Does the liar paradox prove...

[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.

What about:

This sentence has five words.
All sentences have subjects and predicates.​

The first one is plainly true, the second arguably false. I don't see why the LNC doesn't apply to them. Yet they're both sentences-about-sentences or self-referential.

Now you might say, "The LNC doesn't apply to sentences that generate paradoxes," but that would seem a little question-begging in the context of doubting the LNC.

 

[FrenchyBearpaw]

God is love.
Love is Blind.
Stevie Wonder is blind.

Therefore, Stevie Wonder is god.

 

[Ana the Ist]

God is love.
Love is Blind.
Stevie Wonder is blind.

Therefore, Stevie Wonder is god.

This seems very superstitious.

 

[Ripheus27]

Of significant import to the current discussion are some of the reflections (that I just remembered) in the following essay by Timothy Williamson: http://www-personal.umich.edu/~lorm... Understanding and inference (highlights).pdf

 

[Eudaimonist]

What about:

This sentence has five words.
All sentences have subjects and predicates.
​

The first one is plainly true, the second arguably false.

It would be more apt to do something like the following:

A) "I decided to drive home."

Sentence A cannot have five words and six words at the same time and in the same respect.

In that case, there is a clear referent that is separate from its claim. As long as you can conceptually separate the two, you should be fine. If a claim's truth value depends on some other claim's truth value, then you are in trouble.


eudaimonia,

Mark

 

[Ripheus27]

Yes, that may be so, but that doesn't mean that the law of non-contradiction was ever intended to apply in such a self-referential way. It's ripping the law out of its philosophical context and forcing it to be used in ways that it was never designed for.

It seems that if the liar paradox has been around for 2,500+ years (or whatever), and that it is considered a paradox because it results in a contradiction, etc. then the LNC has always been meant to apply to all possible sentences. How familiar would you say you are with paraconsistent logic and dialethism (or dialetheism; not sure if only one spelling is correct)?

 

[Eudaimonist]

It seems that if the liar paradox has been around for 2,500+ years (or whatever)

So?

and that it is considered a paradox because it results in a contradiction

By Aristotelians?

then the LNC has always been meant to apply to all possible sentences.

Did Aristotle or the Peripatetic school say this?

How familiar would you say you are with paraconsistent logic and dialethism (or dialetheism; not sure if only one spelling is correct)?

I'm not.


eudaimonia,

Mark

 

[Ripheus27]

By Aristotelians?

The LNC is not an invention or discovery of Aristotle. He may have been the first to emphasize its significance and come up with a sort of "argument" for it (the denying-the-LNC-is-impossible dialectic), but as a basic principle of thought, it transcends any particular philosopher's teachings. (I don't know if this response is relevant enough to your question, though...)

Did Aristotle or the Peripatetic school say this?

For the reasons outlined above, doesn't matter. Socrates had already been employing reductio ad absurdum arguments before Plato, for instance.

I'm not.

Paraconsistent logic is based, as far as I know, on the denial of explosion. Explosion, logically, is all propositions following from a contradiction. So a paraconsistent logician might think that there are no true contradictions, but his or her thinking this way is not based on a fear of explosion. Dialetheists, on the other hand, do think (or think that they think) that there are true contradictions.

 

[quatona]

Ah, but it's not, "This sentence is wrong or meaningless," it's, "This sentence is false or meaningless."

Sorry - I meant to type "false". Do you see a significant difference?

If there is no genuine law of non-contradiction, then contradictions aren't wrong.

Not sure why you say that - there is a law of non-contradiction.

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.

Yes, I think you are right: referring to"reality" was a bit rashed.
Indeed, I think logical laws are linguistic rules, in the first place. Self-contradictions are "prohibited" because the cause language to implode. I´ll give you an example: If "yes" could be "no" sometimes or always this would eliminate the meaning of those words. They were created exactly for the purpose of forming a dichotomy. In the formal system language "yes=no" is against the rules, just like "3=2" is against the rules in mathematics. Both statements are meaningless in the frame of reference that makes them meaningful.
That´s the same with "This statement is false." It contains a contradiction, hence is a violation of the rules of the formal system it is made in.
If you postulate these things to be meaningful, you necessarily postulate that their frame of reference (language/logic/mathematics) has run its course.

