The Liars Paradox - Resolved

[Brian-M]

Some statements are neither true nor false. Examples include

- The one in the OP (this statement is false).
- The following statement is true. The previous statement is false.
- Everything I say is a lie (which is nothing but a modification of the first example).

One way to explain it is to say they are internally inconsistent.

Actually "everything I say is a lie" is always false. A person who only makes false statements would not be able to make such a statement. A person who only makes true statements would not be able to make such a statement either. Consequently, the statement "everything I say is a lie" must therefore be a false statement made by someone who sometimes, but not always, tells lies.

 

[SezMe]

You're mistaken. The statement "This sentence is false." does not try to deny it's own existence, and so is not false for that reason.

That's the point where the reasoning broke down for me too.

 

[Mirrorglass]

But using external referents doesn't preclude this kind of paradox. For example...

The next sentence is true. The previous sentence is false.

Both sentences are making a truth claim about something external to themselves (the other sentence) yet the liar's paradox remains.

Actually, that paradox only exists if those two sentences are viewed as one entity. Neither sentence in itself is a paradox, but when strung together, neither has a determinable truth value. One of them must be undefined, and the other one false, although interestingly, we can't determine which is which.


By the way, the wiki article lists a few possible 'resolutions' for the paradox. I don't entirely agree with them all, but it is an interesting read for anyone with a casual interest in logic.

 

[Mirrorglass]

Actually "everything I say is a lie" is always false. A person who only makes false statements would not be able to make such a statement. A person who only makes true statements would not be able to make such a statement either. Consequently, the statement "everything I say is a lie" must therefore be a false statement made by someone who sometimes, but not always, tells lies.

You left out some possibilities there. It is also possible for the statement to be made by a person who can lie, tell the truth or make statements of ambiguous truth value. If such a person has only ever made false statements, and now states 'everything I say is a lie', that statement does constitute a liar's paradox, and is neither true nor false.

 

[Wrathernaut]

It's rather easy to resolve this paradox by considering that "Lie" isn't necessarily the opposite of truth, but rather something said, or omitted, that's meant to deceive.

Re-worded, you are left with:
Everything I say is meant to deceive.

 

[Beerina]

It's essentially the halting problem's core issue in a nutshell, at least in the iconic proof by providing a contradictory example.


Some suggest it is proof that the law of excluded middle, as applied in this case (all statements are either true or false) is itself false.

Some suggest, instead, that the law remains true, but that the statement is just a nonsensical, if syntactically correct, sentence, akin to "there's a square with 3 sides."

However, the halting problem itself, which treats it at a much deeper level, suggests there are statements that may actually be true or false, but that there's no way to prove such, within the same system of logic.

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.

 

[shemp]

All your paradox belong to us.

 

[Athyrio]

Two doctors who say, "this will only hurt a little bit."

 

[Ron_Tomkins]

NM

That's a lie

 

[fossilhound]

On top of being an infinite regression, the statement is also untestable as written, without an if statement or more info

 

[SezMe]

On top of being an infinite regression, the statement is also untestable as written, without an if statement or more info

The statement itself is not an infinite regression. One way of analyzing it can lead to an infinite regression of conclusions about its veracity.

I think Beerina nailed it here:

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.

The fact that the sentence is self-referential is central to its paradox. In his Godel, Escher, Bach Hofstadter deals with recursion in some detail but delves specifically with this paradox and many others in two whole chapters in his Metamagical Themas. Fascinating stuff.

 

[fossilhound]

The statement itself is not an infinite regression. One way of analyzing it can lead to an infinite regression of conclusions about its veracity.

I think Beerina nailed it here:



The fact that the sentence is self-referential is central to its paradox. In his Godel, Escher, Bach Hofstadter deals with recursion in some detail but delves specifically with this paradox and many others in two whole chapters in his Metamagical Themas. Fascinating stuff.

Yup. Exactly. It's also a self-referential paradox.

 

[epix]

The Liar's Paradox is most often presented in beginning logic studies to help students understand the need for clear external referents. In sentential logic, propositions are limited to assertions, e.g., A is B. Implicit is that A is also A (the simplest case of identity), but in either kind of statement, the assertion is not about the assertion, it's about A. An assertion about itself is undefined rather than paradoxical.

Exactly. You can see similar response from programmable calculators when you enter a structural equivalent of that "paradox": Undefined.

 

[SezMe]

epix, it's doesn't make any sense to look to a programmable calculator for any help in this matter. Whatever it does when it meets the "structural equivalent" (whatever the hell that means) was built in by engineers constrained by the underlying chip logic.

 

[Brian-M]

You left out some possibilities there. It is also possible for the statement to be made by a person who can lie, tell the truth or make statements of ambiguous truth value. If such a person has only ever made false statements, and now states 'everything I say is a lie', that statement does constitute a liar's paradox, and is neither true nor false.

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.

 

[marplots]

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?

 

[dafydd]

I thought that the Liar's paradox was Epimenides the Cretan's statement that all Cretans are liars.

 

[Robin]

If you're not familiar with the Liars Paradox (which I will refer to as "LP" in this thread), a quick read on google should make it clear.

The paradox is as follows:

"this statement is false".

This is linguistically interesting, since it leads to an eternal regression where the statement ends up being true when it is false and vice versa. So it appears to violate the law of non contradiction.

But I have resolved it.

First, let's be clear on what it is we mean by true and false before we tackle the statement, and also be conscious of how we interpret the statement, as it can be seen in at least two ways, and one must not unconsciously confuse the two perspectives together, as that is what creates the contradiction, the sense of paradox.

So let's look at it this way:

1) The statement is false insofar as the statement itself exists when it tries to say it doesn't. So it is false in that sense.

Where does the statement say it doesn't exist? Saying something is false is not the same as saying it doesn't exist.

2) The statement is true insofar as it correctly denies the validity of it's attempt at denying itself.

But if the statement were invalid then it would not be false. By assertiing it's own falsity it must also by implication be asserting it's own validity.

And in any case, a statement cannot become true by denying it's own validity, since an invalid sentence would not be true either.

(although generally we speak of validity at an argument level, rather than a statment level).

Paradox resolved!

I am afraid not

 

[Robin]

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.

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

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

 

[Robin]

Exactly. You can see similar response from programmable calculators when you enter a structural equivalent of that "paradox": Undefined.

Although you get a lot of really useful stuff if you express a paradox in a digital circuit.