[Guest]

The other day I read a statement saying "This statement cannot be proved".

Is this statement Vrai ou Faux (True or False)?

(Yeah, sorry about the French, I like that little thing, Vrai ou Faux)

Edited: 20:20 03/11/2008 Monday

Edited November 3, 2008 by Hellgate-London

[Guest]

It is false. Of course it can be "proved". Although "proven" would be the most common use as a modifier in a statment of this particular structure, the use of "proved" as a past participle is still acceptable in English grammar.

[Prime]

Same has been posted just recently.

With a number of other variations before. (See "Crocodile Sophism").

[Prime]

Same has been posted just recently.

With a number of other variations before. (See "Crocodile Sophism").

Actually, I should take it back. "Cannot be proved" is not the same as "This statement is false".

[Prime]

I think "This statement cannot be proven" does not lead to the same kind of paradox as "This statement is false".

If we interpret "proof" as a proof of truthfulness (not proof of falsehood), then the statement may be interpreted as asserting impossibility of proof with respect to itself. Which looks plausible, as long as it cannot be proven.

[unreality]

Let's assume "proven" = "shown without doubt to be true", as I assume that's what you mean

'This statement cannot be proved' thus equals: 'This statement cannot be shown without doubt to be true'

If the statement is false, then the statement CAN be shown true

If the statement is true, then the statement CANNOT be shown true... which doesn't mean that it isn't true, it just cannot be proven

going back to if it's false, thus it CAN be shown true... if it is possible to show it being true without doubt, then it must be true. Thus if it's false, then it's also true

Which means the only non-paradoxical option is if the statement is true. Being false means that it's false and also true, but being just true means that it is TRUE but CANNOT BE PROVEN to be true

it is true

[Guest]

Let's assume "proven" = "shown without doubt to be true"

I disagree. We cannot assign greater meaning to "proven" than "shown without a doubt." If I were to state: "Five singing bananas in purple jumpsuits cannot be proven," you would of course immediately respond "Proven to be what?" True? No, that doesn't make sense. Singing? Maybe. Part of a balanced breakfast? ... You get the idea -- it could mean anything. So this, as well as the OP, are simply incomplete statements.

I am reminded of the tired old standby:

If a tree falls in the forest and nobody hears it, does it make a sound?

Who gives a @&!* ?

:-P

[Guest]

The other day I read a statement saying "This statement cannot be proved".

Is this statement Vrai ou Faux (True or False)?

but if we use the olde definition of proof ("test" as oposed to "demonstrate to be true" as in 'the exception that proves (tests) the rule) then we can test the statement and it turns out to be false.

Edited November 7, 2008 by armcie

[Prime]

Upon further reflection:

Let S be a statement. Let P(statement) be a function: Proof of a statement. Lets designate NE(argument) as "Does not exist".

Then S = NE(P(S)). Substituting S, we have: S = NE(P(NE(P(S)))). Whereupon we can substitute S again, and again… An infinite regression. I say, the statement is undecidable.

[Guest]

Upon further reflection:

Let S be a statement. Let P(statement) be a function: Proof of a statement. Lets designate NE(argument) as "Does not exist".

Then S = NE(P(S)). Substituting S, we have: S = NE(P(NE(P(S)))). Whereupon we can substitute S again, and again… An infinite regression. I say, the statement is undecidable.

What the...? I didn't want people to go into so much mathematical detail!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[Guest]

Ok... No one seems interested in this anymore, so I'll just try and remember the answer...

Ok. If the statement "This statement cannot be proved to be true" (Yeah - I know, I forgot to put that bit in and just remembered it!) is true, then the statement is false. If the statement is false then the statement is true. So therefore - There is no answer! It's an endless loop. Unless you added "impossible" into the options of answers, that is!

This was just a puzzle to see if you could figure out if there was an answer or not!

Read the spoiler if you want to know the answer, or PM me the answer YOU think it is, or just PM me if you think I've done it wrong.

[bonanova]

If a tree falls in the forest and nobody hears it, does it make a sound?

My answer to this chestnut has always been:

If a falling tree can be postulated without a corroborating observer, then its sound can equally be postulated without a corroborating listener.

Corollary: If a man speaks in a forest. and no woman is present to listen, is he still wrong?

[Guest]

Let's assume "proven" = "shown without doubt to be true", as I assume that's what you mean

'This statement cannot be proved' thus equals: 'This statement cannot be shown without doubt to be true'

If the statement is false, then the statement CAN be shown true

If the statement is true, then the statement CANNOT be shown true... which doesn't mean that it isn't true, it just cannot be proven

going back to if it's false, thus it CAN be shown true... if it is possible to show it being true without doubt, then it must be true. Thus if it's false, then it's also true

Which means the only non-paradoxical option is if the statement is true. Being false means that it's false and also true, but being just true means that it is TRUE but CANNOT BE PROVEN to be true

it is true

But, unreality, your explanation here is ITSELF a proof: you have just proven that the statement MUST be true, and therefore it is false, which means it is true, which means it is false... Therefore no possible answer.

