The Liars Paradox - Resolved

[Robin]

The example I used is used in uni POL courses, but with your example, it's fairly easy to resolve as well.

If all you ever did was tell the truth, then saying, "I always lie" is simply a single instance of a lie, and hence you statement is merely false, without any truth value.

If all you ever did was lie, then saying, "I always lie" is in fact a single instance of telling the truth, and hence your statement is again merely false, without any truth value.

Firstly, a false statement does have a truth value.

And your "resolution" doesn't make any sense. A single instance of a lie would falsify "all I ever do is tell the truth" and a single instance of a truth would falsify "all I ever do is lie"

 

[Brian-M]

Up is down.

Down, Up, down!
Now Up is down.
Good boy, Up!

Sorry, I couldn't resist.

 

[Mirrorglass]

In that case, as "everything I say" also includes itself, it essentially comes down to saying "(everything I have ever said before this sentence) AND (this sentence) is FALSE". The first part of that statement is effectively irrelevant. You're basically just saying "this sentence is false" in a different way.

Correct. And that sentence has an ambiguous truth value.

 

[drkitten]

I think, however, that your claim that the "This statement" in "This statement is false" cannot refer to itself (nonsensical as per above notwithstanding) cannot be since that is what is intended. You can't just abandon the self-reference by fiat.

It's also easy enough to make the self-reference more formal via Goedel-numbering or something like that.

Essentially, "Consider the string S constructed by procedure P. S is false."

But if the bit in quotation marks above is S itself, then the self-reference is explicit and unavoidable.

 

[kuroyume0161]

But it doesn't fail to refer. It refers rather explicitly to itself.

The words "this statement" refer to the whole statement.

Exactly. You can't take "This statement is false." and apply it like this:

"This statement" = false

and ignore to what "This statement" refers, which is the statement itself (self-referential). "This statement" loses context without reference and just becomes a meaningless set of words, "This statement". So, it must refer to the entire sentence or there is no context, basically (WHAT statement then? That one over there? Oh, this one down here?). See the problem with removing context!?

Logically, we can expand it thus (indefinitely if we're pedantic and can't circumvent the halting problem):

...""This statement is false" is false"...

I agree with the poster who mentioned that we would need to put it into its formal structure possibly with Godel numbering so as to identify references properly and have force of logical application rather than human-level reference mangling.

 

[tsig]

So, the paradox revolves around creating a sentence like this?

Up is down.

And just points out the fact that we can make grammatically correct sentences that are outside of the framework of logic?

Correct. The paradox merely proves that language is ambiguous.

 

[kuroyume0161]

Correct. The paradox merely proves that language is ambiguous.

That isn't quite correct. It doesn't merely prove that language is ambiguous. It proves that ANY formal statement structure is ambiguous (can contain statements that are inconsistent), including meta-structures (and so on). Read up on your Hoftstadter/Godel. This is beyond language - it includes logic and math which are using formal symbols, grammars (structures), and rules.

 

[DallasDad]

There are two cases being discussed here. I'll start with the derivative:

"Unicorns have black hair and golden hooves."

The above statement has no truth value; it applies attributes to an object without a referent. One could as easily assert that unicorns have white hair and silver hooves. Since there is no referent, neither statement can be false; yet both cannot be true by the law of non-contradiction. Any statement without a referent is subject to this process: One has only to imagine the opposite case, conjoin the two propositions, and discover that the first statement must therefore have no truth value.

Now back to the original:

"The following statement is true. The previous statement is false."

Each statement, considered by itself, may have a truth value. The conjunction of the statements has a truth value, too. The conjunction is not paradoxical at all; it is merely false. A conjuction is true only in the case where both conjuncts are true. Since they are mutually-contradictory assertions, at most one of them can be true. Therefore the conjunction must be false; it is the semantic equivalent of A & ~A, even though the logical description is more complex.

A more interesting observation is that the conjunction has no referent, even though each conjunct appears to. When conjoined, the universe of discourse is reduced to the two conjuncts themselves, neither of which has an external reference. So one could make the case that (A & ~A) is obviously false, while simultaneously claiming that (A & ~A) is actually meaningless on its face. Something that is meaningless (has no semantic content) is an artifact of natural language; it cannot be used as a term in formal logic. This demonstrates how the original argument about the two statements is simply another form of the unicorn mistake.

[Typing while recovering from anesthesia; apologies if there are mistakes in the above.]

 

[ttoyooka]

...ANY formal statement structure is ambiguous (can contain statements that are inconsistent), including meta-structures (and so on). Read up on your Hoftstadter/Godel. This is beyond language - it includes logic and math which are using formal symbols, grammars (structures), and rules.

