Soundness

Moliere

I'd be happy for replies to even just contain resources for further reading on my part -- books, papers (behind a paywall or no, doesn't matter), and that sort of thing.

I was thinking today about how in learning baby logic much was made of the distinction between validity and soundness -- and understandably so, given that validity is a counter-intuitive notion -- with primary focus being on the concept of validity because, hey, that's what logic does.

But what about the concept of soundness? If we have a sound argument then we have an argument that is both valid, and contains true premises. But this is what's interesting about that -- if we have an invalid argument the truth-value of all the propositions and the conclusion can all be either true or false. All soundness does is guarantee that insofar that the truth value of the first premises is true, then the conclusion will also be true -- in fact one standard way of demonstrating that an argument is fallacious is to have two true premises that give an obviously false conclusion. So you might even say that soundness is analytically defined as those forms of argument which happen to preserve truth.

Or, given that there are multiple systems of logic, we could also say that soundness is relative to some system of rules of inference, and all arguments which follow said rules of inference and also contain true premises produce sound arguments.

But there's something funny in all that, at least to my ear. It almost seems like we need to already know the truth of our premises in order for the logic to be worked out. But if that's the case then how does logic retain its usefulness in the cases where we do not know the truth value of some conclusion? We may have good reason to believe the premises, but couldn't a novel argument actually be a case where we are proving that the form of our argument is, in fact, fallacious because it leads to a false conclusion?

In which case, what is the point of soundness anyways?

Anyway, a bit of a ramble. Really I'm just fishing for any resources that might detail the concept of soundness more fully to satisfy an intellectual itch I have. I'm trying to avoid castles in the sky and building opinions on my own flim flam.

Shawn

As Wittgenstein said, "Logic takes care of itself."

As to how that is so is a mystery to my mind, and invokes metaphysical abstraction.

Metaphysician Undercover

But there's something funny in all that, at least to my ear. It almost seems like we need to already know the truth of our premises in order for the logic to be worked out. But if that's the case then how does logic retain its usefulness in the cases where we do not know the truth value of some conclusion? We may have good reason to believe the premises, but couldn't a novel argument actually be a case where we are proving that the form of our argument is, in fact, fallacious because it leads to a false conclusion?

In which case, what is the point of soundness anyways? — Moliere

It's not that the logic cannot be worked out without knowing whether the premises are true. But the conclusion cannot be said to be sound without knowing that the premises are true. This is evident from the fact that the conclusion may be valid but not sound.

If we have good reason to believe that the premises are true, and also that the conclusion is false, then we have good reason to doubt the soundness of the argument. Each aspect, the premises and the logical proceeding, ought to be analyzed for the possibility of mistake.

apokrisis

Really I'm just fishing for any resources that might detail the concept of soundness more fully to satisfy an intellectual itch I have. — Moliere

No good resources spring to mind. I don't think this is a well-explored question.

But the answer seems obvious enough. Soundness is based on a mechanical or atomistic ontology. You have some bunch of stable parts. The parts can be rearranged according to some set of rules with nothing essential being changed about those parts. There is no leakage or spoilage as the parts get shuffled around. And any rule-based shuffle of the parts is an open possibility. No patterns or states that can be accessed by those syntactic rules are off-limits.

So it is classic mechanical thinking. All the accidents are ruled out at the level of the atomistic parts. There is closure right there. And then the combinations of those parts is made the opposite - as open and unconstrained as you like. Provided the combinations follow the rules. Which you can freely create. And so even the rules have no rules. Syntax is arbitrary. Or at least the only meta-rule is that the rules are capable of being followed. There is a soundness requirement in that. Which in turn refers you back to the notion of a part. A part, being atomistically stable, guarantees that timeless syntax, abstract rules, can be the case. The determinism is grounded in that assumption. Soundness thus follows as any dynamics or change can only be caused by "the rules".

MindForged

So you might even say that soundness is analytically defined as those forms of argument which happen to preserve truth. — Moliere

That's validity. Broadly, validity is defined as "Truth preservation over all cases", or if one wanted to be a douche, "Preservation of the designated value across propositional transformations" (I wrote this in class once and the professor indirectly told me to chill the fuck out, lol). Soundness requires a valid argument and true premises.

