Does the liar paradox prove...

[Ripheus27]

... or at least provide evidence that the law of non-contradiction is, well, not really a law? That there are some contradictions that are built into reality itself?

For the uninitiated, the paradox can be derived quickly as follows:

1. Start with, "This sentence is false."
2. Test agreeing with that sentence. Then, "'This sentence is false,' is true."
3. But if it's true, it is what it says it is, and it says it's false. But if something is false, it's not what it says it is.
4. Okay, so say that, "This sentence is false," is false.
5. But then that sentence is still what it says it is; and that means it's true.
6. So if it's true it's false (and so on), and if false it's true (and so on). So it's true and false (true and not true).​

At a glance, I would suggest that in the case of the liar paradox's starting sentence, the assertion of falsity (and reciprocal verity) has a slightly different function than standard ascriptions of falsity (and verity). That is, in this one case, at least, the truth function just is the untruth function, wherefore, "'This sentence is false,' is true and false," somehow means the same thing as, "'This sentence is false,' is true and true," or, "'This sentence is false,' is false and false." So the starting sentence is not A and not-A, despite appearances; and the law of non-contradiction is preserved (though the related law of bivalence, to wit that all assertions are either true or false, is slightly incorrect, here).

But what do you think?

ADDENDUM: And if the law of non-contradiction is not true, what would that entail?

 

[essentialsaltes]

But what do you think?

I would say that it demonstrates that some statements are not truth-apt. They may look like propositions that have a truth-value (either true or false), but for whatever reason, they just don't have a truth-value.

ADDENDUM: And if the law of non-contradiction is not true, what would that entail?

That everything is true. And everything is false.

 

[quatona]

Most paradoxes are caused by category errors or unprecise use of language.
Plus, self-referential statements are particularly tricky.

 

[quatona]

ADDENDUM: And if the law of non-contradiction is not true, what would that entail?

...that it could be true, as well. Or that contradictions can be non-contradictions. You don´t want to go there.

 

[Ripheus27]

...that it could be true, as well. Or that contradictions can be non-contradictions. You don´t want to go there.

Paraconsistent logic is an extremely intriguing idea, to my mind, though. To be honest, I fully accept the law of non-contradiction, but I'm still enchanted by the act of questioning it.

 

[quatona]

Paraconsistent logic is an extremely intriguing idea, to my mind, though. To be honest, I fully accept the law of non-contradiction, but I'm still enchanted by the act of questioning it.

My point: You don´t question the law of non-contradiction. You are pointing out problems caused by unprecise use of language. Just because language allows for making meaningless statements doesn´t mean the law of contradiction is refuted.
Language (particularly if used in a deliberately obsfucating manner) doesn´t determine reality.

 

[Ripheus27]

My point: You don´t question the law of non-contradiction. You are pointing out problems caused by unprecise use of language. Just because language allows for making meaningless statements doesn´t mean the law of contradiction is refuted.
Language (particularly if used in a deliberately obsfucating manner) doesn´t determine reality.

Does that mean, in your eyes, that paraconsistent logic/dialethism is meaningless (because based on questioning or rejecting the law of non-contradiction, or at least trying to question/reject it)?

 

[quatona]

Does that mean, in your eyes, that paraconsistent logic/dialethism is meaningless (because based on questioning or rejecting the law of non-contradiction, or at least trying to question/reject it)?

No, I am not sufficiently familiar with those ideas to make such a general statement.
I am merely talking about the examples you have given.

 

[Ripheus27]

No, I am not sufficiently familiar with those ideas to make such a general statement.
I am merely talking about the examples you have given.

Alright. I guess to continue with analysis of the liar paradox, then, my next step would be to question whether language is used imprecisely in formulating it. I remember reading an essay about this by Robert Bass, but unfortunately I can't find it right now. But there was an argument in the essay that

This sentence is false or meaningless.​

leads to a paradox, too, so it is not enough to attribute meaninglessness to the original liar sentence to avoid the liar paradox.