On another level, my question in this thread is: what proof is there, or evidence at any rate, for the LNC?

There is not proof. The LNC is axiomatic.
If you want to question it you shouldn´t use logic nor language because the LNC is the foundation of both. None of your sentences or question would have any intellegible meaning if we abandoned the LNC. Each word you use could mean the opposite of what you intended it to mean. "Proof" could mean "non-proof" (and both), evidence could mean "non-evidence" (and both) etc. etc.
When you tackle the law of non-contradiction you are chopping off the very branch you are sitting on.

If it doesn't admit of proof or evidence or justification or whatever, then if someone rejects it, what do we say to them?

I would say: "You reject the LNC? So that means you accept it?"

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.

If the LNC doesn´t apply, anything a person says can mean the opposite of what it means and both simultaneously. In order to demonstrate that the person denying the NLC hasn´t thought it through, I´d just have to show how each of his statements relies heavily on the acceptance of the LNC.

Finally, here is an additional observation about the sentence "This statement is false." Language is a means of communication, of transferring a message. When someone talks to me I naturally assume that he wants to communicate a message about something.
This sentence is a message about itself, and apart from creating a self-contradiction it communicates nothing, and doesn´t even pretend to. The only purpose of this sentence is to create a self-contradictory sentence. It´s deliberate obsfucation, and nothing else. It has no intelligible message, it contains no information.
The appropriate response would be: "I have no idea what you are trying to say. Please reword this statement for me."
Or, probably more to the point: "Your sentence states that it´s false. So why should I spend a second thought on an admittedly false statement?"

 

[quatona]

Now you might say, "The LNC doesn't apply to sentences that generate paradoxes," but that would seem a little question-begging in the context of doubting the LNC.

The LNC does apply to sentences that generate self-contradictions - therefore these sentences are violating the very system they are utilizing (or: the very frame of reference that allows sentences to be meaningful), and hence are not permitted.
Just like the laws of mathematics do apply to "3=2": they render it false.

 

[Ripheus27]

@ quatona: you seem to be saying that the LNC is somehow part of the very meaning of negation, of saying that something is not a certain way. If this is the case, then what to make of a philosopher like Graham Priest (I think's the name)? If anyone has been able to clearly deny the LNC, it might be him. Let that be an open question for a second.

Think, instead, for a little anyway, about modus ponens. Does that inference rule express the logical function of our concept of "if" in a way like the LNC expresses "not"? Or consider intuitionistic logicians, who deny the law of the excluded middle and double-negation elimination. Are they purely confused about the logical value of "or" and "not"? (Are deniers of modus tollens similarly deluded about both "if" and "not"?) This is what Timothy Williamson asks in section I of his, "Blind Reasoning: Understanding and Inference":

Is making inferences by modus ponens of the form 'If A then B; B; therefore B' a precondition for having the concept if? Van McGee, a distinguished logician, has published purported counterexamples to modus ponens. Presumably he refuses to make some inferences by modus ponens. Does McGee lack the concept if?

He goes on to point out McGee's apparent ordinary-language fluency with the word if. It is suggested that some of McGee's purported counterexamples to modus ponens have a measure of intuitive plausibility upon first (if not later) reflection.

You might say: well, how else are we gonna come up with rational justification for accepting the law of non-contradiction, et. al. from besides the pure function of various words for logical constants in our language? I do have my own answer to that question, one that denies only that grounding the LNC and like axioms in abstract linguistic space is a complete solution to the problem of how to knowingly set out from the true foundations of our epistemic lives. That is to say, it's half the answer; the other half is going to be found in erotetic logic, one of the more exotic forms of logic alongside the modal, deontic, etc. kinds. Consider it like this: there is perhaps a way that the very ability to ask questions can be shown to imply fundamental principles about answers. Perhaps there is something in the act of asking a question that can be reflected in a sentence that we otherwise know as the law of the excluded middle, a sentence expressing this law at any rate. (You yourself have indicated what this erotetic function might be.)