Or, given that there are multiple systems of logic, we could also say that soundness is relative to some system of rules of inference, and all arguments which follow said rules of inference and also contain true premises produce sound arguments. — Moliere

No no, validity is indeed defined relative to a logic. After all, what counts as a "case" isn't fixed across logics. Soundness is the same concept across the board basically. So long as the designated value is preserved from the start to the end and the premises are true, it's a sound argument. The conclusion is veridical.

It almost seems like we need to already know the truth of our premises in order for the logic to be worked out. — Moliere

Not for the validity. That's the logic part. Soundness (at the object language) does need the argument to be valid but the truth value of the premises isn't a question of logic. The logic is the machinery guiding the inferences, soundness is, like, whether or not the machine is doing a good job.

Moliere

That's validity. Broadly, validity is defined as "Truth preservation over all cases", or if one wanted to be a douche, "Preservation of the designated value across propositional transformations" (I wrote this in class once and the professor indirectly told me to chill the fuck out, lol). Soundness requires a valid argument and true premises. — MindForged

You are right.

Here's where I'm getting tripped up in talking about soundness. While validity does not rely upon soundness for its conceptual clarity, soundness does rely upon validity -- since it must satisfy both that the argument is valid and the argument uses true premises. That's what I'm trying to get at at least, though I do not think I'm putting it well since I basically just restated validity as you note.

The conclusion is veridical. — MindForged

Is veridicality the same as soundness?

Not for the validity. That's the logic part. Soundness (at the object language) does need the argument to be valid but the truth value of the premises isn't a question of logic. The logic is the machinery guiding the inferences, soundness is, like, whether or not the machine is doing a good job. — MindForged

Well, I don't know. At least with informal logic usually the procedure is to show some kind of argument that uses a form with true premises and a false conclusion to demonstrate that some form of argument is invalid. Granted that's not the same as validity, but that's where I'm coming from in my (admittedly rambly) ruminations; that the demonstration of invalidity, rather than validity, relies upon true premises reaching a false conclusion.

So I keep on thinking -- might it be the case that we use a (now believed valid) form of argument with true premises that then comes to false conclusions? That doesn't seem correct at all, because when I read formal arguments i feel they are persuasive and they hold regardless of content, and so we can then get on with arguing over the content as long the argument is valid. But then there is this fairly common procedure of demonstrating invalidity that crops up in my mind.

Something like the problem of induction.

Though I admit I'm feeling really unclear right now. I just thought that clarifying soundness might help in getting around the conceptual confusion I've created for myself.

javra

So I keep on thinking -- might it be the case that we use a (now believed valid) form of argument with true premises that then comes to false conclusions? — Moliere

Conclusions can be deemed erroneous due their being contradictory—either relative to themselves, as in some obtained conclusion that affirms A and not-A at the same time and in the same respect, or else in relation to some set of known truths that are contextual to the obtained conclusion. I can’t currently think of other means by which concluded truths can be deemed erroneous which do not themselves reduce to the presence of inconsistency. If someone else can, this would complicate my argument.

What you bring up can, to my mind, be exemplified relatively well by Zeno’s paradoxes. Their reasoning is logically valid, as far as we can tell. Yet their conclusions are contradictory to experience. Hence: That it’s impossible for a runner to outrun a turtle as long as the turtle has an initial lead, or that it’s impossible for an arrow to hit its target, are logically valid arguments (as Zeno argues them). But, because they contradict with experience (which is—as an ever changing awareness of givens—after all required to make sense of these paradoxes of change/motion which conclude that no change can occur), we then know that something is false somewhere along the way with the argument. It might be that some of the premises, thought they seem intuitively true, are false or that the arguments, which seem logically valid, are in fact not valid.

But generally speaking, if both the premises and the conclusions hold consistency to all other related truths and are obtained via reasoning that is not demonstrably invalid, then the argument gives all indications of logical soundness. It’s how we know that the premises are true to begin with: they’re consistency to all other relevant truths. It’s not until contradictions occur that we hold reason to question the soundness of arguments.

I’m not sure, but maybe this serves to address the issue you’re enquiring into.