If I can reconstruct the argument from memory/intuition, and if this is worthwhile enough a topic to keep discussing I'll continue with this line of reasoning. What I may or may not be building up to is my own, as far as I know unique, solution to the liar paradox. (I say "may or may not" inasmuch as I am confident enough in my solution that I'm considering submitting it in essay form to a peer-reviewed journal. I would be really happy to either outright solve a 2,500+ year-old philosophical problem, or at least contribute strongly to its solution, assuming that no solution so far come up with is satisfactory enough.)

 

[Eudaimonist]

... or at least provide evidence that the law of non-contradiction is, well, not really a law?

I don't think that the law of non-contradiction was ever meant to refer to claims-about-claims or self-referential claims. It's meant to refer to direct claims about reality.

For example, a dog cannot be a cat at the same time and in the same respect. Such direct claims are always consistent with the law of non-contradiction.

The liar's paradox isn't that sort of claim. It acts almost like a computer program that recursively bounces back and forth between different truth values, and it can only do so because it involves claims about claims or a self-referential claim. It shouldn't really be surprising that paradoxes can result, because we are getting more into the realm of reasoning than statements about metaphysical reality. Ideas (or claims) can contradict each other, but only because the human brain doesn't automatically resolve conflicts, and the ideas (or claims) might not have much sense behind them. Is there any reason to believe that such claims are simply true? Is there any reason to think that "This sentence is false"? False about what?

I don't think that this should be particularly troubling. The liar's paradox doesn't mean that your dog is going to meow.


eudaimonia,

Mark

 

[bricklayer]

"This sentence is not true." is a self-defeating statement.

Any explicit attempt to deny an axiom will implicitly affirm it also resulting in self-defeating statements.
For example, any explicit attempt to deny the statement "truth exists" will implicitly affirm it and result in a self-defeating statement.

 

[Eudaimonist]

"This sentence is not true." is a self-defeating statement.

Any explicit attempt to deny an axiom will implicitly affirm it also resulting in self-defeating statements.
For example, any explicit attempt to deny the statement "truth exists" will implicitly affirm it and result in a self-defeating statement.

Huh? What do axioms have to do with this discussion?


eudaimonia,

Mark

 

[bricklayer]

Huh? What do axioms have to do with this discussion?


eudaimonia,

Mark

Used them as an example of how self-defeating staements come about.
I used them hoping to be clear and familiar.

 

[Ripheus27]

I don't think that the law of non-contradiction was ever meant to refer to claims-about-claims or self-referential claims. It's meant to refer to direct claims about reality...

I don't think that this should be particularly troubling. The liar's paradox doesn't mean that your dog is going to meow.

I think what concerns me most, if anything, is that if the law of non-contradiction is mistaken, then reductio ad absurdum arguments of the "showing a contradiction results from the premises" kind would be invalid. The LNC can be interpreted as an imperative, to wit:

For any argument, if the premises lead to a contradiction, deny one or more of the premises.​

So if the LNC is off the mark, then when it comes to quantum mechanics, for instance, rather than try to find a solution to the problem of wave-particle duality that is consistent, we might just accept that light is inconsistently, but truly, wave-like and particulate.

The liar's paradox isn't that sort of claim. It acts almost like a computer program that recursively bounces back and forth between different truth values, and it can only do so because it involves claims about claims or a self-referential claim.

This seems to be the standard way of addressing the liar sentence, or is a sentiment reflected in standard solutions. For it amounts to claiming that the liar sentence is meaningless somehow, or that truth-predicates must vary between the object-language and... Uh, egads, I don't know what the corollaries to object-languages are called. But you catch my drift, I assume... Of course there are more exotic solutions, such as those involving denial of bivalence or the law of the excluded middle, solutions such as 3+ truth-values, so that the liar sentence is neither true nor false but whatever the third truth-value is supposed to be. However, I want to advance a solution that neither denies meaningfulness to the liar sentence (if it's actually meaningless, then how can we analyze it to come up with a paradox in the first place?) nor voids our acceptance of something as obvious (to me) as bivalence/LEM (bivalence seems axiomatic, in my book, because I define falsity as the opposite of truth, but which opposition amounts on a propositional level to the absence of truth for a proposition--that is, logically, a proposition and its negation are opposites, even though not all things are logically contrary to their absences).

