1.  2 is false.
2.  1 is false.

Then you have two invalid axioms.  In neither case is any logic actually applied, you're just starting with nonsense and so the trip to impossible is pretty short.  If your starting axioms are all solid, anything you derive from them is at least as solid as the axioms.  If one of your starting axioms is weak, all subsequent work with them is just as weak.

Now, if you can construct a self contradicting proof from self consistent axioms, I'll be impressed.

Offline jknilinux

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« Reply #35 on: November 26, 2008, 02:56:25 PM »
But nums-

It IS valid.

btw: It's not an axiom. Axioms are rules you build your system on. This is just a statement, and we're trying to figure out whether it's true or false. But, we can't even do that simple little thing without getting a contradiction!

Anyway, the file I uploaded - Tarskis___Proof - shows a self contradicting proof from self consistent axioms.

Offline Numsgil

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« Reply #36 on: November 26, 2008, 07:06:51 PM »
Quote from: jknilinux
But nums-

It IS valid.

btw: It's not an axiom. Axioms are rules you build your system on. This is just a statement, and we're trying to figure out whether it's true or false. But, we can't even do that simple little thing without getting a contradiction!

Anyway, the file I uploaded - Tarskis___Proof - shows a self contradicting proof from self consistent axioms.

No it's not.  Logic is built from axioms- things you take as true without proof.  If you take the statement "This statement is not true", and start a proof, you can reach an contradiction, and using the idea of contradiction by proof you can prove that something you assumed isn't a true statement, and since the only thing we assumed was the axiom, we know the axiom isn't self consistent and therefore isn't a valid axiom.  Same thing with the two statement version, etc. etc.

Don't know anything about a Tarskis proof, can't look at it now.  I'll check it out later if no one else comments.

Offline jknilinux

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« Reply #37 on: November 27, 2008, 04:04:40 AM »
Quote from: Numsgil
No it's not.  Logic is built from axioms- things you take as true without proof.  If you take the statement "This statement is not true", and start a proof, you can reach an contradiction, and using the idea of contradiction by proof you can prove that something you assumed isn't a true statement, and since the only thing we assumed was the axiom, we know the axiom isn't self consistent and therefore isn't a valid axiom.  Same thing with the two statement version, etc. etc.

Don't know anything about a Tarskis proof, can't look at it now.  I'll check it out later if no one else comments.

That's the point- Logic is built from axioms, like "if a then b", "a", therefore "b". The thing was that the liar might show these basic assumptions about reality to be wrong.

Also, you're partly right in your second point- what you mentioned is known as indirect proof. This is where you assume x to be true, and if you get a contradiction, then x must be false. It also works in the opposite way; assume ~x (or "x is false") and if you get a contradiction, then you can conclude ~~x, aka "x is false is false", aka x is true.
Take, for example, "the sky is on the ground". Let's define sky as anything that is not on the ground. So, the sky is on the ground and not on the ground. So, it is not the case that the sky is on the ground.
If the sky is not on the ground, then, well, it's not on the ground. No contradiction. So, it must not be on the ground, meaning it is false.

With the liar, though, if it is false then we have a contradiction, and if it's not false we have a contradiction. So, no matter what, we have a true contradiction.

Offline d-EVO

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« Reply #38 on: November 27, 2008, 03:19:30 PM »
Quote from: jknilinux
It can't be false, by definition of falsehood

btw, logic allows curry's paradox to be constructed.

I said forget about it
1:      2 is true
2:      1 is false

Offline jknilinux

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« Reply #39 on: November 28, 2008, 06:03:18 PM »
That's easy for you to say.  