That's exactly right, and I suspect (though I don't have the math background to demonstrate it) that any attempt to disallow self-referential statements by simply casting them out of the language (which seems to be what's going on here) will get you into just as much trouble as Godelian Incompleteness does.

Tak

 

[I Am The Scum]

A conjuction is true only in the case where both conjuncts are true. Since they are mutually-contradictory assertions, at most one of them can be true. Therefore the conjunction must be false; it is the semantic equivalent of A & ~A, even though the logical description is more complex.

But as soon as you begin to explore this option, it becomes obvious that this also fails. If you are saying that it is not the case that the conjunction of two proposition is true, it follows that it must be the case that one or both must be false. Let's analyze the premises in this way:

A. B is true.
B. A is false.

Since you're saying they're not both true, that leaves us with three options: A is true and B is not. B is true and A is not. Or, both are false. Let's analyze them in reverse order.

If they are both false, then this would contradict A, so that is not the case.

If B is true and A is not, then that doesn't work either. It would make B's claim accurate, in that it is claiming A is false. This coincides with our current analysis. But if that is the case, then A is now stating, "it is not the case that B is true." This contradicts our current possibility, so that cannot be the case.

And finally, A is true, and B is not. This might initially appear to be a possibility, but it does not work either, for the same reasons as the previous analysis. To look at it another way, A is stating that B is true. If that is the case, then we cannot say that B is not true. We would be contradicting ourselves yet again.

 

[rocketdodger]

I have always said the Epimenides paradox is the closest we can get to a human Godel string, I.E. something the truth value of which is impossible to establish using human logic.

EDIT by "human logic" I mean something humans can intuitively understand.

 

[kuroyume0161]

But as soon as you begin to explore this option, it becomes obvious that this also fails. If you are saying that it is not the case that the conjunction of two proposition is true, it follows that it must be the case that one or both must be false. Let's analyze the premises in this way:

A. B is true.
B. A is false.

Since you're saying they're not both true, that leaves us with three options: A is true and B is not. B is true and A is not. Or, both are false. Let's analyze them in reverse order.

If they are both false, then this would contradict A, so that is not the case.

If B is true and A is not, then that doesn't work either. It would make B's claim accurate, in that it is claiming A is false. This coincides with our current analysis. But if that is the case, then A is now stating, "it is not the case that B is true." This contradicts our current possibility, so that cannot be the case.

And finally, A is true, and B is not. This might initially appear to be a possibility, but it does not work either, for the same reasons as the previous analysis. To look at it another way, A is stating that B is true. If that is the case, then we cannot say that B is not true. We would be contradicting ourselves yet again.

Nicely done. This is how one must evaluate logic statements or proofs - by assignment and tabulature of the results. "This statement is false" amounts to a single statement proof since it is self-referential. That is, it doesn't need a separate statement in proof format to establish a premise and conclusion as they are encapsulated in the single statement nor does it need external reference. It almost becomes tautological but not quite (as in "God exists", for example, which must be taken as true or false based upon some external criteria or data). It sort of amounts to:

A. A is false.

If you assign true to A. then the result contradicts the statement. If you assign false to A. then the result agrees with the statement but it contradicts itself (saying "'A is false' is false" is A is true). Remember that (not)A inverts the meaning (contrapositive or converse).

 

[Brian-M]

"Unicorns have black hair and golden hooves."

The above statement has no truth value; it applies attributes to an object without a referent. One could as easily assert that unicorns have white hair and silver hooves. Since there is no referent, neither statement can be false; yet both cannot be true by the law of non-contradiction.

Let's try that with something that does have a referent.
"Swans are white."
"Swans are black."

Two contradictory statements, but both are true. So...
"Unicorns have black hair and golden hooves."
"Unicorns have white hair and silver hooves."

Could both be true, for different species of unicorn.

[Typing while recovering from anesthesia; apologies if there are mistakes in the above.]

That's okay. I'm just pointlessly nitpicking here. If you said "all unicorns" instead of just "unicorns" your point would be valid.

 

[Robin]

Exactly. You can't take "This statement is false." and apply it like this:

"This statement" = false

and ignore to what "This statement" refers, which is the statement itself (self-referential). "This statement" loses context without reference and just becomes a meaningless set of words, "This statement". So, it must refer to the entire sentence or there is no context, basically (WHAT statement then? That one over there? Oh, this one down here?). See the problem with removing context!?

I don't see how that is a problem. There is no ambiguity about what "this statement" refers to - it refers to the four words "this statement is false".

There is no a priori reason why the reference cannot be contained in it's referent.

Logically, we can expand it thus (indefinitely if we're pedantic and can't circumvent the halting problem):


...""This statement is false" is false"...

