As usual with Stanford articles, that's some article. But what's the sense of it, or lack of sense of it? My own view of the subject - of many subjects - is that existence claims are best supported by direct demonstration or exhibition of the thing claimed to exist. True contradictions exist? Great! Show one, at least one.

Well, there's Godel's undecidable proposition G. But this at least is tricky. What G says is that neither G nor ~G is provable from within the system that G lives within. That's neither-nor, not a contradiction. But by reasoning from without of the system, usually called meta-mathematically, G is easily seen to be true. True bit unprovable: not a contradiction and the view from two very different positions (mathematical and metamathematical). Pretty clearly, then, Godel's undecidable G is not of use in thinking about contradictions.

What is a contradiction? A creature of language and not of nature. And if it devolves to arbitrary definition, then anyone can define it any way they want. That merely leaves whether that person's definition is 1) internally consistent with itself, 2) consistent with anything else, and 3) useful in any way. Consistency being just the requirement that a given system not blow itself up.

is there anything about contradictions that seems not arbitrary? They seem to defy sense, and consequently if sense is desired, then contradictions are ruled out. And this makes sense. Whatever A exactly is, it is not the case, in that case, that A exactly isn't.

And it's with the "exact," it seems to me, that dialetheism fails. And this because with exactness it must affirm and then deny what it affirms. Without exactitude, and apparent contradictions grow like weeds. Most often through category error: e.g., the bad man who is also a good man because he did one good thing.

Therefore I'm somewhat surprised, and incredulous, — Harry Hindu

Correct usage: mods please note. :wink:

What use is a contradiction? To what use could dialetheism be applied? — Harry Hindu

Vagueness. For example, non-vague discourse requires a non-vague syntax, provided by alphabetic characters of some kind. But these are always vague around the borders, causing judgements of syntactic identity to conflict. In Languages of Art, Goodman pointed out that the conflict is systematic, and maintains a suitable margin for error between the characters.

Some discourses also aspire to a non-vague semantics, in like fashion. Zones of permitted controversy concerning correct usage of a word maintain zones of unanimity concerning unacceptable usage.

And it's with the "exact," it seems to me, that dialetheism fails. And this because with exactness it must affirm and then deny what it affirms. — tim wood

Agreed.

Vagueness — bongo fury

What use is vageness other than to keep others ill-informed or to prevent one's self from being trapped by the rules of logic (especially when appealing to one's own emotions)?

Mistakes can be made with language - which is the whole point in following logical rules - to avoid those mistakes. Dialetheism doesn't seem to recognize this, as if all contradictions in language are true - whatever that means as it seems to blur the lines between what is true and false. — Harry Hindu

The only dialetheist I've read at any length is Graham Priest, and he, at least, does not maintain that all contradictions are true. Rather, he argues that some contradictions are true. More precisely, he argues that there are actual situations containing statements that are both true and false, namely, those situations that fit the semantics of the enclosure schema. These situations involve self-reference and/or reference to absolute limits. Examples from his book "Beyond the Limits of Thought" include the limit of what can be known, the limit of what can be expressed and the limit of what can be conceived. He argues that all thought about such limits is intrinsically contradictory insofar as the very act of thinking such limits requires one to transgress them at the same time. Thought draws a boundary and then crosses it in the act of thinking it.

What use is a contradiction? To what use could dialetheism be applied? What problems does it attempt to solve? — Harry Hindu

Priest would probably maintain that the mere prospect that dialetheism is true should be sufficient warrant for pursuing it, but he also thinks it is useful. In his book "One", he develops a dialetheic metaphysics that he calls "gluon theory" and employs it in solutions to various metaphysical puzzles around identity, unity, universals, being and nothingness and intentionality.

Admittedly, his theories and solutions have not been widely adopted in western analytical philosophy. And while I find them fascinating and worth learning about, I wouldn't say that I endorse them.

A couple of points:

(1) First, I'd just like to second 's claim that Priest (and most dialetheists I know) does not claim that all contradictions are true, just that some of them are. One compelling example of an alleged true contradiction is, of course, the Liar sentence. It is surprisingly difficult to develop a classical account of the Liar that satisfies everyone and that is not prey to revenge paradoxes. Dialetheism provides a very straightforward solution to this and related paradoxes.

