Brainteaser for those who like to have their brains bent

[Moonstone Man]

This statement is false.

Is that statement (^^^^^ up there) a true or false statement? Please show your working out, i.e prove your answer.

[xenopeek]

It's a false statement.

Let's assume for a minute this statement is true, truth value being 1. "This statement is false" can then be simplified to 1 == 0. The question whether it's a false or true statement can then be calculated. In bash test 1 -eq 0 && echo true || echo false answers this is a false statement.

[Hoser Rob]

It's neither true nor false. It may look like a categorical statement but it isn't, it just makes no sense.

[Moonstone Man]

It's a false statement.

Let's assume for a minute this statement is true, truth value being 1. "This statement is false" can then be simplified to 1 == 0. The question whether it's a false or true statement can then be calculated. In bash test 1 -eq 0 && echo true || echo false answers this is a false statement.

I'm sorry, no kewpie doll for you. I'll post the answer and proof later however...

whether it's a false or true statement can then be calculated

You are on the right track. It is calculable.

[Moonstone Man]

It's neither true nor false. It may look like a categorical statement but it isn't, it just makes no sense.

It is either true or false, and it is categorical, and it will make sense when it gets correctly answered. It seems like it makes no sense due to it operating on two distinct levels, so if it's not making sense then you're only seeing one of the two levels of meaning. That missing level of meaning is where the brain bend is.

What is the phrase 'this statement' referring to?

[rene]

What is the phrase 'this statement' referring to?

Either it refers to some statement we are not cognisant of whence it can not be decided by us whether or not the statement "This statement is false" is true or false or it refers to the statement "This statement is false" itself whereby this is the classical Liar's paradox --- and if you think that you have a proof of that one being either true or false (but not both) then you are mistaken.

Also see e.g. https://en.wikipedia.org/wiki/Liar_paradox; within classical logic neither truth-value can be assigned (and famously, a variant of this lies at the heart of the in mathematics / mathematical logic famous incompleteness theorems by Goedel). As the Wikipedia page says, one possible way out is to assign both truth values but we then still run afoul of double negation elimination hence of classical logic.

I expect you'll point out something particularly corny, but otherwise, no, do note that this is a very thoroughly studied paradox and any "proof" you have will not in fact be one.

[Moonstone Man]

and if you think that you have a proof of that one being either true or false (but not both) then you are mistaken.

And all that before seeing the proof.

Also see e.g. https://en.wikipedia.org/wiki/Liar_paradox;

Wikipedia to your rescue.

I expect you'll point out something particularly corny, but otherwise, no, do note that this is a very thoroughly studied paradox and any "proof" you have will not in fact be one.

Tell that to Kurt Gödel and waffle your way out.

[rene]

<shrug>

I'm sure you know better than 2000 years worth of logicians. And as to Goedel, yes I mentioned him already; I'm well aware of his result.

[DAMIEN1307]

I'm sure you know better than 2000 years worth of logicians. And as to Goedel, yes I mentioned him already; I'm well aware of his result.

rene, you know not who you are really in a "battle" of wits with...lol...just roll over and give it up now...You'll be happier that you did so...lol...DAMIEN

[Portreve]

Anyone looking at my location can see I'm fully qualified to deal with bends and curves.

The sentence given at the top of this thread is not a declaration, but rather a circular contradiction. True declarations cannot be self-referential.

As a result, the value I would assign to this is mu.

To give a different example: Assume you arm-wrestle with yourself, and one of your arms succeeds in forcing down your other arm. Did you win, or did you lose? While this is not an example of circular contradiction, it is an example of the production of an irrational answer (which, in essence, is what the sentence at the top of this thread effectively produces) and the proper response is, again, mu.

Probably the best-known broadcast example of carrying out the "This statement is false" circular contradiction (or, certainly, the most cheesy and absurd) is in the Star Trek: The Original Series episode "I, Mudd".

YouTube: Kirk, Spock, Scotty, and McCoy outwit the Androids

[rene]

rene, you know not who you are really in a "battle" of wits with...