The skeptic or nihilist about the axioms of logic, such as they can be shown to sincerely lack knowledge of the axioms we claim self-evidence for, can be best replied by showing them that the capacity for doubt that they formulate their doubts about everything from, instead present to us a small space for axiomatic certainty at our absolute epistemic depths. This is basically the Cartesian cogito; ergo sum but rendered explicitly in erotetic terms. The power to question is the power to correctly try to arrive at answers. And those standards of correctness are the ultimate epistemological background criteria, like the LNC.

 

[quatona]

@ quatona: you seem to be saying that the LNC is somehow part of the very meaning of negation, of saying that something is not a certain way.

To be more precise, I am saying that it is the foundation of any distinction we are conceptualizing and expressing in language.

And to everyone who wants to talk to me under the premise that "false" and "true", or "meaningful" and "meaningless" (or any other contradicting terms) might mean the same I am saying: Find yourself someone else to talk to. Or make a surreal movie or something.

If this is the case, then what to make of a philosopher like Graham Priest (I think's the name)? If anyone has been able to clearly deny the LNC, it might be him. Let that be an open question for a second.

Think, instead, for a little anyway, about modus ponens. Does that inference rule express the logical function of our concept of "if" in a way like the LNC expresses "not"? Or consider intuitionistic logicians, who deny the law of the excluded middle and double-negation elimination. Are they purely confused about the logical value of "or" and "not"? (Are deniers of modus tollens similarly deluded about both "if" and "not"?) This is what Timothy Williamson asks in section I of his, "Blind Reasoning: Understanding and Inference":



He goes on to point out McGee's apparent ordinary-language fluency with the word if. It is suggested that some of McGee's purported counterexamples to modus ponens have a measure of intuitive plausibility upon first (if not later) reflection.

You might say: well, how else are we gonna come up with rational justification for accepting the law of non-contradiction, et. al. from besides the pure function of various words for logical constants in our language? I do have my own answer to that question, one that denies only that grounding the LNC and like axioms in abstract linguistic space is a complete solution to the problem of how to knowingly set out from the true foundations of our epistemic lives. That is to say, it's half the answer; the other half is going to be found in erotetic logic, one of the more exotic forms of logic alongside the modal, deontic, etc. kinds. Consider it like this: there is perhaps a way that the very ability to ask questions can be shown to imply fundamental principles about answers. Perhaps there is something in the act of asking a question that can be reflected in a sentence that we otherwise know as the law of the excluded middle, a sentence expressing this law at any rate. (You yourself have indicated what this erotetic function might be.)

The skeptic or nihilist about the axioms of logic, such as they can be shown to sincerely lack knowledge of the axioms we claim self-evidence for, can be best replied by showing them that the capacity for doubt that they formulate their doubts about everything from, instead present to us a small space for axiomatic certainty at our absolute epistemic depths. This is basically the Cartesian cogito; ergo sum but rendered explicitly in erotetic terms. The power to question is the power to correctly try to arrive at answers. And those standards of correctness are the ultimate epistemological background criteria, like the LNC.

This all sounds very intriguing (and eventually I might read into this stuff), but - to be honest - I have no idea what it has do with anything discussed here, and particularly not how it´s meant to address my points.

 

[Ripheus27]

And to everyone who wants to talk to me under the premise that "false" and "true", or "meaningful" and "meaningless" (or any other contradicting terms) might mean the same I am saying: Find yourself someone else to talk to. Or make a surreal movie or something.

... [The bulk of the rest] all sounds very intriguing (and eventually I might read into this stuff), but - to be honest - I have no idea what it has do with anything discussed here, and particularly not how it´s meant to address my points.

