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The Liars Paradox - Resolved

Robin said:
I like "This senctence has contains three mistaks".
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My favorites from real life:

The placard on an otherwise bare wall that says, "Post no signs here."

And the MI-5 phrase that informs you: "This page intentionally blank."
 
Robin said:
I like "This senctence has contains three mistaks".
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You need one less mistake to make that work. As it stands, it has two spelling errors and one grammatical one. I thought the point of it was to have two obvious mistakes (spelling etc) and the third being the sentence is wrong in saying it has three mistakes, creating the paradox.
 
Also, simply saying it is poor syntax, undefined, etc, doesn't solve the fundamental problem, for example:

"This statement is either false or undefined"

The other problem is that there are plenty of examples of sentences that have no external reference yet can be true or false statements on their own, so it is not accurate to say it is poor syntax or gibberish, for example:

"This statement has 5 words."
 
CaptainManacles said:
Also, simply saying it is poor syntax, undefined, etc, doesn't solve the fundamental problem, for example:

"This statement is either false or undefined"

The other problem is that there are plenty of examples of sentences that have no external reference yet can be true or false statements on their own, so it is not accurate to say it is poor syntax or gibberish, for example:

"This statement has 5 words."
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Actually, it has four words and one number. ;P

Still, yes, one cannot argue logical consistency based upon linguistic ambiguity or syntactical ineptitude as these are imperfections of inference. The entire idea of symbolic notation is that it encapsulates idealistic factors of some underlying important principles. Math is not powerful because it simply abstracts counting. It is powerful because the abstraction has transcended counting to include notions which give sight to our deepest understandings of reality. Therefore, the principle has meaning and must be fundamental in some respect. When such an abstraction can be incontrovertibly shown to be flawed (or consistently imperfect) then it is not just a quibble over perspective (syntax, linguistics, semantics, context). It is a fundamental flaw which, as of yet, has not been shown to be circumventable. It is a paradox. Maybe our greatest achievement in existence is to solve dolely one paradox at a time until we have some fundamental system that is incorruptible? I doubt it but it is an interesting idea of what knowledge is and how it evolves.
 
CaptainManacles said:
Also, simply saying it is poor syntax, undefined, etc, doesn't solve the fundamental problem, for example:

"This statement is either false or undefined"
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I don't have a problem with this one. I read it as "undefined" and hence, neither true nor false. It doesn't give me a sense of paradox.

I've never read Wittgenstein, and now I wish I had. (You may envy me, having this experience awaiting me, ready to consume like a fine cut of Kobe beef.)
 
marplots said:
I don't have a problem with this one. I read it as "undefined" and hence, neither true nor false. It doesn't give me a sense of paradox.
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The problem is that if the sentence S is in fact, undefined, then the sentence "S is false or undefined" is true.

Which means that if S is "S is false or undefined," then if it's undefined, it's true -- and no longer undefined.
 
Excuse me, but there is no problem, or paradox. Allow me to demonstrate:

Bins weird have to gloom.

The simple fact is that the conglomeration of words we call a sentence does not have to have a meaning, even if it is grammatically and/or semantically correct.

Thus, what appears a paradox is merely a sentence without meaning.

Likewise, we can make mathematical statements that are meaningless:

2+2=7

It follows that since symbols are freely interchangeable, a paradox cannot be represented solely in a symbolic form, because a symbolic expression needs not have meaning.

A true paradox must have a material form.

Hans ;)
 
MRC_Hans said:
Thus, what appears a paradox is merely a sentence without meaning.
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Not at all. The sentence is clearly meaningful, or we wouldn't understand it. It's simply impossible to evaluate for truth conditions.


It follows that since symbols are freely interchangeable, a paradox cannot be represented solely in a symbolic form, because a symbolic expression needs not have meaning.
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That doesn't follow at all. Just because a symbolic expression need not have meaning doesn't mean that any particular type of expression must not have meaning.

