And the next U.K. Prime Minister will be?

[A Toy Windmill]

I suspect this post is another waste of time. Unlike Angra Mainyu, I don't think anyone is reading these posts other than thread participants.

One of my first posts to Speakpigeon gave the definition of syllogism, and explained how, by definition, there are no syllogisms with contradictory premises. Speakpigeon responded:

Arguments with contradictory premises, or generally false premises, are easily formalised in Aristotelian logic. Such arguments are not valid. And intuitively, they are not.

This was the extent of their rebuttal. No attempt to formalize an argument as a legal syllogism. No engagement with my post. Just a bare assertion.

In this post, I showed how weakening and modus tollens lead to contradictions entailing everything, via a detailed proof. Speakpigeon responds:

I accept Modus Tollens and weakening. But, no, that doesn't imply that contradiction implies everything.
EB

That is the extent of their rebuttal. No attempt to show where my proof is invalid. No engagement with my post. Just another bare assertion.

In a different thread, Speakpigeon writes

LOL. This shows you are just ignoring two of the premises. Sorry, you can't do that.

This completely contradicts their earlier claim to accept weakening. Angra Mainyu seemed to have some fun pointing out this blatant self-contradiction, and finally, Speakpigeon responded by changing the definition of "weakening" that I had originally provided

I do accept weakening but only as A implies A or B, as I indeed explained somewhere, and not as A implies B implies A and C implies B.

By this time, I concluded that Speakpigeon's idea of debate is to just say "nuh-uh" without engaging with people's arguments and that they get confused easily over proofs and definitions. I wanted to debate weakening. That's why I brought it up! I gave Speakpigeon an opportunity to engage in a discussion about it, but all they did was reply "nuh-uh" and then later, when caught in a contradiction, changed their mind about what was meant by "weakening" all along. This is not good faith debate, and I gave up trying to engage Speakpigeon in debate from then on. They are anti-intellectual and a crackpot.

 

[Angra Mainyu]

I suspect this post is another waste of time. Unlike Angra Mainyu, I don't think anyone is reading these posts other than thread participants.

I don't know whether anyone is reading these posts other than thread participants, or whether someone else will read them eventually. Maybe, or maybe not. I do know that at least two thread participants other than Speakpigeon and the two of us are reading these posts.

 

[DBT]

I'm reading the posts. There are interesting comments made, but I can't see that it's worthwhile for me to add anything. Maybe there are other readers that feel the same.

 

[Speakpigeon]

I suspect this post is another waste of time. Unlike Angra Mainyu, I don't think anyone is reading these posts other than thread participants.

One of my first posts to Speakpigeon gave the definition of syllogism, and explained how, by definition, there are no syllogisms with contradictory premises. Speakpigeon responded:

This was the extent of their rebuttal. No attempt to formalize an argument as a legal syllogism. No engagement with my post. Just a bare assertion.

In this post, I showed how weakening and modus tollens lead to contradictions entailing everything, via a detailed proof. Speakpigeon responds:

That is the extent of their rebuttal. No attempt to show where my proof is invalid. No engagement with my post. Just another bare assertion.

In a different thread, Speakpigeon writes

LOL. This shows you are just ignoring two of the premises. Sorry, you can't do that.

This completely contradicts their earlier claim to accept weakening. Angra Mainyu seemed to have some fun pointing out this blatant self-contradiction, and finally, Speakpigeon responded by changing the definition of "weakening" that I had originally provided

I do accept weakening but only as A implies A or B, as I indeed explained somewhere, and not as A implies B implies A and C implies B.

By this time, I concluded that Speakpigeon's idea of debate is to just say "nuh-uh" without engaging with people's arguments and that they get confused easily over proofs and definitions. I wanted to debate weakening. That's why I brought it up! I gave Speakpigeon an opportunity to engage in a discussion about it, but all they did was reply "nuh-uh" and then later, when caught in a contradiction, changed their mind about what was meant by "weakening" all along. This is not good faith debate, and I gave up trying to engage Speakpigeon in debate from then on. They are anti-intellectual and a crackpot.

LOL, you're getting into the right mood.

I'm not interested in teaching you logic. You're not interested.

I'm not interested publishing here and you yourself claimed whatever I might want to publish would have no value. You can't have it both way, Sir.

Find me a decent journal of logic for a maths drop-out to publish in and I might condescend to consider it and who knows even find the option acceptable.

I will remind you of why we are even talking here.

I started different threads on the validity of various logical arguments and AM (and Bomb#20) started here and there to insist that any assessment of logical validity required knowledge and understanding of mathematical logic, dismissing the answers other posters had given on the validity of my arguments and suggesting these posters didn't understand the arguments or didn't have the mathematical background necessary to make an informed judgement.

