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An unusual resolution of the liar's paradox.

Posted: Tue Mar 13, 2007 6:57 am UTC
I imagine you are all familiar with the liar's paradox:

This sentence is false.

I first propose the following resolution of the paradox: the logical value of the sentence exists in a quantum superposition of states, and the sentence is true (with a 50% probability) and false (with a 50% probability). Therefore, half of the time the sentence is true, and therefore false, and half of the time it is false and therefore true, so it is true with 50% probability, and false with 50% probability.
Yes, this is meant as humor, and not an actual resolution to the liar's paradox. Stop bugging me with your logic!
So, what is the probability that the following sentence is true?
This sentence exists in a quantum superposition of states, and is true with 50% probability and false with 50% probability.

Posted: Tue Mar 13, 2007 7:53 am UTC
Don't dis your idea; it's called "fuzzy logic". Although the term is usually used by people who don't know what it is and think they're clever, fuzzy logic is actually a well-defined branch of mathematics, useful for spam filters and rice cookers. It's a superset of boolean logic, wherein every statement is either true (with value 1) or false (with value 0). In fuzzy logic, statements can have any truth value between these inclusive. The usual boolean operators and, or, and not are replaced with functions of two and one variables. not(p) is almost always given by 1-p. So although the Liar's Paradox has no solution in boolean logic, because it would require p = 1-p, it clearly has the solution p = 0.5 in fuzzy logic.

Posted: Tue Mar 13, 2007 8:57 am UTC
It's easy to win the game when you change the rules.

Posted: Tue Mar 13, 2007 8:59 am UTC
The answer to your proposed question is it's false, all the time. Hence the probability it is true is 0.

Reasoning is simple, it can't be true all the time, because that would cause a paradox with what the sentence says. It can't be true/false 50% of the time, because that would make it true all the time. Therefore it must be blatantly false.

There can not be any percentage being true, lets say it's 1% true, and 99% false. Then at some time where the 1% makes it true, makes it actually false, because it isn't 50/50. Therefore it can never ever be true.

Edit:

Didn't explain that well.

1) It's true all the time - This can't be because then it wouldn't be true 50% of the time, and false 50% of the time.
2) It's true 50% of the time, and False 50% of the time - this can't be, because that would make the statement true in every instance.
3) It's true X percent of the time and false Y percent of the time, where X=/=Y and neither = 0 - This can't be, because when the instance in which it is true occurs it causes a paradox as in (1)
4) It's false all the time - this is the only one that works. The sentence is false.

And hence the probability of it being true has to be 0.

Posted: Tue Mar 13, 2007 5:37 pm UTC
Oh, I meant to answer your question as with fuzzy logic: "This sentence exists in a quantum superposition of states, and is true with 50% probability and false with 50% probability."

Actually, I don't know fuzzy logic, but here's my guess as to how this would work. This statement has truth value p, where:

p = (p eqv 0.5)

I'm not sure how the "eqv" operator would be defined, but I guess in analogy with boolean logic, you could define it as:

p eqv q = (p and q) or (~p and ~q)

The "and" operator is defined as min, and the "or" operator is defined as max, so you get that your statement requires:

p = max(min(p, 0.5), min(1-p, 0.5))

The right-hand side evaluates to 0.5 regardless of the value of p, so the solution is p = 0.5.

Posted: Wed Mar 14, 2007 6:33 am UTC
Fuzzy logic isn't about probability though; it's about how true a statement is, not the likelyhood of the statement being true. For example, a random person has an indeterminate height so he has some unknown probability of being at least 6 feet tall, but a person with a definite height may or may not be "tall", and one might use fuzzy logic to say how true it is to say they are tall. If they are very tall, it's very true, and if they're short, it's false, but if they are slightly taller than average, it's sort of true. So I don't think fuzzy logic really applies. If something similar applied, you probably wouldn't use min/max definitions of and and or, but you'd rather use the product to define and, define or by De Morgan's law, with some modification for when two statements are not independent.

I think gelsamel is right to the extent that the question even makes sense, but I think the question itself is pretty meaningless. I mainly just posted it to be amusing.

Posted: Wed Mar 14, 2007 8:32 am UTC
I'm really quite unsure what a "logical value" is... As far as I can tell, it's either a value assignment to a proposition (or something thereabouts), or some kind of abstract object (perhaps the state of affairs of the property truth inhering in something). Either way, it just doesn't seem to be the sort of thing that can be described by quantum physics. It's either a construct, or non-physical.

