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#1 octopuppy

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Posted 14 February 2011 - 01:10 PM

I had a dream the other night which got me thinking. In the dream I was taking part in a pub quiz. The 1st question was to write down the entire lyrics to Led Zep's Immigrant Song (actually, being a dream it wasn't really Immigrant Song but something equally epic). I didn't do so well on that question, after the 1st few lines I was a bit unsure. Anyhow, the 2nd question was this: "Evaluate the following statement: 'This is a mystery' "
Now that's obviously a bit open to interpretation but the way I saw it was that I needed to determine the truth value of the statement, interpreted self-referentially. For clarity you could reword it as "This statement cannot be evaluated as true or false" *.
I woke up thinking about that and carried on thinking about it all morning.

For sure, the statement can not be evaluated as being definitely true. That would lead to a contradiction.

If it were false, no contradiction arises. The trouble is, can we be sure that it is false?

EDIT:
* To be clear, by "evaluate as x" I mean to ascertain that x is the only possible value.
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#2 Chokfull

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Posted 14 February 2011 - 08:24 PM

I had a dream the other night which got me thinking. In the dream I was taking part in a pub quiz. The 1st question was to write down the entire lyrics to Led Zep's Immigrant Song (actually, being a dream it wasn't really Immigrant Song but something equally epic). I didn't do so well on that question, after the 1st few lines I was a bit unsure. Anyhow, the 2nd question was this: "Evaluate the following statement: 'This is a mystery' "
Now that's obviously a bit open to interpretation but the way I saw it was that I needed to determine the truth value of the statement, interpreted self-referentially. For clarity you could reword it as "This statement cannot be evaluated as true or false".
I woke up thinking about that and carried on thinking about it all morning.

For sure, the statement can not be evaluated as being definitely true. That would lead to a contradiction.

If it were false, no contradiction arises. The trouble is, can we be sure that it is false?


I'd say it's false, as there is no alternative. If it is a declarative statement it has to be either one or the other.
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#3 octopuppy

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Posted 14 February 2011 - 08:38 PM

I'd say it's false, as there is no alternative. If it is a declarative statement it has to be either one or the other.

A statement like "This statement is true" cannot be evaluated as true or false. It's simply undefined, as it works either way. I was thinking about whether the statement "This statement cannot be evaluated as true or false" might fall into the same category.
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#4 Chokfull

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Posted 15 February 2011 - 05:43 AM

Let me clarify a bit.

If the statement was true, it could be evaluated as true. And yet, according to the statement, it could not be evaluated as true. Since it cannot be both possible and impossible to evaluate it as true, the statement cannot be true. THerefore the statement is false.

I have just proven that the statement can be evaluated as false, by doing so.

Your question is whether the statement falls into the category of statements which cannot be evaluated as true or false.

This statement can be evaluated as false, it does not fall into the category of statments that cannot be evaluated as true or false.

Simple logic.
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#5 octopuppy

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Posted 15 February 2011 - 08:05 AM

Let me clarify a bit.

If the statement was true, it could be evaluated as true.

Maybe I haven't been clear enough. Taking "this statement is true" as an example, while an assumption that it is true or false leads to no contradiction, in order to ascertain that it is false, we would need to show that it cannot be true, and vice versa. We can't do that so we can't evaluate it either way. Maybe your interpretation is different, but by "evaluate as x" I mean to ascertain that x is the only possible value. I'll think about rewording the OP for more clarity...

Anyway, bearing that in mind and going back to "This statement cannot be evaluated as true or false", our inability to evaluate it as true doesn't preclude the possibility that it may be true anyway. We just know that we can't be certain that it is true.

Moving things forward a little, if we use "undefined" to denote the state where neither truth nor falsehood can be ascertained, we then run into difficulties when we consider this as being the answer. If we know that the statement is undefined, then the claim made by the statement is true. That takes us back into a contradiction. So the statement cannot be evaluated as true or as undefined. Can we now evaluate it as false?

We cannot evaluate (prove) the statement to be true.
We cannot evaluate (prove) that the statement even might be true.

Does that equate to falsehood? I don't think it does. The statement is not IMO proved false and yet we cannot even state with any certainty that it might possibly be otherwise. :wacko:

If I'm right, we have a question without an answer. Even saying that we cannot know the answer is not a valid answer! Dang those pub quizzes, they can be pretty tough sometimes :lol:
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#6 Chokfull

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Posted 15 February 2011 - 05:34 PM

I see what you're saying, but I disagree. I suppose we just have to agree to disagree. However, when a statement is about itself' why does it matter whether it's true, false, or whatever? :D
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#7 octopuppy

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Posted 15 February 2011 - 08:26 PM

I see what you're saying, but I disagree. I suppose we just have to agree to disagree. However, when a statement is about itself' why does it matter whether it's true, false, or whatever? :D

That's probably why it was a pub quiz in the dream. In a pub quiz, everything matters.
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#8 plasmid

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Posted 24 February 2011 - 06:21 AM

This took quite a bit of thought, but think I finally got a reasonable grasp on it. It's trying to lead us into the following trap.