MindForged

Here's where I'm getting tripped up in talking about soundness. While validity does not rely upon soundness for its conceptual clarity, soundness does rely upon validity -- since it must satisfy both that the argument is valid and the argument uses true premises. That's what I'm trying to get at at least, though I do not think I'm putting it well since I basically just restated validity as you note. — Moliere

Well, what's wrong with soundness relying on validity? :) For validity, we just want to ensure the logic cannot take us from truth in to falsity out because the point of logic is to have this clean rules for reasoning. We want to move from truth to further truths. Soundness of an argument is, really, just a confirmation that the conclusion is actually true.

Is veridicality the same as soundness? — Moliere

I'm just saying that in the case of a sound argument the conclusion is really true. "Veridical" isn't really a logic term, I was just trying to use a word besides "true" lol.

At least with informal logic usually the procedure is to show some kind of argument that uses a form with true premises and a false conclusion to demonstrate that some form of argument is invalid. Granted that's not the same as validity, but that's where I'm coming from in my (admittedly rambly) ruminations; that the demonstration of invalidity, rather than validity, relies upon true premises reaching a false conclusion. — Moliere

Ahhh, I see. An argument for can be shown to be invalid in the way you mention. But that isn't because of soundness is prioritized (though it is relevant in an indirect way). By showing that an inference pattern can in at least one case take you from truth to falsity, you're showing it isn't valid because, remember, validity is defined as "Truth preservation over all cases". So if you show that there's one case where it doesn't, the argument isn't valid. It doesn't really have to do with the truth of the premises, the exercise just shows a flaw with the logical machinery.

Perhaps an example is in order. Take the following Aristotlean syllogism:

All As are Bs
All As are Cs
Therefore some Bs are Cs

In this very abstract formulation, it might seem reasonable to think it valid. It kind of looks like it might be. And a simple substitution of terms seems to give it credence:

All Mammals are life forms.
All Mammals have blood.
Therefore some life forms have blood.

But as it turns out, Aristotle's logical system is flawed because we know this argument form fails to go from truth to truth in some cases, like:

All winged horses are horses
All winged horses have wings
Therefore some horses have wings

And obviously there are no winged horses in reality. The problem is with the actual argument form, we are merely using real world things and truth value to bring the flaw in the argument form to light. Lack of soundness here just shows us that the argument form cannot be valid since soundness requires validity. It's about this truth-in, truth-out requirement.

tim wood

And obviously there are no winged horses in reality. The problem is with the actual argument form, we are merely using real world things and truth value to bring the flaw in the argument form to light. — MindForged

I think the language got away from you. The - your - argument above is called "Bramantip," A-A-I in the fourth form. It's valid (with an existential qualification usually not made explicit). But with winged horses it's not sound. But i'm pretty sure you already know this.

MindForged

↪tim wood

No, that was not Bramantip, it was Darapti. Like Bramantip, Darapti is also not a valid argument in modern logic because as we see is Syllogistic, it can lead from truth to falsity due to existential quantification.

javra

All winged horses are horses
All winged horses have wings
Therefore some horses have wings — MindForged

Isn’t a winged horse by definition not a horse? (Its proper term being a Pegasus.) In parallel, I’m thinking that a unicorn is not a horse. Or, a sphinx is not a lion. I’m here addressing my sense of proper categories as regards the validity of the argument—this rather than the issue of fictitious beings v. real beings.

MindForged

↪javra

Surely a winged horse is just a type of horse? I mean, Pegasus is just a horse with wings...

javra

↪MindForged

Ea, maybe I’m picking at straws. But then, if a Pegasus is just a horse with wings, wouldn’t the conclusion preserve truth? As in, because horses can both have and not have wings as a category of entity/being—winged horses being a type of horse—then some horses (members of the category just expressed) do have wings.

Yet we acknowledge that horses do not have wings ... I'm guessing due to our implicit understanding of the category horse as something distinct from the categories of Pegasus or unicorn.

tim wood

No, that was not Bramantip, it was Darapti. Like Bramantip, Darapti is also not a valid argument in modern logic — MindForged

You're right, Darapti, A-A-I, third form! I goofed. If possible in just a couple of sentences, why is it invalid? Is it because of the existential implication of some s is p? If that, then you cannot get from all s is p to some s is p. Or is it something else?