It's easy enough to develop a formal framework under which the Goedel sentence is demonstrably meaningful. Following Hofstadter's development :

Define "quining" a string as making two copies of a string, the first in quotations (and emending capitalization appropriately).

E.g., the string "x," quined, yields "x" x.
The string "bicycle," quined, yields "bicycle" bicycle.

If the string is grammatically in the form of a predicate, the resulting string will be a grammatical sentence.

E.g. quining "played quarterback for the Detroit Lions" yields "Played quarterback for the Detroit Lions" played quarterback for the Detroit Lions.

This is a sentence that claims a particular sentence fragment was an NFL player. It's obviously false, because NFL players are humans, not sentence fragments.

But quining "is a sentence fragment" yields "Is a sentence fragment" is a sentence fragment, which is true.

Quining "is a five word phrase" yields "Is a five word phrase" is a five word phrase, which is also true.

Quining "when quined, is a fourteen word phrase" yields "When quined, yields a fourteen word phrase" when quined, yields a fourteen word phrase. Also true. Also meaningful. Also self-referential.

And quining "when quined, yields a falsehood", produces -- meaningfully, the sentence "When quined, yields a falsehood" when quined, yields a falsehood.

This is clearly meaningful, but also impossible to analyze in terms of truth conditions.
 
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Semantics is an essential part of sentential logic. Without it, one cannot evaluate the terms or discern why (A & ~A) is logically meaningless.

Let A equal "squircle."

Sentential logic is unable to operate on A (a thing that is simultaneously both a circle and a square) because A is semantically null. Logic is therefore unable to generate ~A, and any proposed conjuction of A with anything else (including ~A) is an undefined operation.
 
drkitten said:
The problem is that if the sentence S is in fact, undefined, then the sentence "S is false or undefined" is true.

Which means that if S is "S is false or undefined," then if it's undefined, it's true -- and no longer undefined.
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I got that, but I read it this way: "S is false or undefined" is equivalent to: "S barometer starbucks." In other words, the fact that it is undefined has to do with it's lack of meaningful content, not its own commentary on that fact. This removes 'true' as an option, leaving only 'undefined' and silence.

But then, you would say, "Ah, but to know it is undefined means you have to make sense of it. Without understanding it, you can't say it is nonsense."

That objection I cannot get past yet. So, for instance, "Left of center, reducto plum wheel" is nonsense, but do I have to understand its meaning to know there is a lack of meaning? I'm befuddled on this question. My guess is the answer involves some meta set, but I'm not convinced.

It does remind me of the 'tip of the tongue' question. How is it, when we are searching to recall some bit of trivia, that we know "Tom Jones" is not the right answer? We do not know the right answer yet, so how do we know when we have the wrong one?
 
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Robin said:
This senctence has contains three mistaks
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Definitely false!
Code: 
B:\test\test.c(1) : error C2061: syntax error : identifier 'senctence'
B:\test\test.c(1) : error C2059: syntax error : ';'
B:\test\test.c(1) : error C2061: syntax error : identifier 'contains'
B:\test\test.c(1) : error C2059: syntax error : ';'
B:\test\test.c(1) : error C2061: syntax error : identifier 'mistaks'
B:\test\test.c(1) : error C2059: syntax error : ';'

As far as the statement in the OP, I prefer looking at it as loaded, i.e. not unlike:

"X has stopped beating his wife"

it carries at least one implicit assertion, in this case: "A & ~A".
 
drkitten said:
Not at all.
*snip*

This is clearly meaningful, but also impossible to analyze in terms of truth conditions.
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And:

I'm baffled by this. Do you have an example?

Tak
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I realize I am probably out of bounds here, philosophically. My point is that using symbolic language any message can be constructed.

"The door is open and closed"

is just a sentence. It is gramatically correct and fully comprehensible, but it does not convey any meaning.

However, a real paradox would be a door that was open and closed.

Velbekomme ;)

Hans
 
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