[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

[bonanova]

"This statement is false" is an invalid statement in the same way that "Cow touch throw at mail" is an invalid statement.

What invalid means to you is still unclear.

Two comments:

1. Your simile seems shaky:
   
   "This statement is false." is formally valid, both logically and syntactically; equally valid in a formal sense as "This statement is true." is.
   Given the formal validity of the statement, it becomes possible to discuss its meaning, its truth value, its decidability, etc...
   That the statement can be discussed cannot be even questioned: It is being discussed in this forum.
   To disqualify it as being not discussable ignores that fact.
   
   To disqualify it by saying it's invalid, is to beg the question: invalid is defined by its usage, producing justification by assumption: viz, a circular argument.
   (a) It's disqualified because it's invalid.
   (b) It's invalid, because to say so lets me disqualify it.
   
   By stark contrast, "Cow touch throw at mail." is not formally valid in either sense; and cannot be discussed.
   
   
   
2. You use the term invalid to disqualify a formally valid statement, rather than discuss it.
   But neither your simile nor your disqualification of the statement sheds light on why disqualification is permissible.
   If your meaning of invalid were out in the open, one could take the statement to that definition and justify [or not] disqualification.
   
   To that end,
   

A term may be defined in [at least] three ways.

1. By giving examples - as in "Cow touch throw at mail.", but that likeness is in dispute.
   
2. By explaining its usage - but how to use invalid, once understood, is probably not in dispute.
   
3. By substituting another word [already understood] or phrase - with sufficient precision to justify the intended use of the term.
   

[Guest]

What invalid means to you is still unclear.

Two comments:

1. Your simile seems shaky:
   
   "This statement is false." is formally valid, both logically and syntactically; equally valid in a formal sense as "This statement is true." is.
   Given the formal validity of the statement, it becomes possible to discuss its meaning, its truth value, its decidability, etc...
   That the statement can be discussed cannot be even questioned: It is being discussed in this forum.
   To disqualify it as being not discussable ignores that fact.
   
   To disqualify it by saying it's invalid, is to beg the question: invalid is defined by its usage, producing justification by assumption: viz, a circular argument.
   (a) It's disqualified because it's invalid.
   (b) It's invalid, because to say so lets me disqualify it.
   
   By stark contrast, "Cow touch throw at mail." is not formally valid in either sense; and cannot be discussed.
   
   
   
2. You use the term invalid to disqualify a formally valid statement, rather than discuss it.
   But neither your simile nor your disqualification of the statement sheds light on why disqualification is permissible.
   If your meaning of invalid were out in the open, one could take the statement to that definition and justify [or not] disqualification.
   
   To that end,
   

A term may be defined in [at least] three ways.

1. By giving examples - as in "Cow touch throw at mail.", but that likeness is in dispute.
   
2. By explaining its usage - but how to use invalid, once understood, is probably not in dispute.
   
3. By substituting another word [already understood] or phrase - with sufficient precision to justify the intended use of the term.
   

I've given lots of examples. Just because the sentence is written correctly doesn't mean it's logical.

1 + 1 = 1 + i

Is this written in proper mathematical notation? Sure. Does that mean its logical or correct? No.

If you wrote 1 + 1 = 1 + i in a math class I was teaching, you'd get an F for being pretty bad at math.

If you wrote "This sentence is false." in an English class I was teaching, you'd get an F for being pretty bad at English.

When I say invalid I mean that the statement was constructed wrong, much like an unequal equation. In mathematical terms, the statement presents an inequality.

[bonanova]

I've given lots of examples. Just because the sentence is written correctly doesn't mean it's logical.

1 + 1 = 1 + i

Is this written in proper mathematical notation? Sure. Does that mean its logical or correct? No.

If you wrote 1 + 1 = 1 + i in a math class I was teaching, you'd get an F for being pretty bad at math.

If you wrote "This sentence is false." in an English class I was teaching, you'd get an F for being pretty bad at English.

When I say invalid I mean that the statement was constructed wrong, much like an unequal equation. In mathematical terms, the statement presents an inequality.

1 + 1 = 1 + i is formally valid.

It is also decidable. Only because of that can we say that it's incorrect, or logically it's false.