MindForged

↪tim wood

Yeah it's existential implication. Because I'm going from talking about a class of things and proceeding to a conclusion about a particular. But of course in the real world, categories can fail to have members that populate it (empty sets). Hence, in modern logical systems, the translation of the Darapti mood is invalid in virtue of committing the existential fallacy. You cannot go from a universal claim to a conclusion about a particular thing without first asserting that there is at least one member which falls into that class. So,

All winged horses are horses,
All winged horses have wings,
There is at least one winged horse,
Therefore some horses have wings.

That's valid. But we know it's not sound since the third premise is clearly false.

Some people try to defend Syllogistic on this point by saying Aristotle thought logic was only concerned with existing things and so it's not really invalid. Of course, this is just stupid. If you try to keep this as valid and say that, for example, Syllogistic doesn't get anything wrong you end up invalidating other argument forms that are considered valid (I can go into this if you want) and you might as well exhaust mathematics since mathematics cannot function with the limited inference resources of Syllogistic (and besides, lots of math (pure mathematics) isn't "real" so this is a death knell). With the creation of Classical Logic, Frege set up a nice foundation for standard mathematics and its underlying reasoning.

tim wood

↪MindForged

It seems - seems - that you bury Darapti and kin because they don't "work" outside of their home ground, Aristotelian logic. And this seems similar to burying Newtonian physics because of relativity. But where Newtonian physics works, it works just fine and is usually preferred for simplicity's sake. The question then is, does Aristotelian logic work within its proper sphere? I'm thinking it works just fine.

Looking a littler closer:

1. All winged horses are horses,
2. All winged horses have wings,
~~3. There is at least one winged horse,~~
4. Therefore some horses have wings. — MindForged

Your 3 isn't part of the syllogism. There is always an implied if before every proposition; for the conclusion, it might be if 1 and 2 are true, then....

Maybe a problem for modern logic, but not-so-much for what it is and always was. What am I missing?

MindForged

↪tim wood

The problem is logic is supposed to work everywhere. There's no way to demarcate fictional or formal objects as a realm where logic need not apply. So the comparison to the practical use of Newtons an dynamics (despite technically not holding in all cases (relativistic speeds)) is seems like a false comparison to me. As I said, unless you're willing to severely limit and falsify much of mathematics (namely pure mathematics, which bear no known relationship to the actual world) then this seems doubly unacceptable. The transition away from Aristotle's logic to Frege's "classical" logic was Precisely because of these and other issues with the former.

Your 3 isn't part of the syllogism. There is always an implied if before every proposition; for the conclusion, it might be if 1 and 2 are true, then.... — tim wood

That's not true. Syllogistic doesn't have conditionals at all. I added number 3 to show the argument as it would be rendered in modern logic if it is the be valid. If you remove the 3rd premise in a modern logic, the resulting argument is Darapti and that commits the existential fallacy.

Syllogistic uses universal and particular declarations (although one should note that an actual theory of quantifiers did not exist until Frege created one). It doesn't have conditionals, connectives and such. So there is no implied "If" in Aristotle's logic. He actually mentions knowing (maybe in his work "Metaphysics", can't remember off hand) that there are examples of necessary relationships between premises and conclusions but which he does not consider syllogisms. He names what we call conditionals as one such non-syllogism.

tim wood

So there is no implied "If" in Aristotle's logic. — MindForged

I find much to disagree with in your post, but this can stand for most of it. There is an implied if in all logic. What the exact content of the if-statement is depends on the context; usually it's not important; and usually it's only a problem when people forget it's there.

And hypothetical syllogisms are discussed in (by) Aristotle.

MindForged

I find much to disagree with in your post, but this can stand for most of it. There is an implied if in all logic. — tim wood

I said in Aristotle's logic. I'd agree if you said the rest of logic has an implied conditional. It's in the definition of logical consequence, after all.

And hypothetical syllogisms are discussed in (by) Aristotle. — tim wood

I did not deny that. What I said was that Aristotle says conditionals are not syllogisms, and even goes so far as to say hypothetical syllogisms are not reducible to syllogisms.