Maybe what they would mean is that in a very few instances, contradictory terms do apply to the same thing. Thus they would accept the LNC, but as a rule with, like any other rule, an exception. (I myself don't think all rules have exceptions.)

And the point of most of what I said was that the LNC is meant to express the very concept of one thing not being another thing. The very concept of difference, even. And though I agree with this claim, I also recognize that without a deeper backing from the direction of erotetic logic, the argument-from-the-concept-of-negation is not completely sufficient to "prove," if you will, the LNC. (Now maybe you really do believe that no one can deny it; so maybe draw the obvious conclusion, that no one does deny it. So maybe if someone claims to deny it to you, you deny that they are in denial. I don't know, I'll admit... But just in case you're wrong and maybe, just maybe, it is possible for exceptions to the LNC to obtain, or at least possible to believe in such a thing, what else can be said on behalf of the LNC?)

 

[quatona]

Maybe what they would mean is that in a very few instances, contradictory terms do apply to the same thing. Thus they would accept the LNC, but as a rule with, like any other rule, an exception. (I myself don't think all rules have exceptions.)

I think we can throw logic completely out the window when we assume that are exceptions to the LNC.
Just like we can forget about mathematics altogether when we allow for the idea that 7 may, in some exceptional cases, be 8.

Now, I won´t keep people from doing that...

And the point of most of what I said was that the LNC is meant to express the very concept of one thing not being another thing. The very concept of difference, even.

Yes.

And though I agree with this claim, I also recognize that without a deeper backing from the direction of erotetic logic, the argument-from-the-concept-of-negation is not completely sufficient to "prove," if you will, the LNC.

Yes, the NLC can´t be proven.
Look at it this way: Even if I could - hypothetically - string together a logical proof for the NLC, the NLC denier will refuse it because he doesn´t accept logic as the frame of reference.
OTOH, the NLC denier can talk a lot of illogical nonsense in support of his denial, and I couldn´t even say "This is illogical" - after all, from his pov logic isn´t binding.

(Now maybe you really do believe that no one can deny it; so maybe draw the obvious conclusion, that no one does deny it. So maybe if someone claims to deny it to you, you deny that they are in denial.

No, that was not the point of my hypothetical response. The point was that - once we allow for the idea that there may be exceptions to the LNC - "denial" might possibly be the same as "acceptance".

I don't know, I'll admit... But just in case you're wrong and maybe, just maybe, it is possible for exceptions to the LNC to obtain, or at least possible to believe in such a thing, what else can be said on behalf of the LNC?)

Do you want me to answer this question based on logic and the NLC, or am I, at this point, already invited to give illogical answers?

 

[Ripheus27]

I think we can throw logic completely out the window when we assume that are exceptions to the LNC.

... Now, I won´t keep people from doing that.

To repeat my caveat: though I agree that the LNC admits of no exceptions, my acquaintance with paraconsistent logic among other topics has persuaded me that I need a deeper answer to doubts about the LNC than just "in the end, no one really is able to deny it." (Other topics include the debate over foundationalism, infinitism, coherentism, and other positions in epistemology.)

Look at it this way: Even if I could - hypothetically - string together a logical proof for the NLC, the NLC denier will refuse it because he doesn´t accept logic as the frame of reference.

But paraconsistent logic seems capable of steering these murky waters in a way that casts a reasonable shadow of doubt over the island of the LNC. (Of course, when you study paraconsistent logic, you might notice a basic absurdity in its arguments. I don't know. I just know that the points some opponents of the LNC have made against my original position convinced me that I needed to try harder to support my claim that the LNC did not admit of cogent dispute.)

No, that was not the point of my hypothetical response. The point was that - once we allow for the idea that there may be exceptions to the LNC - "denial" might possibly be the same as "acceptance".