Mathematics, mathematical logic included, is not ever a battle but the literally one and only part of humanity's doings in which no matter how loud one or the other speaks or how soft or hard one or the other's sword is fact can in fact simply be fact. In this case it is very much a fact that without some creative use of language his stated problem is the classical Liar's Paradox and that it is as such within classical logic not assignable either true or false as a truth value. Period.

[Moonstone Man]

True declarations cannot be self-referential.

On the subject of references, do you have any reputable sources to support that claim?

[Moonstone Man]

Mathematics, mathematical logic included ... as such within classical logic not assignable either true or false as a truth value. Period.

and if you think that you have a proof of that one being either true or false (but not both) then you are mistaken.

... any "proof" you have will not in fact be one.

Using Gödel's first incompleteness theorem:

All consistent axiomatic formulations of number theory which include Peano arithmetic include undecidable propositions. (Hofstadter 1989).

Consequently in any consistent formal system /L/ where some arithmetic can be performed, there are statements in the language of /L/ that are not capable of being proved or disproved using the language of /L/.

Given a set of axioms that are computably generated, let PROVABLE be the set of numbers to encode statements that are provable from the given axioms.

Let /s/ = any statement
Let /P/ = PROVABLE

Thus for any statement /s/:

1) </s/> is in /P/ iff /s/ is provable.

Since the set of axioms is computably generable, then so is the set of proofs that use these axioms, as is the set of provable theorems, thus /P/ is also computably generable, ie the set of encodings of provable theorems in the language /L/ are computably generable.

IMPLICATION: Since computable implies definable in adequate theories, /P/ can be defined therefore /P/ is definable.

Let /s/ be the statement "This statement is false". Using Tarski's Self-Reference Lemma: For any formula p(x) in an adequate theory, there is a statement /s/ such that /s/ iff p(</s/>) where </s/> is the number encoding /s/, ie in adequate theories, such equations always have solutions, thus /s/ exists since it is the solution of:

2) /s/ iff </s/> is not in /P/, therefore

3) iff /s/ is not provable then /s/ iff </s/> is not in /P/

Consequently, and applying excluded middle, /s/ is either true or false.

If /s/ is false, then by 3), /s/ is provable, which is impossible because provable statements are true therefore /s/ is true. It necessarily follows then that by 3), /s/ is not provable thus /s/ is true but unprovable.

Conclusion: There are statements in language /L/ that are true but cannot be proved.
Corollary 1: Truth lies beyond provability.

The correctness of the above is demonstrable in language /L/ via a simple analogy:

rene: "I am me."

KM: "Prove it."

[Moonstone Man]

... you know not who you are really in a "battle" of wits with...lol...

I don't fight with the unarmed, besides, I wrote the proof in contemporary first order predicate calculus, not "Mathematics, mathematical logic included ... as such within classical logic" [*1] so the proof is a waffling-piffle-proof proof. I'll just let the proof stand exactly as it is and rene can throw waffling piffle at it, completely devoid of reliable, reputable references and also devoid of both evidence and proof, until the cows come home. I will add though that the theorem is testable in real life.

[*1] Classical logic is completely undefined. Also note the lack of reference to contemporary logic, and the complete absence of any reference to truth and falsity values in propositional logic. Worst of all, the assertions are just that, absolutely no proof whatsoever, and not a single reliable reference cited. In short, it says "Trust me!"

If you say to me that you are feeling unwell, I cannot prove or disprove your claim because I cannot enter your mind so as to directly experience being you as if I were you, which is what would be required to either prove or disprove your claim thus I am forced to rely on assumptions that are underpinned by observations measured against how I assume I might feel if the observations applied to me, and those that might apply to me are usually further underpinned by my own direct experience of my own internal states; we might call this empathy. Consequently, any statements about the self's internal states are only provable within the self. Not only is that statement subject to yet another Gödelian proof (statements by the system about the system are only provable within the system itself), it is what forces and causes us to make far too many assumptions before, during and after, every waking split-second moment of our lives. It is these constant assumptions that trip so many people up; they become so used to taking assumptions as absolute, undeniable facts that they are incapable of perceiving actual reality distinct and separate from their self-created perception of an assumptive universe existing in some nebulous place out there --->

And the corollary of all that is that the only actual reality is that conceived and perceived in the mind. It is not out there ---> What is out there ---> is a massive set of collectively agreed on, and often disputed, set of assumptions.