Oh, to add to my previous post. I found the bit of Aristotle I was thinking of, but it was in the Prior Analytics. Relevant to this discussion (emphasis added):

In some arguments it is easy to see what is lacking, but others escape our notice and appear to syllogize because something necessary results from what is
supposed. For example, if one had assumed that if a non-substance is destroyed then
a substance will not be destroyed, and that if those-things-out-of-which are destroyed,
the-thing-out-of-them also perished—when these things have been laid down, it is necessary indeed that the part of a substance should be a substance, but this has not been syllogized through the things taken, but rather premises are left out.
[...]
Again, if it is necessary, if a human is, for an animal to be and, if an animal, a substance, then it is necessary, if a human is, for a substance to be; but it has not yet been syllogized, since the premises are not related as we have said. We are misled in cases like these by the fact that something necessary results
from what is supposed, because a syllogism is also necessary. But ‘necessary’ is more extensive than ‘syllogism’: for every syllogism is necessary, but not everything necessary is a syllogism. Consequently, if something does result when certain things have been
posited, one should not try straight off to lead it back <into the figures>. Instead, one
must first get the two premises and next divide them this way into terms, and that
term which is stated in both the premises must be put as the middle (for the middle
must occur in both of them in all of the figures) — Aristotle

tim wood

↪MindForged

I appreciate your effort typing. I read this as a discussion as the need for a middle term in a syllogism, and for the middle term to be distributed.

But I'd like to stay within the ten ring of the target in this part of this thread. I read you as saying that certain arguments that presuppose existential import are out-of-court because the presupposition is not in all cases justified, and therefor the form of the argument is no good. My side is simply that as "machine," such arguments are indeed efficacious, but as with most things, some discernment in use is desirable lest their product be misapprehended. Or, Aristotelian logic not much use for modern logic, which says noting about its usefulness in Aristotelian logic. That is, I'm defending Aristotelian logic - because I suppose your attack on parts of it are an error.

MindForged

I read you as saying that certain arguments that presuppose existential import are out-of-court because the presupposition is not in all cases justified, and therefor the form of the argument is no good. — tim wood

Hm, I think that is exactly what I'm saying. The usual defense I see of this is that Aristotelian logic was intended for real things so existential import is sort of a given, but as I said I believe this severely limits the use for that logic. That's why Frege made classical logic, the old logic just wasn't good for mathematical reasoning. I'm not saying the logic is useless or something, but that modern logic is more useful.

Andrew M

All winged horses are horses,
All winged horses have wings,
There is at least one winged horse,
Therefore some horses have wings.

That's valid. But we know it's not sound since the third premise is clearly false.

Some people try to defend Syllogistic on this point by saying Aristotle thought logic was only concerned with existing things and so it's not really invalid. Of course, this is just stupid. If you try to keep this as valid and say that, for example, Syllogistic doesn't get anything wrong you end up invalidating other argument forms that are considered valid (I can go into this if you want) and you might as well exhaust mathematics since mathematics cannot function with the limited inference resources of Syllogistic (and besides, lots of math (pure mathematics) isn't "real" so this is a death knell). — MindForged

I'm curious about what the problems really are. It seems easy enough to modify the above argument so that it is sound. For example, in Greek mythology all winged horses are horses ... in Greek mythology there is at least one winged horse, namely Pegasus.

Also it seems to me that your first premise is problematic in the same way that the King of France is bald is problematic. If winged horses don't exist, then you can't predicate anything of winged horses. If so, then the argument is invalid as well as unsound.

MindForged

It seems easy enough to modify the above argument so that it is sound. For example, in Greek mythology all winged horses are horses ... in Greek mythology there is at least one winged horse, namely Pegasus.[/quote]

I should note the argument you quoted is valid in modern logic, but if the third premise is dropped it's not. It will become valid in Aristotle's logic but will render it unsound. I wasn't sure if you were disputing that, I'm reading this in a hurry at the moment.

That said, as I said, Aristotle does not take logic to regard fictional things so I don't think adding "In Greek Mythology" will render it a valid syllogism. He takes it to be that while something might be a necessary relation, that doesn't make it a syllogism. See above for my quote of him in Prior Analytics.