Is there any reason to think that "This sentence is false"? False about what?

Here you come perilously close to anticipating my proposed solution to the paradox.

 

[essentialsaltes]

when it comes to quantum mechanics, for instance, rather than try to find a solution to the problem of wave-particle duality that is consistent, we might just accept that light is inconsistently, but truly, wave-like and particulate.

Light is consistently, and truly, both wave-like and particulate.

 

[Ripheus27]

Light is consistently, and truly, both wave-like and particulate.

I agree, but I (think I) know that Hilary Putnam, for instance, seems to think that wave-particle duality is either contradictory or at least a violation of Leibniz's principle of the identity of indiscernibles.

 

[Eudaimonist]

I agree, but I (think I) know that Hilary Putnam, for instance, seems to think that wave-particle duality is either contradictory or at least a violation of Leibniz's principle of the identity of indiscernibles.

Really? Well, I don't see how QM contradicts the law of non-contradiction. It may be counter-intuitive, but that's not the same thing.


eudaimonia,

Mark

 

[quatona]

Alright. I guess to continue with analysis of the liar paradox, then, my next step would be to question whether language is used imprecisely in formulating it. I remember reading an essay about this by Robert Bass, but unfortunately I can't find it right now. But there was an argument in the essay that

This sentence is false or meaningless.​

leads to a paradox, too, so it is not enough to attribute meaninglessness to the original liar sentence to avoid the liar paradox.

"This picture is all white and all black" is contradictory, as well. I´m not sure I understand why the ability of language to create self-contradictions points to reality allowing for self-contradictions.

"This sentence is wrong or meaningless" should prompt us to find out what´s wrong with the sentence, not with reality.

 

[Eudaimonist]

"This picture is all white and all black" is contradictory, as well. I´m not sure I understand why the ability of language to create self-contradictions points to reality allowing for self-contradictions.

"This sentence is wrong or meaningless" should prompt us to find out what´s wrong with the sentence, not with reality.

That is well put. Agreed.


eudaimonia,

Mark

 

[Ripheus27]

"This picture is all white and all black" is contradictory, as well. I´m not sure I understand why the ability of language to create self-contradictions points to reality allowing for self-contradictions.

"This sentence is wrong or meaningless" should prompt us to find out what´s wrong with the sentence, not with reality.

Ah, but it's not, "This sentence is wrong or meaningless," it's, "This sentence is false or meaningless." If there is no genuine law of non-contradiction, then contradictions aren't wrong.

And why isn't the law of non-contradiction a linguistic matter instead of a "real" one? It might be that in reality, it's not that there are or are not contradictions, but rather that application of the category "logically contradictory or not" to extralinguistic reality is a category mistake. --But now I put "real" in quotation marks in the first sentence of this paragraph, also, to signal that I'm not as convinced that "reality" and "language" are so separate as contrasting the two would imply.

On another level, my question in this thread is: what proof is there, or evidence at any rate, for the LNC? If it doesn't admit of proof or evidence or justification or whatever, then if someone rejects it, what do we say to them? The "denying the LNC is meaningless" route is the classical response, which is to say that we (philosophers) have traditionally looked at "denials" of the LNC as not really denials at all. We've held the LNC to be impossible to deny, so that those who claim to do so are at best confused, at worst liars. But I think this might be an intolerant point of view to adopt, especially in light of paradoxes like the one under discussion in this thread, and I'm looking for a deeper reason to accept the LNC.