[bonanova]

Ok... No one seems interested in this anymore, so I'll just try and remember the answer...

Ok. If the statement "This statement cannot be proved to be true" (Yeah - I know, I forgot to put that bit in and just remembered it!) is true, then the statement is false. If the statement is false then the statement is true. So therefore - There is no answer! It's an endless loop. Unless you added "impossible" into the options of answers, that is!

This was just a puzzle to see if you could figure out if there was an answer or not!

Read the spoiler if you want to know the answer, or PM me the answer YOU think it is, or just PM me if you think I've done it wrong.

As others have noted, this is different from the Liar paradox "I am lying." or "This statement is false."

The assertion "This statement cannot be proved." is a close approximation to a proof of Gödel's first incompleteness theorem.

Logicians up through Hilbert, but starting with the Greeks, envisioned a system of proof which, given a useful set of axioms,

could compute proofs of all valid theorems, therefore creating a complete and self-consistent system of logic.

Didn't happen.

Gödel blew that dream up by showing that any system of inference, sufficiently rich to be useful, had a fatal flaw:

It was either incomplete or it was inconsistent. For example, it is impossible to prove the statement in the OP.

Being incapable of proof, it is true. But to be complete, the system must be able to prove it.

Thus the system is incomplete.

So this is an example of a Gödel statement.

It's true by the fact the mere fact that it cannot be proved.

And that fact - that the system contains a true statement that eludes proof - makes the system incomplete.

It takes a moment or two of thought: it's almost as if Gödel is saying you simply have to say the system is incomplete for it to be so.

Which doesn't seem intuitive.

But it stands.

P.S.

Gödel was not liked when he did this.

To show their disdain for Gödel, logicians have simply gone ahead and used the system to achieve useful results, anyway

[Guest]

The existence of a true paradox is a paradox in itself. For a paradox presents a logical impossibility and therefore it is paradoxical for something which cannot exist to exist.

[Guest]

Take a step back from "This statement is false." Consider that True and False are not the only possibilities in terms of this statement. This statement is neither. This statement is merely invalid.

An invalid statement is neither true nor false.

[bonanova]

Take a step back from "This statement is false." Consider that True and False are not the only possibilities in terms of this statement. This statement is neither. This statement is merely invalid.

An invalid statement is neither true nor false.

It's clearly undecidable.

When you say [not] valid do you mean [not] decidable?

Validity usually has a formal sense, in which statements of the form "This A is B." are permitted, independent of A and B.

By your definition, is "An invalid statement is neither true nor false." valid?

[Not] provable or [not] decidable are more useful terms to use here I think.

You must be permitted to make a statement; then decide on [a] it's truth or provability.

[Guest]

It's clearly undecidable.

Do you mean valid to convey decidability?

Validity has a formal definition, which includes statements of the form This A is B.

What is your definition of validity?

"This statement is false" is an invalid statement in the same way that "Cow touch throw at mail" is an invalid statement. Neither is true, nor false. They're just invalid statements. The latter is an abuse of grammar, the former an abuse of logic. A "paradox" is just wordplay of language, an attempt to mask or veil the invalidity of the statement by trapping you in a mindset where a statement must be true or false. This is why parodoxes do not occur in nature, only language.

[Guest]

I would equate it to the "2 = 0" problem posted here (and many other places).

Quadratically speaking, the solution could be X = 2 or X = 0. The author used invalid math and arrived at a semi-believable answer (2 = 0). However, if the same problem were to occur in language, instead of math, it may appear to some people as a logical impossibility (2 = 0 is impossible).

This equation is not paradoxical; It's just invalid. In the same sense, language is misused just as math is misused.

In our math "paradox", our tool (mathematics) was used improperly.

In our language "paradoxes", our tool (language) is used improperly.

Edited February 3, 2009 by Llam4

[Guest]

Perhaps the statement is quadratic; It is true and it is false.

(Just kidding)

Edited February 3, 2009 by Llam4

[Guest]

Consider the following mathematical equations. Exclude zero as it does not have a polarity. Are they paradoxes? What would you call these? I would call these inequalities, not paradoxes:

x = -x


and


x = y

y = -x

A "paradox" is nothing more than an invalid equation. To put this in terms that are very easy to understand (in my opinion), consider a true/false statement as a mathematical equation. Each statement is a variable (x, y, z, etc). A true statement is positive, a false statement is negative. A statement is equal to what it is referring to.

"The below statement is true."

"The above statement is true."

x = y

y = x

This one requires little explanation.

VALID LOGIC

"The below statement is true."