Your "Cow" statement is tantamount to something like 1 + = x 4 - x 3 - / i 13 / = 45 / -.

That is formally invalid and cannot be evaluated or discussed as to truth, meaning, decidability.

You can't say it's false, because you can't even compose an assertion from it.

That is the point.

This discussion is about pinning down the meaning of valid.

You are being a bit slippery about it because on the one hand you say if a statement is false or if it's undecidable, then it's invalid.

Then you say if a statement is invalid, that means you can't even discuss it. [i.e. carry on a discussion of OP]

Because calling it invalid is to have discussed it, you can't have both meanings.

------------

Consider your ability to say that someone had just told you a lie.

Well, you couldn't.

Because by your definition, it would be an invalid statement.

Being invalid, you would be unable to determine whether it was true.

If you say no, first you determined that is was not true, then called it invalid, you're no better off:

You still have an invalid statement on your hand and you'd have to apologize to yourself for having determined it to be false.

Because, of course, you can't analyze invalid statements.

Net:

I agree that you cannot analyze an invalid statement. A statement has to have formal validity for it to be analyzed as to meaning.

I do not agree - it seems it only needs to be stated - that a false [or undecidable] - statement is not therefore invalid.

False statements can [must] be formally valid.

[Prime]

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

Some opinions are more equal than others.

[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

~HL~ says: This above statement is not dissimilar to my own. I mean, look at this: ""The below statement is true." "The above statement is false."" (the reason I used 2 sets of quotes is because I'm quoting what you said) If the below statement IS true, and the below statement says the above statement is false, that would mean that the above statement is true, ok - i forgot what I was muttering to myself as I was saying it! Well - I'm surprised that people actually ARE interested in my topic now... now to think of a new one! see you all next time for Statement Version 2.086! 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

"This statement is true."

x = x

VALID LOGIC.

"This statement is false."

x = -x

INVALID LOGIC.

[bonanova]

Let's get my definition of "formal validity" out into the sunlight, for understanding and discussion.

Then I'll invite you to do the same.

Let x = "This statement is false" be an element of the set S of all self-referential statements of the form "This statement is B".

Let y = "This statement is true" be another element of S.

Let T be the property of Truth.

Let F be the property of Falseness.

Then x becomes x = F(x).

and y becomes y = T(y).

Then, because

1. F and T are formally defined over the set S, and
   
2. x and y are elements of S,
   

both statement are formally valid.

y is formally valid and decidable: T(y).

x is formally valid, but not decidable: F(x) leads to ^x or ^F(x).

So we cannot conclude F(x), and we also cannot conclude T(x).

x is formally valid and undecidable.

---------------------

In the realm of arithmetic,

Let p = "1 + 1 = 1 + i" be an element of the set E of equations of the type A = B where A and B are expressions.

Let q = "1 + 1 = 1 + 1" be another instance

And r = "1 + = x 4 - x 3 - / i 13 / = 45 / -" be considered as be a third.

Let F and T be the properties defined above over the set E.

p is formally valid and decidable: F(p).

q is formally valid, and decidable: T(q)

r is not of the form A=B [formally invalid]; it's not an element of E.

Therefore F and T do not formally apply to r.

r cannot be discussed with respect to F and T.

Now you have my definition of formal validity.

In this case it means + and x and / and - are binary operators, taking prefix and postfix arguments.

It means elementary terms, 1, i, 4, 3, 13, 45 are proper arguments of these binary operators.

It means = is a binary relationship taking two numbers or expressions and asserting their equivalent value.

p and q follow these formal rules. One happens to be false, the other true.

r does not; it is therefore formally invalid.

It's not the case that r is undecidable with respect to truth: T(r) and F(r) don't apply.

From this, see that r and x [above] are not similar.

In a previous post, you said in effect that because r is invalid, so is x.

Since my definition of valid does not permit this, let me ask for a definition that does:

What's your definition of validity?

You seem to define valid as: if F(a), then a is invalid.

Is that how you define valid: if a is false then it's invalid?

Why use two terms [false and invalid] confusingly, in different ways and for different purposes

[false means not true and invalid means we can dismiss it] that in the end have the same definition?