(2) Second, paraconsistent logics in general are concerned with controlling the trivialization that follows from the principle of explosion. That is, such logics provide a workaround for when we find contradictions in our belief set or in our model. Now, you may say, why would we want such a workaround? Shouldn't we just jettison the contradiction and be done with it? Well, yes, but the problem is, how do we do this? Suppose I have beliefs $A_1, \dots, A_n$, and from these beliefs I eventually derive a contradiction, say $B \& \neg B$. This means that I should give up one of the $A_i$'s, but which one? There may be no obvious way of selecting such an $A_i$, since there may be equal evidence for each of them. In that case, a reasonable course of action would be to investigate further into the source of the contradiction so that I can eventually revise my beliefs. In the meantime, however, do I need to act irrationally, as if I believed everything (which would follow from the explosion)? Of course not. But this means that I will need to employ a paraconsistent logic, since I will need to ignore explosion. So paraconsistency may be a useful tool in "controlling" a contradiction during belief revision.

And per regarding belief revision, dialetheism could also have practical applications within the field(s) of artificial intelligence and machine learning. In fact, now that I look at it, the SEP article linked in the OP actually contains a short section devoted to this topic.

To say that a contradiction is true implies the following:

1. The definition of the AND logical connective is different from that in classical logic because p & ~p has to evaluate to true which isn't possible with the classical logic AND which is true only when both propositions connected with it are true

AND/OR

2. The equivalence rule of ADDITION has to be discarded because allowing it leads to the principle of explosion

AND/OR

3. The definition of NEGATION is different if the AND connective has the same definition as in classical logic so that both p and ~p would be true and then (p & ~p) would evaluate to true

Since the law of noncontradiction (LNC), expressed as ~(p & ~p), is critically dependent on the definitions of negation and the AND logical connective, it follows that paraconsistent logic or dialetheism has different definitions for both of them.

More precisely, he argues that there are actual situations containing statements that are both true and false, namely, those situations that fit the semantics of the enclosure schema. These situations involve self-reference and/or reference to absolute limits. — Theorem

One compelling example of an alleged true contradiction is, of course, the Liar sentence. It is surprisingly difficult to develop a classical account of the Liar that satisfies everyone and that is not prey to revenge paradoxes. Dialetheism provides a very straightforward solution to this and related paradoxes. — Nagase

Thank you both for your informative posts.

It appears that Priest is confusing a misuse of language as a new logical system. This is why I pointed out that the fact that we can organize words that follow grammatical rules doesn't necessarily mean that you've also applied the LNC. As I have said, all logical rules - where applicable - need to be followed, and you can't have a sentence that is grammatically correct, yet logically inconsistent, be true.

The fact that we can place scribbles next to each other on paper, on a computer screen, or sounds in the air, that obey grammatical rules of some language, doesn't follow that what the scribbles mean together obeys the rules of logic, namely the LNC.

The fact that I can arrange the scribbles, "This sentence is false" in a way that obeys the grammatical and spelling rules, but when the words are read and interpreted (also a necessary part of language-use, not just arranging scribbles in a way that follow grammatical and spelling rules) it is found to be a contradiction, doesn't mean that the sentence is both true and false. It is a misuse of language.

Second, paraconsistent logics in general are concerned with controlling the trivialization that follows from the principle of explosion. That is, such logics provide a workaround for when we find contradictions in our belief set or in our model. Now, you may say, why would we want such a workaround? Shouldn't we just jettison the contradiction and be done with it? Well, yes, but the problem is, how do we do this? Suppose I have beliefs A1,…,AnA1,…,An, and from these beliefs I eventually derive a contradiction, say B&¬BB&¬B. This means that I should give up one of the AiAi's, but which one? There may be no obvious way of selecting such an AiAi, since there may be equal evidence for each of them. In that case, a reasonable course of action would be to investigate further into the source of the contradiction so that I can eventually revise my beliefs. In the meantime, however, do I need to act irrationally, as if I believed everything (which would follow from the explosion)? Of course not. But this means that I will need to employ a paraconsistent logic, since I will need to ignore explosion. So paraconsistency may be a useful tool in "controlling" a contradiction during belief revision. — Nagase

And per ↪Nagase regarding belief revision, dialetheism could also have practical applications within the field(s) of artificial intelligence and machine learning. In fact, now that I look at it, the SEP article linked in the OP actually contains a short section devoted to this topic. — Theorem

This seems to be very rare occasions where two contradictory beliefs have the same amount of information. I am finding it difficult to even think of an actual example.