Einstein fell into that trap, and often. Most notably when he wrote, complaining about the implications of quantum mechanics, "I like to think that the moon is there even if I am not looking at it." Quantum mechanics aside, the moon is there when he isn't looking at it only because almost the entire human population has agreed to rely on the assumption that it is there even when not one single person is looking at it. Besides that, and conversely since the probability wave of the moon has been collapsed by a first observer into actuality (Copenhagen Interpretation), it exists when no one is looking at it even if we all agree that it does not exist when no one is looking at it.

His waffling piffle cut no ice and he floundered on the stage. The loud laughter that accompanied the trust question finished him off. Yet incredibly by the next day the footage of that defining moment in the debate was edited out by the BBC so that viewers watching the Saturday bulletins concerning the previous night’s debate were prevented from witnessing the loud laughter directed at [him] as he pleaded to be trusted.

https://astutenews.com/2019/11/the-rich ... s-raining/

[rene]

I really quite literally hardly know where to start --- but let us then I guess. You explicitly adhere to excluded middle, so good.

At 1 we have, for any statement s,

Code: Select all

1	(<s> in P) <=> (s is provable)

We can take this as mere definition of P (or of <s>) but fine, let us take it at face value. It is then claimed that the specific statement "s: This statement is false" is the solution of

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2	s <=> (<s> not in P)

which we with the help of 1 find equivalent to

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2'	s <=> s is not provable

That is, what you have claimed is that our statement s is true if and only if it is not provable. This follows from absolutely nothing. Moreover and in fact you note only a bit later that a provable statement is true, i.e., s is provable => s, which contradicts that very claim.

But still, let us again have another attempt to take things at face value and keep on trying. You then say that, given 2, "therefore",

Code: Select all

3	(s is not provable <=> s) <=> <s> not in P

which we with again 1 find equivalent to

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3'	(<s> not in P <=> s) <=> (<s> not in P)

This means you have just now simply defined s to be true since this is otherwise absolutely not the case. In conclusion you then find that, surprise, s is true; that tends to happen if you define it so indeed.

I'm relatively sure you'll not find these problems to be so but do note that I from your reaction up to now know that you are neither a mathematician nor a mathematical logician; the above mentioned thing as to objectivity is fundamentally engrained in either and specifically my comments up to now would not have been interpreted combative (and which indeed they were not intended as, certainly not on a personal level); any even halfway serious mathematician loves to be properly proved wrong. Of course, there's also that thing about the already 2000+ years worth of discussion of this very paradox which you would have not swept aside...

I'll keep it at this but for the interested: the only rebuttal of the Liar's Paradox I find interesting is Buridan. He states that any statement other than what it in fact states also implicitly states its own truth simply by virtue of being stated. In this case this would be to mean that

Code: Select all

This statement is false

in is fact equivalent to

Code: Select all

(This statement is false) AND (This statement is true)

which as a logical statement is not a paradox any more but simply false.

Problems with this approach to the paradox is the either ad hoc nature of this introduced virtual implication, i.e., needing a specific rule for a specific case, or if alternatively NOT ad hoc leading to a truthless logic since if anything that is implied by some given statement, infinitely many things, must be decided upon then eventual truth can not be --- whence we still haven't an answer to the specific question posed by the paradox. But still; hmm.

The Liar's Paradox really is nasty: there's more than 2000 years worth of literature documenting attempts to come to grips with it.

[Termy]

As a programmer, I see it as true, because it's a value, regardless of that value simply saying it's false; that being said, this isn't code, and we have no idea what the limitations are, if any, for what is or isn't true. Interesting thread.

[Moonstone Man]

As a programmer, I see it as true, because it's a value, regardless of that value simply saying it's false; that being said, this isn't code, and we have no idea what the limitations are, if any, for what is or isn't true. Interesting thread.

Code: Select all

class Base {
   public:
    bool true() {
        return false;
    }
};

class Derived : public Base {
   public:
    bool true() {
        return false;
    }
};

int main() {
    Derived derived;
    cout << "True: " << derived.true() << endl;
   
    return 0;
}