But still. I'm not sure how fuzzy logics are supposed to help us in the liar case. Graham Priest has some good work on this... He begins with classical, 2-valued logic, where (1) is a liar sentence:
(1) This sentence is false.
So suppose we add "Neither true nor false" as a 3rd truth value. We still have liar sentences. Consider (2):
(2) This sentence is either false or neither true nor false.
So what do we do now, add in "both"? This is Priest's suggestion. And it seems highly unintuitive. But one thing his work has shown is that so long as the truth values are considered as some sort of range over 0 to 1, either just the integers in [0,1], or the rationals in [0,1] or whatever, we can construct liar sentences. They just get more complex the more values we have. That is, at least, until the value "both" or "all" is countenanced.

Posted: Wed Mar 14, 2007 12:23 pm UTC
As far as fuzzy logic goes, The book "probability theory: the logic of science" by jaynes is good in that it shows that there is one, and only one, way to sensibly develop that theory, starting with some very intuitive axioms.

Posted: Wed Mar 14, 2007 4:05 pm UTC
Well, okay, but quantum superpositions aren't exactly about probability either. If something is in a superposition of A and B, it's very (though subtly) different from being "50% probability of A, and 50% probability of B". Just dealing with the probabilities doesn't make it a resolution to the paradox. If so, then you could write "this coin is heads up" on its tails side, write "this coin is tails up" on its heads side, flip it without looking at it, then claim that the statement on the up-side of the coin is true. That doesn't work. Neglecting Schrodinger's-cat-like effects, a 50% probability that the coin is heads up is not the same as the coin being in a quantum superposition of heads and tails.

Rather, the thing is in a state something like 1/sqrt(2) A + 1/sqrt(2) B with 100% probability. So I think that even though you used the word probability, the meaning of your statement is actually closer to fuzzy logic, than either quantum superpositions or fuzzy logic are to probability.

Posted: Wed Mar 14, 2007 8:11 pm UTC
Okay, clearly we need to be using Quantum logic. Anyone know anything about it? The wikipedia page kind of sucks in terms of explaining things...

Posted: Wed Mar 14, 2007 9:00 pm UTC
Okay, so the statement should have said "I am 1/rt2 true, 1/rt2 false." (Call this A) The orthogonal statement is "I am 1/rt2 true, -1/rt2 false." (Call this B)
We know that the truth value of the statement is r1*A+r2*B where r1^2+r2^2=1, and if so it is (r1+r2)/rt2 true+(r1-r2)/rt2 false. But it's true to the extent that it's A, and false to the extent that it's B, so we must have r1=(r1+r2)/rt2, r2=(r1-r2)/rt2. In other words, the truth value of the statement must be an eigenvector of the matrix
[1/rt2 1/rt2]
[1/rt2 -1/rt2]
with eigenvalue 1. The characteristic polynomial is x^2-1, so it does indeed have 1 as an eigenvalue, with corresponding (normalized) eigenvector .
That makes it's logical value true + false. Assuming that this is actually how one does quantum logic, which I have no idea. Can you have negative truth values? If this is right, and we observed the superposition, we would find the statement true with 85% probability, and false with 15% probability.

Of course, taking logical values to be points in a seperable hilbert space is beyond weird.

Posted: Thu Mar 15, 2007 4:24 am UTC
Actually I totally ignored the fact that that the liar's paradox is not like quantum superposition at all (and can't be solved like that).

I basically ignored the whole start bit and just answered his last question.

Posted: Wed Mar 28, 2007 11:54 am UTC
The whole problem with the liar's paradox is that it never actually evaluates.

This isn't because there's a 50% probability of it being incorrect.. it's because it sets into motion a sort of cascading logical path...

Code: Select all

this statement is false {
this statement is false {
this statement is false {
this statement is false...

and the closing braces } never actually appear, of course.

It seems to me like it's infinitely recursive.

It's hard to gather my thoughts on the matter..

but the statement, in describing itself, seemingly has to create a copy of itself in the logical 'data' space, which it then attempts to evaluate. This copy, in being evalutated, has to create a copy of ITSELF.. This is because all conclusions require premises - except axioms, which I don't really understand. So in evaluating itself, its like.. it has to create a copy of itself as a premise..