If the expansion is valid then it is only a tautology that says no more than the original. So I don't really see why it would be a problem that there would be an infinite set of expansions that would say the same thing.

I agree with the poster who mentioned that we would need to put it into its formal structure possibly with Godel numbering so as to identify references properly and have force of logical application rather than human-level reference mangling.

Unfortunately, doing this only confirms that there is a problem, rather than resolving it.

Principia Mathematica was an attempt to provide a formal structure that avoided the problems posed by paradoxes like this.

Goedel showed that the problems posed by self-reference still existed in PM, and ultimately in any axiomatisation that was capable of properly describing mathematical truths.

 

[Robin]

Let's try that with something that does have a referent.
"Swans are white."
"Swans are black."

Two contradictory statements, but both are true.

No, if they are both true then they are not contradictory.

If they are contradictory then they are not both true.

So it comes down to the interpretation of the words.

If it means "Some swans are white. Some swans are black" then they are both true and non-contradictory.

If it means "All swans are white. All swans are black" then the are contradictory and neither of them is true.

 

[kuroyume0161]

I don't see how that is a problem. There is no ambiguity about what "this statement" refers to - it refers to the four words "this statement is false".

There is no a priori reason why the reference cannot be contained in it's referent.

Yes. But it was how the OP 'solved' the paradox (in one mentioned way) that involved changing the context of the reference:

Another approach, which is just as reasonable, is to say that it is neither true nor false because of a failure to refer. The words 'this statement' refers to something outside of the sentence which is not specified.

You cannot do that without specifying the outside reference. Or, at least, make the fact that the referent is external unambiguous: "That statement is false." Same as 'God exists' (tautology). It is only true or false if God is well-defined, otherwise its truth value is undefined (or indeterminate). Without a referent, such a statement's truth value is indeterminate.

If the referent is well-defined, as long as it does not directly or indirectly make a statement about 'this' statement, the paradox (most likely) is avoided. But that isn't the statement being examined, is it? This just makes it a completely different statement (in logical grammar) unrelated to the paradoxical one ("THIS statement, self-referentially, is false").

If the expansion is valid then it is only a tautology that says no more than the original. So I don't really see why it would be a problem that there would be an infinite set of expansions that would say the same thing.

Unfortunately, doing this only confirms that there is a problem, rather than resolving it.

Exactly. It shows the result and the result of the result (etc.) in a meta-context to illustrate that one cannot escape the paradox simply by moving up a level. It exists from the beginning and trying to up (or down) the context won't solve the problem.

Principia Mathematica was an attempt to provide a formal structure that avoided the problems posed by paradoxes like this.

Goedel showed that the problems posed by self-reference still existed in PM, and ultimately in any axiomatisation that was capable of properly describing mathematical truths.

QFT

 

[marplots]

That isn't quite correct. It doesn't merely prove that language is ambiguous. It proves that ANY formal statement structure is ambiguous (can contain statements that are inconsistent), including meta-structures (and so on). Read up on your Hoftstadter/Godel. This is beyond language - it includes logic and math which are using formal symbols, grammars (structures), and rules.

My favorite from Hoftstader (GEB):
"This sentence, no verb."

And one I claim cryptomnesia on:
"This sentence is not written in English."

 

[elipse]

In this case this statement is just true: it's not written in English; it's typed.

 

[Robin]

My favorite from Hoftstader (GEB):
"This sentence, no verb."

And one I claim cryptomnesia on:
"This sentence is not written in English."

I like "This senctence has contains three mistaks".

 

[I Am The Scum]

Nicely done. This is how one must evaluate logic statements or proofs - by assignment and tabulature of the results. "This statement is false" amounts to a single statement proof since it is self-referential. That is, it doesn't need a separate statement in proof format to establish a premise and conclusion as they are encapsulated in the single statement nor does it need external reference. It almost becomes tautological but not quite (as in "God exists", for example, which must be taken as true or false based upon some external criteria or data). It sort of amounts to:

A. A is false.

If you assign true to A. then the result contradicts the statement. If you assign false to A. then the result agrees with the statement but it contradicts itself (saying "'A is false' is false" is A is true). Remember that (not)A inverts the meaning (contrapositive or converse).

Clearly identifying the claims does a great job of making it clear precisely where paradoxes lie. In my previous example, we can substitute the presence of a letter with the claim that the letter represents. Allow me to illustrate:

A. B is true.
B. A is false.

So our first line is saying that B is true. Well, what is B? B is the claim that A is false. Let's substitute the B variable with its claim. We are left with this:

A. "A is false" is true.

In other words, A is claiming that A is false. In other words yet again, "A" is "not A." We're left with the same result as the original liars paradox, with a more convoluted method.

The same result holds true if you solve for B instead of A.