Computer programmers still have to program a machine to make decisions based on something, and when contradictions arise in memory, other factors can help determine which path the program should take.

This also seems to conflate our ignorance as a new logical system. Sure, there are times when we need to act on incomplete, or even inconsistent, information. I don't understand how one could act as if they were all true, or what that would even look like, but I can understand acting as if one is true one moment and the other true at another moment. It doesn't follow that what I am acting on is completely true as I may make a mistake, and then that is when I learn which contradictory belief is actually false.

Since the law of noncontradiction (LNC), expressed as ~(p & ~p), is critically dependent on the definitions of negation and the AND logical connective, it follows that paraconsistent logic or dialetheism has different definitions for both of them. — TheMadFool

Then dialetheism and LNC are talking past each other when using these terms?

Then dialetheism and LNC are talking past each other when using these terms? — Harry Hindu

The whole thing reminds me of the political entity we so fondly call "government" of, say, the USA. At the end of every term, the entire team that is the government is changed (barring the times when a party gets re-elected) - the president is different, the vice president is different, and so on, and yet, we still refer to every one of these disparate entities as the government of the USA. Likewise, every little essential detail of the LNC has been altered and all that remains of the real LNC is just the label - "the law of noncontradiction." This term is empty of meaning.

The whole thing reminds me of the political entity we so fondly call "government" of, say, the USA. At the end of every term, the entire team that is the government is changed (barring the times when a party gets re-elected) - the president is different, the vice president is different, and so on, and yet, we still refer to every one of these disparate entities as the government of the USA. Likewise, every little essential detail of the LNC has been altered and all that remains of the real LNC is just the label - the law of noncontradiction. It's an empty word if you ask me. — TheMadFool

There isn't much difference between the political parties in the U.S. They both promote bigger government. And the reason that we will always switch back and forth (and a reason for the Electoral College) is that if one party gains to much power and the other party is never able to take the majority temporarily, then many states will secede. To appease the masses and keep the union unified, both parties take turns being the majority.

Three examples of "true" contradictions come to mind

i) Self-negating universal imperatives, i.e. hypocritical statements such as "Don't live by rules!".

ii) When a semantic distinction is more fined grained than is expressible in the language used, such as when standing in a doorway and thereby "being in the kitchen but not in the kitchen".

iii) When a semantic distinction is vague or uncertain, such as "a heap of sand" that isn't defined in terms of a particular numeric range of sand grains Hence "heaps of sand" exist, but no particular collection of sand grains constitutes a heap.

ii + iii are contradictions that programmers have to deal with, but they also present challenges for self-learning autonomous agents, that like human beings must somehow internalise a truth-language distinction.

I suspect that like humans, AI agents will also behave in a logically inconsistent fashion relative to their self-knowledge.

You can't actually picture a married bachelor, or a square-circle in your mind. — Harry Hindu

Are you saying that things can only be the case if we can picture them in our mind?

There isn't much difference between the political parties in the U.S. They both promote bigger government. And the reason that we will always switch back and forth (and a reason for the Electoral College) is that if one party gains to much power and the other party is never able to take the majority temporarily, then many states will secede. To appease the masses and keep the union unified, both parties take turns being the majority. — Harry Hindu

I don't know how politics works but the words "law of non contradiction" as used in (some) paraconsistent logical systems is like the word "government" in being nothing more than a label - all the essential components change but the label remains the same.

I maybe completely wrong on this but that's how I feel about it.

If the LNC is indeed violated or rejected then only one option remains for anyone developing such a system and that's to remove the ADDITION equivalence rule. If not the principle of explosion is going to wreak havoc in such a system; interestingly, it seems that in paraconsistent logic the aim is to allow contradictions but avoid a system that's trivial in the sense one in which any and all propositions can be proved. One wonders what it means to not want a system in which any and all propositions can be proved if not that the aim is not to have situations where p & ~p i.e. a logical system is most trivial, if triviality comes in degrees, precisely when contradictions become possible.

:chin:

It appears that Priest is confusing a misuse of language as a new logical system. — Harry Hindu

Priest (and other dialetheists) would obviously disagree. They present arguments. You should study them sometime.

Priest (and other dialetheists) would obviously disagree. They present arguments. You should study them sometime. — Theorem

And while I find them fascinating and worth learning about, I wouldn't say that I endorse them. — Theorem

Obviously you weren't moved by their arguments or else you would endorse them. Why aren't you endorsing them? What is it that you find lacking, or unreasonable, in their arguments?