Essentially, my arguments were all ones with contradictory premises and most posters correctly answered that these arguments were not valid or were nonsensical, which of course implies not valid. So, you have to understand why AM and Bomb#20 wanted to claim these answers were wrong.

However, instead of merely providing the accurate information that the arguments may not be seen as valid according to common sense but are considered valid in mathematical logic, they were adamant that "valid" was the only correct answer (which is just a stupid falsehood), and this on the ground that it was the answer universally accepted in mathematical logic. These two are obviously merely ignoramuses. They probably genuinely believed that logical validity can only be assessed on the basis of mathematical logic. Well, I hope they've learned the lesson now.

Given their attitude, I asked for a justification by logicians that the definition of validity used in mathematical logic would be correct. I'm not interested in AM's justification. I was asking for a published justification. One that would have been subjected to the criticism of all logicians. I'm confident there is none whatsoever. I looked for one and couldn't find any.

This shows the attitude of both AM and Bomb#20 to have been at best that of two ignoramuses, and I will assume that's the situation. Essentially, these are people with a modicum of expertise on a subject which was entirely irrelevant to my thread and who thought the answers provided by other posters were wrong. Instead of taking the time to think about it, they just started to lecture people, pretending to have the expertise to tell that these answers were wrong, when in fact, they didn't have the expertise at all to do that since all the expertise they may have is entirely irrelevant. I had asked people if they thought the arguments were valid. I hadn't ask whether they were valid according to mathematical logic.

AM doesn't even understand the situation. You yourself clearly didn't take the time to see what it was. Your pontificating is merely pathetic. You don't know what you are talking about. I'm not interested in mathematical logic. I'm not here to teach you logic or argue anything. How can you argue with people who employ a notion of validity which is not correct? I merely pointed out the fact that there is no correct definition of validity in mathematical logic. Not one.

This is a fact and you're welcome to waste your time trying to prove me wrong by providing a link to such a justification, one recent enough, say after 1950, published, and published by well-known logicians and in widely circulated journals.

The rest is of no interest whatsoever.
EB

 

[Speakpigeon]

Here is a typical answer.

I choose that the argument does not make sense. The Tories will choose the next PM and they will not choose Corbyn. And I believe there is yet someone else in the running for PM as a Tory. So the argument reduces to BJ or JH or ____will be the next prime minister. There is no logical reason that any of the 3 outcomes will necessarily occur. From I read, the odds are that BJ will be chosen, even though by most standards Mr. Johnson is likely to be a complete and utter disaster compared to the other choices.

Yeah, i know all this and all too well, I listen to BBC Radio Four every day! They've been talking about the Brexit and only the Brexit for the last two years at least.

Why did you vote the argument doesn't make sense? You seem to understand the English of it well enough. What is the problem exactly?
EB

I find
1) the premises do not encompass all of the alternatives, and
2) the conclusion does not necessarily follow from the premises.

Now, from what I have read, I think Mr. Johnson is the expected selection because the Tories appear dead set on destroying their party and England.

That's how most people feel about arguments with contradictory premises. Either that they are not valid or that they are nonsensical, which obviously implies not valid.
EB

 

[Speakpigeon]

Here is another typical reply:

Here is a "U.K. current affairs" argument.

P1 - Jeremy Corbyn is not Boris Johnson;
P2 - Boris Johnson is not Jeremy Hunt;
P3 - Jeremy Hunt is not Jeremy Corbyn;
P4 - The next U.K. Prime Minister will be either Boris Johnson or Jeremy Hunt;
P5 - The next U.K. Prime Minister will be Jeremy Corbyn;
C - Therefore, the next U.K. Prime Minister will be Boris Johnson.

Thank you to say whether you consider this argument valid or not.

Invalid. The five premises cannot all be true at the same time.

These are the kind of answers that AM dismissed with all the vacuous assurance of the little expertise he may have in mathematical logic, which is irrelevant to this thread: Do you think the argument is valid?
EB

 

[Speakpigeon]

Again, same here:

Invalid. The conclusion doesn't necessarily follow from the premises.

You have both the answer and the proper justification for it. Absolutely perfect.

Yet, apparently, AM thinks they can authoritatively assert such answers are wrong.
EB

 

[Angra Mainyu]

I suspect this post is another waste of time. Unlike Angra Mainyu, I don't think anyone is reading these posts other than thread participants.