"In some cases," some anti-LNCers might say. But not necessarily all. Remember, we're forsaking explosion if we're thinking in terms of paraconsistent logic. Now either a host of people who've survived university philosophy rigors are total fools about something as obvious as the LNC, or there's maybe, just maybe, something to the inquisitors, here and now. I'm opting to investigate and support, while also supporting the LNC, the opposition. If this were a political campaign, I'd say I was looking for a compromise.

 

[quatona]

To repeat my caveat: though I agree that the LNC admits of no exceptions, my acquaintance with paraconsistent logic among other topics has persuaded me that I need a deeper answer to doubts about the LNC than just "in the end, no one really is able to deny it." (Other topics include the debate over foundationalism, infinitism, coherentism, and other positions in epistemology.)




But paraconsistent logic seems capable of steering these murky waters in a way that casts a reasonable shadow of doubt over the island of the LNC.

Ripheus, then why don´t you present an example of that kind?
Or am I to assume that it´s stuff like "this sentence is false" (i.e. the mere fact that language allows for forming self-contradictory sentences) is what gives you reason to doubt the NLC?

(Of course, when you study paraconsistent logic, you might notice a basic absurdity in its arguments. I don't know. I just know that the points some opponents of the LNC have made against my original position convinced me that I needed to try harder to support my claim that the LNC did not admit of cogent dispute.)

Why, then, don´t you give me those points? It´s possible they make me reconsider. The liar paradox certainly doesn´t.

"In some cases," some anti-LNCers might say. But not necessarily all. Remember, we're forsaking explosion if we're thinking in terms of paraconsistent logic.

Indeed, that would be the ghist of my argument so far: When postulating that there are exceptions to the NLC (and be it even only one) you are exploding the very system you are utilizing. I understand that this sounds like circular reasoning but I don´t think it is. After all, it´s the very system by which they arrive at and communicate their hypothesis.

Now either a host of people who've survived university philosophy rigors are total fools about something as obvious as the LNC,

...which I think is entirely possible.

or there's maybe, just maybe, something to the inquisitors, here and now.

Then, Ripheus, why don´t you present their case? Or is the liar paradox really a typical example of what they have in store?

I'm opting to investigate and support, while also supporting the LNC, the opposition. If this were a political campaign, I'd say I was looking for a compromise.

I may have missed it, but so far I haven´t even seen any attempt from you to paraphrase the arguments against the absolute validity of the NLC.

And I repeat: As long as I see the deniers of the NLC - in support of their case - utilizing the very systems they want to explode in support of their case, I see a fundamental self-contradiction in their entire case. I do understand that pointing out a self-contradiction is not a good argument against someone who doesn´t consider self-contradictions a problem, though.

 

[Ripheus27]

Or am I to assume that it´s stuff like "this sentence is false" (i.e. the mere fact that language allows for forming self-contradictory sentences) is what gives you reason to doubt the NLC?

It doesn't give me reason to directly doubt the LNC. It gives me reason to think that it's possible to doubt it, though. And since, if an "argument" for the LNC is ever given, it often goes something like, "The LNC is true because it's impossible to actually deny it," the fact that it might be possible to deny it undermines the LNC's status as axiomatic.

Believing that the regress of beliefs and evidence and so on ends with axioms seems intuitively the default position when it comes to justifying the claims we make to ourselves and each other, granted. And I'm not saying that, "Why?" or, "Prove it," never rationally come to an end. I'm just saying that it's better to be open-minded about the chances of foundationalism being wrong, at least in the case of something like the LNC.

Indeed, that would be the ghist of my argument so far: When postulating that there are exceptions to the NLC (and be it even only one) you are exploding the very system you are utilizing.

Well, then, the other side in this debate would say: no, we're not using a system of logic based on the LNC. It is intrinsic to Aristotelian logic that no contradictions are ever true, but that school of logic is out-of-date and there are a host of other formal systems available that can meet the dialetheists' needs.

Then, Ripheus, why don´t you present their case? Or is the liar paradox really a typical example of what they have in store?