Are you saying that things can only be the case if we can picture them in our mind? — Michael

Sure. Why not? We, and our minds, are governed by the same laws of the universe as everything else. It seems to me that the burden is upon those that claim otherwise - to explain why a mind that is governed by the same laws of the universe wouldn't be able to understand the laws of the universe.

Doesn't solipsism, idealism and panpsychism endorse the idea that things only exist when present in some mind, primarily because mind is a fundamental feature of reality?

i) Self-negating universal imperatives, i.e. hypocritical statements such as "Don't live by rules!". — sime

A contradictory statement says nothing at all, and is therefore useless. It is basically asserting something and then walking back that assertion at the same time resulting in a net zero amount of information. It is basically scribbles on a page, or sounds in the air.

ii) When a semantic distinction is more fined grained than is expressible in the language used, such as when standing in a doorway and thereby "being in the kitchen but not in the kitchen". — sime

There is also the problem of ignorance of the language being used. I could just say, "being in the doorway between the kitchen and dining room". Languages are typically malleable and new things can be said using the same words (or even new words) in different patterns in different contexts, given the intelligence and wittiness of the person using some language. It's one of the ways that languages evolve.

iii) When a semantic distinction is vague or uncertain, such as "a heap of sand" that isn't defined in terms of a particular numeric range of sand grains Hence "heaps of sand" exist, but no particular collection of sand grains constitutes a heap. — sime

Vagueness or uncertainty don't count as a contradiction. A contradiction is a set of clearly defined assertions that stand in direct opposition to each other. There is no vagueness or uncertainty there, except as an effect of the contradiction, as per my response to i). A contradiction provides zero information, and zero use.

ii + iii are contradictions that programmers have to deal with, but they also present challenges for self-learning autonomous agents, that like human beings must somehow internalise a truth-language distinction.

I suspect that like humans, AI agents will also behave in a logically inconsistent fashion relative to their self-knowledge. — sime

If a contradiction provides zero information, and zero use, then what would it look like to act on zero information? What information would it be using to act on? When a contradiction arises, alternative reasons for acting or not acting a certain way will be searched for, so the reason for acting or not will actually have nothing to do with the contradiction.

Are you saying that things can only be the case if we can picture them in our mind? — Michael

Sure. Why not? — Harry Hindu

Do you think that this undermines realism? Can you picture a cup without picturing the look of a cup, or the feel of a cup, i.e. how a cup is experienced?

Of course not.

Can you picture a cup and non-cup in the same mental space and moment? I can picture the words, "cup and non-cup", but not what they refer to.

So you're an idealist now? Or are you saying that cups have some innate appearance, irrespective of what they look and feel like to us?

Can you picture a cup without picturing the look of a cup, or the feel of a cup, i.e. how a cup is experienced? — Michael

Or are you saying that cups have some innate appearance, irrespective of what they look and feel like to us? — Michael

The innateness of the look, or appearance, of the cup lies in the mind. The way the cup is, is irrespective of what they look/appear and feel like in our mind.

The way the cup is, is irrespective of what they look/appear and feel like to us. — Harry Hindu

So you can picture the way a cup is, irrespective of its appearance?

To say that the cup has an appearance is to project "appearances" onto the cup, when the appearance lies only in the mind, as an effect of the existence of the cup and it's interaction with light and your eyes and brain. The cup is (one of) the cause(s), the appearance in the mind is the effect. Effects carry information about the cause, hence we can get at what the cup is via it's appearance in the mind.

To say that the cup has an appearance is to project "appearances" onto the cup, when the appearance lies only in the mind, as an effect of the existence of the cup and it's interaction with light and your eyes and brain. The cup is (one of) the cause(s), the appearance in the mind is the effect. Effects carry information about the cause, hence we can get at what the cup is via it's appearance in the mind. — Harry Hindu

This doesn't answer my question. Can you picture a cup without just picturing how a cup appears to us? Can you picture just the innate nature of the cup?

This doesn't answer my question. — Michael

Sure it does.

Can you picture a cup without just picturing how a cup appears to us? — Michael

Like I said, pictures/appearances/looks only exist in minds, so no you can't picture a cup without picturing how a cup appears in the mind. The question is nonsensical. It's like asking, can you think about a cup without thinking about it?