I think this is a nice idea but maybe it isn't completely logical ~~ my brain is actually exploding trying to think my way through this, in reality.

I should get back to my programming assignment =/

Posted: Wed Mar 28, 2007 12:21 pm UTC
Well if the first is 50/50, then 2nd is 50/50 or 50/50 or 25/25(??) then both the chance of it being true, and the chance of it being false tend to 0. It is neither true or false(??).

Posted: Wed Mar 28, 2007 6:29 pm UTC
Cynic wrote:but the statement, in describing itself, seemingly has to create a copy of itself in the logical 'data' space, which it then attempts to evaluate. This copy, in being evalutated, has to create a copy of ITSELF.. This is because all conclusions require premises - except axioms, which I don't really understand. So in evaluating itself, its like.. it has to create a copy of itself as a premise..

But it's not simply the self-reference that causes this problem. It's the phrase itself that makes it paradoxical. Self-reference alone is actually quite necessary in any kind of reasonable system of communication. In the introduction to a book, it's perfectly fine to talk about why I wrote "this book" and what "this book" will say, despite the fact that "this book" refers to something that includes the introduction in which the book is being discussed.

"This statement is being made on Wednesday, March 28, 2007" is not a paradox. It's just a true statement, even though it refers to itself. "This statement is not self-referential" is also not a paradox. It's just a false statement that happens to refer to itself.

"This statement is false" is a paradox, because there's no way to avoid a contradiction. If it's true, it's false. So it can't be true. But if it's false, it's not the case that the statement is false, which means it must be true. So it can't be false. Here, we're fucked either way, because it can be neither true nor false.

Re: An unusual resolution of the liar's paradox.

Posted: Wed Dec 12, 2007 12:22 pm UTC
i need some inspiration to make an essay about the consequence of liar paradox (and some other logical paradox if any) to breakthrough of fuzzy logic (or dynamic logic). i must make this essay to complete my english course.

btw, this is my first exploration to fuzzy world of logic.

so what's wrong with sentence like

this sentence contain thrae mistoke.

Re: An unusual resolution of the liar's paradox.

Posted: Wed Dec 12, 2007 10:14 pm UTC
glup.up wrote:so what's wrong with sentence like

this sentence contain thrae mistoke.

This isn't contradictory; it can be successfully evaluated as true or false. Counting up the mistakes:
1) "this" should be capitalized
2) "contain" is the plural form--should be "contains"
3a) "mistake" is misspelled
3b) "mistake" is the singular form--should be "mistakes"
4a) "three" is misspelled
4b) Three is the wrong number of mistakes. Should be four, five, or six, depending on how we count them.

That is, it could be considered one or two mistakes that the word "mistakes" has been misspelled "mistoke". If so, it could also be counted as one mistake that the word "four" has been misspelled "thrae". Or else, we could count the misspelling of "three" as a mistake, then say that the sentence is false, because there are more than three mistakes.

For paradox though, let me suggest: "This sentence contain three mistokes."
I've corrected 1, 3b, and 4a--so we still have 2 and 3a, but the third mistake is that there are only two mistakes. That is, the fact that it's false is the third mistake, but that means it's true....

Re: An unusual resolution of the liar's paradox.

Posted: Wed Dec 12, 2007 11:04 pm UTC
Consider:
We have heard the Administration say "There is no scientific consensus on Global Warming." This statement is false.

Here, "This statement is false," is actually a true statement. Without any context, "this" becomes an ambiguous term.

Re: An unusual resolution of the liar's paradox.

Posted: Thu Dec 13, 2007 6:30 am UTC
thank you EricH for the correction: it's suggest me to be more careful with english sentences.

i want to make such a good introduction to the Incompleteness Theorem of Godel that break Hilbert faith that mathematics is complete and consistent, so that motivate other mathematician to make unusual system of logic: fuzzy logic.

this is the famous barber paradox of Russell:

there is a town with a male barber who shaves precisely all those men in the town who do not shave themselves. does the barber shaves himself?

either the answer is yes or no, it will lead to a contradiction. first i want to investigate: the set whose elements were exactly all those sets that do not contain themselves as an element.
i believe that there is a connection to infinite set, but i still can not make such a good connection. any idea...

Re: An unusual resolution of the liar's paradox.