I was referring more to the fact that paraconsistent logic exists at all. How could people talk intelligibly about it if intelligible discourse derives in part from the idea that anything can be inferred from a contradiction? Yet:

A most telling reason for paraconsistent logic is the fact that there are theories which are inconsistent but non-trivial. Once we admit the existence of such theories, their underlying logics must be paraconsistent. Examples of inconsistent but non-trivial theories are easy to produce. An example can be derived from the history of science. (In fact, many examples can be given from this area.) Consider Bohr's theory of the atom. According to this, an electron orbits the nucleus of the atom without radiating energy. However, according to Maxwell's equations, which formed an integral part of the theory, an electron which is accelerating in orbit must radiate energy. Hence Bohr's account of the behaviour of the atom was inconsistent. Yet, patently, not everything concerning the behavior of electrons was inferred from it, nor should it have been. Hence, whatever inference mechanism it was that underlay it, this must have been paraconsistent. (Graham Priest and Koji Tanaka, "Paraconsistent Logic," sec. 2.1, in the Stanford Encyclopedia of Philosophy​

Dialetheism might be founded on stuff like the liar paradox, but the paradox of the set of all sets that are not members of themselves, or some of Zeno's problems, are less linguistic or abstract--more concretely or substantively real--yet difficult to resolve and intuitively comprehensible.

And I repeat: As long as I see the deniers of the NLC - in support of their case - utilizing the very systems they want to explode in support of their case, I see a fundamental self-contradiction in their entire case. I do understand that pointing out a self-contradiction is not a good argument against someone who doesn´t consider self-contradictions a problem, though.

So then you'd have to consider that those who deny the LNC might also deny that their reasons for doing so depend on a system of logic in which the LNC is implicit.

 

[quatona]

It doesn't give me reason to directly doubt the LNC. It gives me reason to think that it's possible to doubt it, though. And since, if an "argument" for the LNC is ever given, it often goes something like, "The LNC is true because it's impossible to actually deny it," the fact that it might be possible to deny it undermines the LNC's status as axiomatic.

I don´t think that making the mere claim "...but it might be possible to deny axiom X" undermines its status as axiomatic.

Believing that the regress of beliefs and evidence and so on ends with axioms seems intuitively the default position when it comes to justifying the claims we make to ourselves and each other, granted. And I'm not saying that, "Why?" or, "Prove it," never rationally come to an end.

The problem here is that asking "Why?" or "Prove it!" is meaningful in Aristotelean logic, but not without it. Challenging the LNC by asking for a line of reasoning that is based on the LNC is but a joke.

I'm just saying that it's better to be open-minded about the chances of foundationalism being wrong, at least in the case of something like the LNC.

I am open minded about it. In order to let this open-mindedness to result in doubts, however, I would have to be presented a reason to doubt it.
Semantics trickery like "This statement is false" surely is not such a reason. Moreso, it seems to confirm the LNC.

Well, then, the other side in this debate would say: no, we're not using a system of logic based on the LNC. It is intrinsic to Aristotelian logic that no contradictions are ever true, but that school of logic is out-of-date and there are a host of other formal systems available that can meet the dialetheists' needs.

1. Do you not see how this line of reasoning itself is based on Aristotelean logic, and thus - if constructed for the purpose of being a compelling argument - implicitly affirms Aristotelean logic?

2. What I would say in response: Ok, give me the rules of the formal system you want to replace Aristotelean rules by, and let´s see where a discussion based on these rules will get us.

I was referring more to the fact that paraconsistent logic exists at all. How could people talk intelligibly about it if intelligible discourse derives in part from the idea that anything can be inferred from a contradiction?

I don´t know. I have yet to see an intelligible discourse that isn´t based on LNC.