Can you picture just the innate nature of the cup? — Michael

Like I said, the picture is an effect of the innate nature of the cup, innate nature of light, innate nature of your brain and eyes. You can get at the innate nature of all of these things via the innate nature of the picture. Turn out the lights and that changes the picture of the cup. Grow a tumor in your brain, or on the eye stem, and that changes the picture of the cup.

Like I said, pictures/appearances/looks only exist in minds, so no you can't picture a cup without picturing how a cup appears in the mind. — Harry Hindu

Then you're contradicting yourself. Here you claimed that things can't be the case if we can't picture them in our mind, and now you're saying that the innate nature of cups can't be pictured in our mind; only their appearances can.

You need to drop one of these claims. Either only appearances are the case or things can be the case that can't be pictured in our mind.

A contradictory statement says nothing at all, and is therefore useless. It is basically asserting something and then walking back that assertion at the same time resulting in a net zero amount of information. It is basically scribbles on a page, or sounds in the air — Harry Hindu

Yes in the sense of contradictory propositions. Nobody of course, experiences contradictory propositions - which goes to show that the general meaning of "contradiction" isn't to refer to propositions but to conflicts, such as the conflict between the definition of a language and it's application, or the rules of a sport and the moral notion of fair-play etc.

Yes in the sense of contradictory propositions. Nobody of course, experiences contradictory propositions - which goes to show that the general meaning of "contradiction" isn't to refer to propositions but to conflicts, such as the conflict between the definition of a language and it's application, or the rules of a sport and the moral notion of fair-play etc. — sime

The reason we don't experience contradictory propositions is precisely because what we experience is information, and if there is no information, then there is nothing to experience - except for the visual experience of the seeing scribbles on a screen or hearing sounds spoken - which is information, but about something else that isn't about what is being written or said.

Then you're contradicting yourself. Here you claimed that things can't be the case if we can't picture them in our mind, and now you're saying that the innate nature of cups can't be pictured in our mind; only their appearances can.

You need to drop one of these claims. Either only appearances are the case or things can be the case that can't be pictured in our mind. — Michael

Right, and what I was talking about when it comes to picturing things in the mind were contradictions - like cups and non-cups. Do cups and non-cups exists as one entity either in your mind as an appearance (can you imagine a cup and non-cup entity?) or outside of your mind as an innate object of the universe? From where do contradictions come from - from somewhere out in the world for to be experienced by a mind that observes them, or are they created by the mind as a misuse of language, and then projected onto the world as if they existed outside of the mind?

Obviously you weren't moved by their arguments or else you would endorse them. Why aren't you endorsing them? What is it that you find lacking, or unreasonable, in their arguments? — Harry Hindu

I don't look at it that way. Dialetheism provides a way solving certain problems. Like any proposed solution to a problem, it has costs and benefits relative to other proposals. Which solution to use depends on the solution context. If I were writing a belief revision algorithm for an AI program, I would use whatever algorithm worked best within the resource constraints I was facing. If the dialetheic approach proved the most adequate to meet the requirements, I would use it.

When it comes to using dialetheism as the basis for one's personal metaphysics and epistemology, the same considerations apply. The class of situations/propositions to which dialetheism is purported to apply is exceedingly small. Again, I can only speak for Priest, but he applies it only to propositions/situations satisfying the semantics of the "enclosure schema". He feels that dialetheism provides the most "satisfying" solution for dealing with the semantics of such situations - for him, the benefits outweigh the costs.

Of course, things get fuzzy when it comes to evaluating the "truth" of metaphysical theories. Beyond insisting that such theories not be fatally self-contradictory, the criteria are mostly aesthetic and personal, although the behavioral consequences of holding such theories should also be a consideration (e.g. if a such a theory would prompt a person to become a suicide bomber, etc.). For me personally, while I can appreciate the beauty of Priest's metaphysics, and have no problems with the ethical commitments he derives from it, I ultimately find it too lifeless for my tastes.

You seem to think that a picture is a causally closed event. The innate nature of pictures is that they are about other things because they are caused by those things - the innate nature of other things. It wouldn't be a "picture" if the image of the cup wasn't about the innate nature of the cup, just as words are about your mental state, but aren't your mental state, yet I can still imagine the nature of your mental state by you describing it to me.

Here you claimed that things can't be the case if we can't picture them in our mind, and now you're saying that the innate nature of cups can't be pictured in our mind; only their appearances can. — Michael

Would it be better if I had said that things can't be the case if we can't represent, symbolize, or simulate them in our mind?