"The above statement is false."

x = y

y = -x


Let's prove that x = y and y = -x can not both be true:

If (x = y) then

y = -x can be substituted for

x = -x

Which is invalid.

INVALID LOGIC

"The below statement is false."

"The above statement is false."

x = -y

y = -x


x = -y (x is true and y is false)

y = -x (y is true and x is false)

^^ Either of these answers are correct.


Lets prove that either of these answers can be correct:

If (x = y) then

y = x can be substituted for

x = x

Which is valid.

VALID LOGIC

Now take into account the one-liners. These are much simpler.

"This statement is true."

x = x

VALID LOGIC.

"This statement is false."

x = -x

INVALID LOGIC.

Edited February 3, 2009 by Llam4

[Prime]

Take a step back from "This statement is false." Consider that True and False are not the only possibilities in terms of this statement. This statement is neither. This statement is merely invalid.

An invalid statement is neither true nor false.

I don't quite follow the "invalid". Does it violate some rule according to which statements must be constructed? If so, what rule?

I could understand "invalid logic" where you specifically show some derivation rule that was violated. But how does that apply to "invalid statement"? If in trying to resolve the statement we applied some logic rules incorrectly, we must discard our solution, but not the statement.

As others have noted, this is different from the Liar paradox "I am lying." or "This statement is false."

The assertion "This statement cannot be proved." is a close approximation to a proof of Gödel's first incompleteness theorem.

Logicians up through Hilbert, but starting with the Greeks, envisioned a system of proof which, given a useful set of axioms,

could compute proofs of all valid theorems, therefore creating a complete and self-consistent system of logic.

...

Gödel's incompleteness theorem is a proof that certain logical systems will necessarily have undecideable propositions. But it does not mean that those propositions cannot be decided outside the system. For example, in the crocodile and baby paradox, we assume that the crocodile is bound by honor to keep his word. Outside of that nothing stops the crocodile from eating the baby.

d3k3 already introduced one possible outside system where the statement does not present the dilemma, namely "who cares" system. Other than that, did we come to a conclusion that we have no logic tools at our disposal to resolve the statement?

I think, the notion of "proof" is somewhat undefined in our discussion. So "This statement cannot be proven to be true" is ambiguous. Note, in Gödel's work, the "proof", or "demonstration" is defined quite explicitly.

[Guest]

The other day I read a statement saying "This statement cannot be proved".

Is this statement Vrai ou Faux (True or False)?

(Yeah, sorry about the French, I like that little thing, Vrai ou Faux)

Edited: 20:20 03/11/2008 Monday

cannot be proved what? true or false?

Edited February 3, 2009 by Kevin_3.1415

[Guest]

I don't quite follow the "invalid". Does it violate some rule according to which statements must be constructed? If so, what rule?

"This statement is false."

Let's break it down to some basic algebra:

"This statement" is our variable. We will call it X.

This statement refers to itself, and claims that it is not true. If "this statement" (x) is not true, then it can't be equal to "this statement" (x).

Therefore, "This statement is false" translates to x != x.

If you read back a post, I related the "paradox" to the inequality x = -x. My method was an example, but in reality the statement claims that x can be anything except x.

[Prime]

"This statement is false."

Let's break it down to some basic algebra:

"This statement" is our variable. We will call it X.

This statement refers to itself, and claims that it is not true. If "this statement" (x) is not true, then it can't be equal to "this statement" (x).

Therefore, "This statement is false" translates to x != x.

If you read back a post, I related the "paradox" to the inequality x = -x. My method was an example, but in reality the statement claims that x can be anything except x.

In Gödelian basic algebra notation: x ⊃ `x (IF x, THEN NOT x).

The above statement is not consistent (perhaps, what you call invalid.) It states a contradiction. There are some statements, which cannot be proven -- the proof leads to an incosistent (contradictory) statement.

Whereas, x = -x is a valid algebraic statement and it has a solution (x = 0).

So if your proof for the statement "this statement cannot be proven to be true" yielded "true", that would create an inconsistency. On the other hand, if we derived a proof that yielded "false" that would be a contradiction too. The accepted conclusion is rather we cannot have a proof, not that we cannot have the statement.

[Guest]

In Gödelian basic algebra notation: x ⊃ `x (IF x, THEN NOT x).

The above statement is not consistent (perhaps, what you call invalid.) It states a contradiction. There are some statements, which cannot be proven -- the proof leads to an incosistent (contradictory) statement.

Whereas, x = -x is a valid algebraic statement and it has a solution (x = 0).

So if your proof for the statement "this statement cannot be proven to be true" yielded "true", that would create an inconsistency. On the other hand, if we derived a proof that yielded "false" that would be a contradiction too. The accepted conclusion is rather we cannot have a proof, not that we cannot have the statement.

Not all statements are true/false. Consider opinions. If opinions have a place anywhere in such equations, I'd say 0 = opinion.