Yet:

A most telling reason for paraconsistent logic is the fact that there are theories which are inconsistent but non-trivial. Once we admit the existence of such theories, their underlying logics must be paraconsistent. Examples of inconsistent but non-trivial theories are easy to produce. An example can be derived from the history of science. (In fact, many examples can be given from this area.) Consider Bohr's theory of the atom. According to this, an electron orbits the nucleus of the atom without radiating energy. However, according to Maxwell's equations, which formed an integral part of the theory, an electron which is accelerating in orbit must radiate energy. Hence Bohr's account of the behaviour of the atom was inconsistent. Yet, patently, not everything concerning the behavior of electrons was inferred from it, nor should it have been. Hence, whatever inference mechanism it was that underlay it, this must have been paraconsistent. (Graham Priest and Koji Tanaka, "Paraconsistent Logic," sec. 2.1, in the Stanford Encyclopedia of Philosophy​

I hope you don´t take it as a cop out, but I am not well versed in natural science, and I would have to dig deep in order to understand the ideas that are described here, and to investigate if the conclusions drawn are sound. But since the conclusions quite obviously are derived by means of Aristotelean logic ("hence..."), I would have lost either way before I have even started:
- Either the conclusion is logically water-proof.
- Or the conclusion can be shown to be logically inconsistent. In which case the LNC-doubter would tell me that it´s his very position that logical consistency isn´t axiomatic, so poking holes in the conclusion´s consistency doesn´t lead to acceptable counterarguments.
​

Dialetheism might be founded on stuff like the liar paradox, but the paradox of the set of all sets that are not members of themselves, or some of Zeno's problems, are less linguistic or abstract--more concretely or substantively real--yet difficult to resolve and intuitively comprehensible.

Sorry, but as far as I am familiar with them, they aren´t even problems. They are based on abuse of language.

So then you'd have to consider that those who deny the LNC might also deny that their reasons for doing so depend on a system of logic in which the LNC is implicit.

Funnily enough, though, they would make an argument based on Aristotelean logic here. It is an appeal to the very axiom they are trying to abolish.

 

[Ripheus27]

I don´t think that making the mere claim "...but it might be possible to deny axiom X" undermines its status as axiomatic.

Fair enough, except if a claim really is self-evident, then we have to say that those who deny it are somehow deceiving themselves, don't we? So if a person can honestly disagree with a claim, then that claim can't be self-evident. Or so it seems to me...

The problem here is that asking "Why?" or "Prove it!" is meaningful in Aristotelean logic, but not without it. Challenging the LNC by asking for a line of reasoning that is based on the LNC is but a joke.

That's what's in question, though: that meaningful speech requires an exceptionless LNC.

Ok, give me the rules of the formal system you want to replace Aristotelean rules by, and let´s see where a discussion based on these rules will get us.

All I can say here is: read up on paraconsistent logic.

I hope you don´t take it as a cop out, but I am not well versed in natural science, and I would have to dig deep in order to understand the ideas that are described here, and to investigate if the conclusions drawn are sound. But since the conclusions quite obviously are derived by means of Aristotelean logic ("hence..."), I would have lost either way before I have even started...

It's not a cop-out. I myself don't know physics enough to fully confirm or deny the article I cited on this point. I'm just pointing out that perhaps there is some reason to think that we can think in terms of either the LNC being wrong, or the LNC at least admitting of exceptions (unless the LNC does in fact admit of a proof/derivation, wherefore its status as axiomatic is less self-evident).

Sorry, but as far as I am familiar with them, they aren´t even problems. They are based on abuse of language.

I don't know enough about "the set of all sets that are not members of themselves" to defend that as a paradox generator that is extralinguistic. But the question as to whether change intrinsically is inconsistent: that seems to me to be a clearer case of a concept admitting of being thought of in violation of the LNC.

Funnily enough, though, they would make an argument based on Aristotelean logic here. It is an appeal to the very axiom they are trying to abolish.

Are you totally sure, though, that paraconsistent logic derives from Aristotelian logic?

 

[quatona]

You know what: let´s just say parconsistentialism and dialetheism are right and wrong.