Offline abyaly

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« Reply #40 on: December 02, 2008, 10:23:06 PM »
A friend of mine who is proficient with boolean rings deduced that
"A: A -> B" is an member of the proposition set only in case B is universally true.
I've been convinced that curry's paradox isn't really a problem, but I don't think I could explain it well enough to convince you :-/
Lancre operated on the feudal system, which was to say, everyone feuded all
the time and handed on the fight to their descendants.
        -- (Terry Pratchett, Carpe Jugulum)

Offline jknilinux

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« Reply #41 on: December 02, 2008, 11:00:22 PM »
So he agrees with your idea- that it's not a proposition. Are you saying Tarski is wrong when he expresses it as a proposition in the proofs I uploaded?

I realize that's kind of a fallacy- "the super smart guy said x, I don't know why, therefore x" - but anyway, I still think it's a proposition. I mean, Tarski's proofs seem to show that it is logically valid, if not even logically sound...

By the way, what do you mean by "I don't think I could explain it well enough to convince you"? I've been convinced of a new idea before....
« Last Edit: December 02, 2008, 11:33:19 PM by jknilinux »

Offline Numsgil

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« Reply #42 on: December 02, 2008, 11:47:10 PM »
I looked at the proof.  I still say that because you take the statement "this statement is not true" as an axiom (or assume it's true, same difference), and you arrive at a contradiction, you essentially formed a proof by contradiction.  Which means the initial statement is false and can't act as an axiom in the first place.  It also doesn't matter that if you take the statement as false you arrive at another contradiction.  It just means that neither it nor its negation are true.  And if you use contraposition, you can conclude that neither are false.

Which just demonstrates that it's possible for a statement to be neither true nor false.  This isn't a breakdown of logic.  Logic is designed to work in a discreet world of only pure truth and pure falsehood.  So long as your axioms are discreet (completely true or completely false), you can use logic, and any proofs you do will be discrete (purely true or false).  But you can't use logic in a non discreet problem space, which is the problem here.  The proof was flawed before you even started.  You need to use something more like Fuzzy logic.
« Last Edit: December 02, 2008, 11:49:40 PM by Numsgil »

Offline abyaly

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« Reply #43 on: December 03, 2008, 11:44:40 AM »
Quote from: jknilinux
So he agrees with your idea- that it's not a proposition. Are you saying Tarski is wrong when he expresses it as a proposition in the proofs I uploaded?
Tarksi's proof is about formal languages. The first two premises are true about our language, and Tarski showed that applying them to a formal language leads to a contradiction (it seems he later gave up the first premise). I have no problem with Tarski's proof. But if you try to apply the proof in general, the error will be in the first step.

Quote from: jknilinux
By the way, what do you mean by "I don't think I could explain it well enough to convince you"? I've been convinced of a new idea before....
I suppose I could try, but I don't know how many of the terms will be familiar.
He used the idea that a boolean algebra is isomorphic to a boolean ring.
He assumed that
A = "A -> B"
was a member of the boolean algebra, and formulated what I assume was an equivalent statement in the ring.
A = A + AB + 1
+ in this case is symmetric difference. Multiplication is AND, 1 is universally true (multiplicative identity), and 0 is universally false (additive identity).
He then preformed the following operations:
0 = AB + 1   (left cancellation)
1 = AB + 1 + 1 (add 1 on the right)
1 = AB (boolean rings are characteristic 2, so 1 + 1 = 0)
From there, we get that A = B = 1.
So saying
A = "A -> B"
is the same as saying
A = B = 1
So the statement "A = 'A -> B'" being valid is equivalent to B being universally true. In boolean logic, at least.
Lancre operated on the feudal system, which was to say, everyone feuded all
the time and handed on the fight to their descendants.
        -- (Terry Pratchett, Carpe Jugulum)

Offline jknilinux

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« Reply #44 on: December 10, 2008, 04:13:52 PM »
Okay, I never learned boolean algebra before, but now that I sat down and reasoned through what you said, it makes sense. Boolean algebra isn't all that different at all from predicate logic. Anyway, the proof seems valid, but when you say it is equivalent to B being universally true, aren't you just saying the paradox all over again? You seem to be admitting that B must be true, even though we haven't even defined what it is. So are you saying the paradox is correct?