Posted: Mon Dec 17, 2007 4:40 am UTC
I'm with loon radio. You guys have fun

Re: An unusual resolution of the liar's paradox.

Posted: Mon Dec 17, 2007 10:51 am UTC
The problem here is that we want two things:
1. Everything that looks like a statement in English should actually be a statement.
and
2. Every statement should be only true or only false.

The Liar paradox is a sentence that shows that these two are contradictory, at least one must be false.

Either the first is false, so we define self-referencing or contradictory sentences not to be statements.
or
The second is false, some statements are neither true nor false, or are both.

Re: An unusual resolution of the liar's paradox.

Posted: Thu Dec 20, 2007 7:18 am UTC
Macbi wrote:2. Every statement should be only true or only false.

Anybody who's engaged in any serious lying, either as liar or being lied to, knows that this is false. Some statements are merely meant to be placeholders for actual content, conforming to the shape of information whilst saying absolutely nothing.

Re: An unusual resolution of the liar's paradox.

Posted: Thu Dec 20, 2007 10:32 am UTC
jestingrabbit wrote:
Macbi wrote:2. Every statement should be only true or only false.

Anybody who's engaged in any serious lying, either as liar or being lied to, knows that this is false. Some statements are merely meant to be placeholders for actual content, conforming to the shape of information whilst saying absolutely nothing.

Surely they can still be evaluated for truth or falseness? Can you give an example? Although I agree with you that 2 is false, I would say this is because statements such as the liar's paradox are both, and "this statement is true" would be neither.

Re: An unusual resolution of the liar's paradox.

Posted: Thu Dec 20, 2007 10:45 am UTC
I would say that there were a lot of statements made in the speech of politicians is littered with this sort of thing. Direct questions are met with vague references to vague things. There is a lot of vagueness.

For instance, what is a specific policy difference between Hilary Clinton and Barak Obama? You can search through what they say and find a few maybe, but they rarely answer the questions. In the recent Australian elections there was often absolutely no difference on a policy point and yet a voter would pick one or other party on the basis of that policy area.

Political speech is designed to say completely nothing. Either the terms are undefined (working families for the Australian audience), or it is a simple truism. Yet somehow opinions are created by this weird directionless drivel.

Re: An unusual resolution of the liar's paradox.

Posted: Fri Dec 21, 2007 4:54 am UTC
jestingrabbit wrote:Political speech is designed to say completely nothing. Either the terms are undefined (working families for the Australian audience), or it is a simple truism. Yet somehow opinions are created by this weird directionless drivel.

QFT.

Macbi wrote:Surely they can still be evaluated for truth or falseness? Can you give an example?

Ambiguous English sentences are very common. (It occurs to me that the ambiguity of language is one of the themes of xkcd comics...) One of my favorites is "Time flies like an arrow." True? False? It depends on how you choose to interpret it.

Re: An unusual resolution of the liar's paradox.

Posted: Fri Dec 21, 2007 10:53 am UTC
Ah yes, Okay, in that case you're right, those are neither true nor false, which is sufficient to break 2.

Re: An unusual resolution of the liar's paradox.

Posted: Thu Feb 02, 2017 7:14 am UTC
Sorry to necro this thread but I was watching someone talk about this and I disagree that there is a perfect 50/50 involved in this.

When you determine that the statement is both true AND false there are now three (or more) possibilities.

It's true
It's false
It's both true and false (which makes the statement false)

etc

I imagine the actual outcome is kind of complex, but it's almost certainly not 50/50.

Re: An unusual resolution of the liar's paradox.

Posted: Fri Mar 02, 2018 10:50 pm UTC
My standard answer to the Liar's Paradox is that it is neither true nor false; it is non-sense. I would argue that in order for something to be a statement at all or to have a truth value, it must be well formed. For an easy example, consider:

"Wednesday potato blue."

I think everyone would generally agree that this doesn't mean anything and does not have a truth value as it isn't even grammatically correct consisting of "noun noun adjective".

As pointed out above, it isn't just the self-referential nature that makes it non-sense or a paradox. It's that the self-reference has no termination point in an attempted evaluation of it's truth value. So, in addition to requiring a statement is grammatically correct in order to have a truth value, we should also require that an statement's evaluation tree be finite.

Note: if we adopt this rule, I do believe that this makes determining whether a statement has a truth value or is non-sense equivalent to the halting problem in general.