One of my first posts to Speakpigeon gave the definition of syllogism, and explained how, by definition, there are no syllogisms with contradictory premises. Speakpigeon responded:

This was the extent of their rebuttal. No attempt to formalize an argument as a legal syllogism. No engagement with my post. Just a bare assertion.

In this post, I showed how weakening and modus tollens lead to contradictions entailing everything, via a detailed proof. Speakpigeon responds:

That is the extent of their rebuttal. No attempt to show where my proof is invalid. No engagement with my post. Just another bare assertion.

In a different thread, Speakpigeon writes

This completely contradicts their earlier claim to accept weakening. Angra Mainyu seemed to have some fun pointing out this blatant self-contradiction, and finally, Speakpigeon responded by changing the definition of "weakening" that I had originally provided

By this time, I concluded that Speakpigeon's idea of debate is to just say "nuh-uh" without engaging with people's arguments and that they get confused easily over proofs and definitions. I wanted to debate weakening. That's why I brought it up! I gave Speakpigeon an opportunity to engage in a discussion about it, but all they did was reply "nuh-uh" and then later, when caught in a contradiction, changed their mind about what was meant by "weakening" all along. This is not good faith debate, and I gave up trying to engage Speakpigeon in debate from then on. They are anti-intellectual and a crackpot.

LOL, you're getting into the right mood.

I'm not interested in teaching you logic. You're not interested.

I'm not interested publishing here and you yourself claimed whatever I might want to publish would have no value. You can't have it both way, Sir.

Find me a decent journal of logic for a maths drop-out to publish in and I might condescend to consider it and who knows even find the option acceptable.

I will remind you of why we are even talking here.

I started different threads on the validity of various logical arguments and AM (and Bomb#20) started here and there to insist that any assessment of logical validity required knowledge and understanding of mathematical logic, dismissing the answers other posters had given on the validity of my arguments and suggesting these posters didn't understand the arguments or didn't have the mathematical background necessary to make an informed judgement.

Essentially, my arguments were all ones with contradictory premises and most posters correctly answered that these arguments were not valid or were nonsensical, which of course implies not valid. So, you have to understand why AM and Bomb#20 wanted to claim these answers were wrong.

However, instead of merely providing the accurate information that the arguments may not be seen as valid according to common sense but are considered valid in mathematical logic, they were adamant that "valid" was the only correct answer (which is just a stupid falsehood), and this on the ground that it was the answer universally accepted in mathematical logic. These two are obviously merely ignoramuses. They probably genuinely believed that logical validity can only be assessed on the basis of mathematical logic. Well, I hope they've learned the lesson now.

Given their attitude, I asked for a justification by logicians that the definition of validity used in mathematical logic would be correct. I'm not interested in AM's justification. I was asking for a published justification. One that would have been subjected to the criticism of all logicians. I'm confident there is none whatsoever. I looked for one and couldn't find any.

This shows the attitude of both AM and Bomb#20 to have been at best that of two ignoramuses, and I will assume that's the situation. Essentially, these are people with a modicum of expertise on a subject which was entirely irrelevant to my thread and who thought the answers provided by other posters were wrong. Instead of taking the time to think about it, they just started to lecture people, pretending to have the expertise to tell that these answers were wrong, when in fact, they didn't have the expertise at all to do that since all the expertise they may have is entirely irrelevant. I had asked people if they thought the arguments were valid. I hadn't ask whether they were valid according to mathematical logic.

AM doesn't even understand the situation. You yourself clearly didn't take the time to see what it was. Your pontificating is merely pathetic. You don't know what you are talking about. I'm not interested in mathematical logic. I'm not here to teach you logic or argue anything. How can you argue with people who employ a notion of validity which is not correct? I merely pointed out the fact that there is no correct definition of validity in mathematical logic. Not one.

This is a fact and you're welcome to waste your time trying to prove me wrong by providing a link to such a justification, one recent enough, say after 1950, published, and published by well-known logicians and in widely circulated journals.

The rest is of no interest whatsoever.
EB

As usual, that is a gross misrepresentation of what happened. Speakpigeon engages in libelous behavior against Bomb#20, A Toy Windmill, and me. The above is not only full of falsehoods. It is unethical of Speakpigeon to post it. It's not okay to make things up and disparage your opponents, and then even keep insisting in the falsehoods about what one's opponents said and did after they have been repeatedly debunked. Of course, Speakpigeon's position on logic has also been repeatedly debunked. Purely for example, see this previous post in the thread, which contains nice links to several instances of debunking.

 

[steve_bank]

Here is a "U.K. current affairs" argument.

