Darwinbots Forum
General => Off Topic => Topic started by: jknilinux on November 06, 2008, 02:10:17 PM
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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.
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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"
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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...
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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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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.
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Ah, conscious deceit, the true measure of intelligence...
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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.
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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.
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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.
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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 ?
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cond
x x !=
start
stop
The problem with this, or any paradox for that matter, is that it has no useful applications.
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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.
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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.
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?)
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?
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?
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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?
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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.
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.
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.
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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
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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).
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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.
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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.
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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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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...
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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)
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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.
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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...
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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.
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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
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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.
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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...)
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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)
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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.
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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".)
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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.
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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.
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
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It can't be false, by definition of falsehood
btw, logic allows curry's paradox to be constructed.
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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.
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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.
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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.
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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.
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It can't be false, by definition of falsehood
btw, logic allows curry's paradox to be constructed.
I said forget about it
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That's easy for you to say.
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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 :-/
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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....
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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 (http://en.wikipedia.org/wiki/Fuzzy_logic).
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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.
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.
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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?
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These are equivalent:
- "A = 'A -> B'" is a valid statement
- B is universally true
aren't you just saying the paradox all over again?
So are you saying the paradox is correct?
No to both of these.
You seem to be admitting that B must be true, even though we haven't even defined what it is.
Once we know what B is, we will know whether or not B is universally true. When that happens, we will know whether or not "A = 'A -> B'" is in the statement set. Although in the case that B is universally true and "A = 'A -> B'" is in the statement set, we end up with A = B = T.
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Please define universally true. Is this synonymous with being tautologous?
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"Universally true" is a name of the proposition T.
Yes. It's synonymous.
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So, you're saying B is tautologous, so B must always be true. To me, that's the paradoxical conclusion.
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no, he's saying that what you're saying is that B is tautological.
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These are equivalent:
- "A = 'A -> B'" is a valid statement
- B is universally true
Consider each sentence preceded by a - to be a proposition. They are equal.
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Oh, I understand now.
Dang- you might have solved it.
Awwww...... You broke my paradox!