I'm curious about what the problems really are — Andrew M

Now I REALLY have to do this quickly, so I had to just copy paste this from Graham Priest's book "Doubt Truth to be a Liar" where he covers this. Sorry about this, but I have to go to work and I'll likely forget about coming back to it later after work tires me out! Sorry for any formatting errors I missed. I don't know if this forum supports math libs so I tried to make everything look close enough to how the book displays them:

Reveal

That Aristotelian and classical logic are distinct will hardly be denied. But it might
well be suggested that the adoption of classical logic did not revise Aristotelian logic
in any interesting sense: Aristotelian logic was perfectly correct as far as it went; it was just incomplete. Classical logic simply extended it to a more complete theory. Such a suggestion would be false. It is a well-known fact, often ignored by philosophers (though not, perhaps, historians of philosophy) that Aristotelian logic is incompatible with classical logic in just the same way that non-Euclidean geometries are incompatible with Euclidean geometry. A central part of Aristotelian logic is syllogistic, and the most natural translation of the syllogistic forms into classical logic is as follows:

AaB | All As are Bs ∀x(Ax ⊃ Bx)
AeB | No As are Bs ¬∃x(Ax ∧ Bx)
AiB | Some As are Bs ∃x(Ax ∧ Bx)
AoB | Some As are not Bs ∃x(Ax ∧ ¬Bx)

Given this translation, Aristotelian syllogistic gives verdicts concerning the validity
of some syllogisms that are inconsistent with classical logic. Consider the inferences
called by the medievals Darapti and Camestros, which are, respectively:

All Bs are Cs
All Bs are As
Hence some As are Cs

All Cs are Bs
No As are Bs
Hence some As are not Cs

Both of these are valid syllogisms. Both are invalid in classical logic. The problem is, of course, one of existential import. Some syllogisms seem to presuppose that various categories are instantiated. It is sometimes suggested that the problem can be repaired by adding the import to the translations explicitly. Specifically, we add the clause ∃xAx to each of the a and e forms. (It would be redundant in the other two.) This is, indeed, sufficient to render all the syllogistic forms classically valid, but the problem with this is that it invalidates other central parts of Aristotelian logic, notably, the square of opposition. The square is:

AaB| AeB
_________
AiB | AoB

where the claims on the top line are contraries; on the bottom line are sub-contraries; and on both diagonals are contradictories. Now it is clear that, once the a form is augmented with existential import, a and o are not contradictories: both are false if
there are no As. For the same reason, neither are e and i.

Another suggested repair is to add existential import to the a form (but not the e), and take the o form to be its negation (∃x(Ax ∧¬Bx)∨¬∃xAx). This validates all the syllogisms and the square of opposition. The oddity of taking ‘some As are not Bs’ to be true if there are no As is clear enough. But more importantly, this repair invalidates another part of the traditional logic: the inferences of obversion. Specifically, obversion permits the inference from ‘no As are Bs’ to ‘all As are non-Bs’; which fails if the e form
is not existentially loaded. Obversion is not in Aristotle, but it is a perfectly standard
part of traditional logic.

It is sometimes suggested that, rather than adding existential import to the translations explicitly, we should take the instantiation of all the categories involved to be a global presupposition.This is a move of desperation. If it is correct, then we cannot use syllogistic to reason, e.g. in mathematics, where we certainly do not make such presuppositions. I don’t think that the traditional logicians who endorsed syllogistic believed this. Moreover, if we were to allow validity to have contingent presuppositions, pretty much anything could be made to be valid.

More importantly, the suggestion really will not save syllogistic. All winged horses
are horses, and all winged horses have wings. Applying Darapti, we may infer that there are some winged horses. The argument clearly generalizes. All As are As. A fortiori, all ABs are As; and symmetrically, all ABs are Bs. By Darapti it follows that there are some ABs. Thus syllogistic allows us to prove that any two categories intersect.

And if it be replied that this is just one of the global presuppositions, take B to be Ā, the complement of A (non-A). It can hardly be maintained that Aristotelian logic
globally presupposes contradictions. This argument requires the use of compound
terms. Again, these are not in Aristotle, but are an established part of traditional
logic.

What we have seen is that, however one interprets traditional logic in classical logic, something has to be given up. Moreover, this is quite essential. For as the last argument shows, traditional logic is, in fact, inconsistent. At any rate, classical logic is not (just) a more generous framework subsuming traditional logic. Prevarication aside, modern logic has given the thumbs-down to Darapti and its ilk.
— Graham Priest

Moliere

↪MindForged

Alright, cool. So I think that it would be better to focus on validity after reading your comments, then. I suppose my thought was that soundness was more appropriate because I recall with validity the truth-value of some proposition wasn't something we were after as much.

But, yes, I'm interested in how to justify whether a given form is valid. That's a cleaner statement than what I've been saying so far.

Soundness

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