If you suppose that the statement is true, and then try to evaluate the statement under that supposition, then the statement would evaluate as false. If you suppose that the statement is unevaluable, and then try to evaluate the statement, then the statement would evaluate as true. If you suppose that the statement is false, and then try to evaluate the statement, then the statement would evaluate as false. So the only self-consistent answer would be that the statement is false.

Does that prove that the statement is false? Deep down, I don't think so. I suspect that the trap lies in the way a statement is defined as unevaluable. For "this statement is true", if you try to evaluate the function under the supposition that it is true then it evaluates to "true", and if you evaluate it under the supposition that it is false then it evaluates to "false", so both truthfulness and falsehood are self-consistent answers. For "this statement is false", both truthfulness and falsehood are self-contradictory answers. This leads one down the path of thinking that a criteria for evaluability is that there is one and only one self-consistent answer.

My rebuttal would be that sure, there is not a unique self-consistent answer for either "this statement is true" or "this statement is false", but that’s not how one ought to define evaluability. Those statements are both invoking circular logic, and that’s what is really at the heart of what makes them unevaluable.

Now the statement "this statement is unevaluable" is likewise invoking circular logic. Does that mean that it is therefore unevaluable? (Which ironically would make you think that the statement evaluates to "true".) Well, consider a similar statement: "This statement is either true or false." Assuming that it is true produces the evaluation that it is true, and is therefore self-consistent. Assuming that it is false produces the evaluation that it is true, and is therefore not self-consistent. And if you were to assume that it is unevaluable then it would evaluate as false, and therefore not be self-consistent. And for what it’s worth, it seems like most rational people would consider the statement to be true. But does its very nature as a self-referential statement render it unevaluable due to circular logic, despite the fact that "true" is the only self-consistent answer? My gut feeling is that both of these statements ("this statement is unevaluable" and "this statement is either true or false") are in fact unevaluable due to their inherent circular logic, even though working under the assumption that they are unevaluable produces the conclusion that they must evaluate to be true or false, respectively.

The main argument against my stance seems to be "If you evaluate the statement under the supposition that it is unevaluable, then you reach the conclusion that the statement is true, therefore reaching a self-contradiction that proves that the statement cannot be unevaluable". Well, consider "this statement is false" which was proven to be unevaluable because supposing either truthfulness or falsehood would lead to a contradiction. If you try to evaluate that statement under the assumption that it is unevaluable, and slightly rephrase it as "this statement is evaluable and evaluates to false", then the statement would evaluate as being false rather than unevaluable and would therefore make the conclusion that it's unevaluable seem self-inconsistent. But this is the very statement with which we demonstrated the whole concept of unevaluability! That just goes to show that with an unevaluable statement, if you attempt to evaluate it under the assumption that it is unevaluable and thereby reach an apparent self-contradition because it would evaluate to either true or false, that really doesn't prove anything.

Logicians have always had a healthy fear of self-referential statements, as they can from time to time make and Honestant or Swindlecant's head explode. (An example of this is given in part I of my old "How would you cross puzzle land" mini-series.) Set theorists even came up with the rule that no set can contain itself as a member precisely to prevent wise guys from defining "the set that contains all sets that do not contain themself as a member" and asking whether that set contains itself. So I think that despite all the apparent weirdness going on, I'd call them unevaluable.
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#9 octopuppy

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Posted 24 February 2011 - 11:52 PM

...So I think that despite all the apparent weirdness going on, I'd call them unevaluable.

That was an enjoyable read and I'm inclined to agree with you. The trick of making the "unevaluable" evaluation lead to a contradiction doesn't mean it isn't so. But I'm still not sure what I'd write down for the answer in the pub quiz :(

I'll bet this will amuse you. The book that got me thinking about self-referential statements is "Metamagical Themas" by Douglas R. Hofstadter. Here's a passage from it:

What is it like to be asked,
"What is it like to be asked, self-embedded in quotes after its comma?"
self embedded in quotes after its comma?


Here again, you are invited to construct a typographical entity that turns out, when the appropriate operations have been performed, to be identical with the set of instructions. This self-referential question suggests the following puzzle: What is a question that can serve as its own answer?
Readers might enjoy looking for various solutions to it.

Spoiler for Some answers that occurred to me:


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#10 Segul

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Posted 27 May 2011 - 07:46 AM

I agree with chockful's logic and I feel it is false.
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