P1 - Jeremy Corbyn is not Boris Johnson;
P2 - Boris Johnson is not Jeremy Hunt;
P3 - Jeremy Hunt is not Jeremy Corbyn;
P4 - The next U.K. Prime Minister will be either Boris Johnson or Jeremy Hunt;
P5 - The next U.K. Prime Minister will be Jeremy Corbyn;
C - Therefore, the next U.K. Prime Minister will be Boris Johnson.

Thank you to say whether you consider this argument valid or not. P1-P3 are irrelevant. C in no way follows from P4 P5.

Thank you to abstain from commenting before you voted.
EB

It is non squitter. The conclusion does not follow from the premises. The syllogism is not valid, conclusion does not follow from premise.

P1-P3 are irrelevant to C. C does not follow from P4 P5.

Non Squitter

p1 the car is blue
p2 the truck is red
p3 the car is not the truck
p4 either a democrat or republican will get elected.
p5 Biden is not Warren
p6 Warren is not Mayor Pete
c therefore Trump wall not get re-elected.

 

[Speakpigeon]

It is non squitter. The conclusion does not follow from the premises. The syllogism is not valid, conclusion does not follow from premise.

P1-P3 are irrelevant to C. C does not follow from P4 P5.

Non Squitter

Good, but why do you not cast your vote?

Are you afraid you might want to change your mind?
EB

 

[Speakpigeon]

There is always one "guest" or two, beyond members.
Maybe often the same ones but probably not always.
So possibly Chomsky is here with us today.
Indeed any member could be Chomsky.
Hey, for all you know, maybe I am.
Whoa, think of that...
I'm not even sure myself I'm not Chomsky since consciousness is an illusion, according to some here.

OK, thanks to those who took the time to vote, I think the result is overwhelmingly against validity.

I never thought anyone at all would so brainlessly insist such arguments are valid but apparently mathematicians have managed to brainwash themselves into this remarkable piece of dogma and it is now clear that they won't budge from their silly stance.

I have to point out to all those who voted either not valid or nonsensical that their position is in contradiction to mathematical logic. You should be aware that there are millions of mathematicians the world over and that probably at least 95% of them would end up calling you stupid if you dared insist to their face such arguments are not valid. I think this is a remarkable fact of our modern world. 165 years of mathematical logic to get to this point that most mathematicians would say that most non-mathematicians are logically inept. I don't think anyone who take these arguments to be not valid would also disagree with Aristotle about any of his syllogisms: No man is a fish, all sardines are fish, therefore no man is a sardine? There's not one syllogism about which most people wouldn't agree with Aristotle. So, where would be the ineptitude of ordinary people? Why would they agree with Aristotle if people were inept at logic?

The only serious explanation is that mathematical logic is not a correct model of logic. Mathematicians are trained at mathematical logic. Those who disagree at this crucial stage probably don't move on to the next stage and never become mathematicians (indeed, like myself). So we have a system that's co-opting people who mostly will abide by th the dogma. And once you've spent years training, you're not going to admit, even to yourself, that it's crap. And hence we have this extraordinary situation where mathematical logic has become as dogmatically wrong as the Catholic Church in the Middle-Ages. Some mathematicians disagree with the mainstream view but none of them has produced anything like a correct model of human logic.I believe the reason is that they are not even trying. They are not scientists. They can't be bothered with observing the reality of human logic and since nobody is doing it, the charade is allowed to continue. People like Boole and Frege had velleities of doing the correct thing but this was quickly put aside, probably by Russell himself. And everybody coming after him just followed because they were mathematicians or philosophers and not scientists.

I would rather that the situation be different. I looked for possible explanations and even disproof of my thesis. All I found was contempt and self-righteousness of the self-deceiving expert. I haven't found anything looking even remotely like a disproof of my thesis.

So, again, thank you for taking the time to cast your vote. I might come up with some more polls to test my ideas but for now I think we've done a good job given the lack of cooperation of the self-appointed "authorities" on the subject.
EB

 

[A Toy Windmill]

OK, thanks to those who took the time to vote, I think the result is overwhelmingly against validity.

To the extent that there is a single correct notion of validity, and the poll representative, the results show that two thirds of people are confused.

 

[Speakpigeon]

OK, thanks to those who took the time to vote, I think the result is overwhelmingly against validity.

To the extent that there is a single correct notion of validity, and the poll representative,

The poll is representative of people who think they have an opinion.

As to the notion of validity, most people give broadly the same justification for their votes, namely that the conclusion doesn't follow from the premises.

the results show that two thirds of people are confused.

Excellent. Thank you to confirm what I said mathematicians think...

I have to point out to all those who voted either not valid or nonsensical that their position is in contradiction to mathematical logic. You should be aware that there are millions of mathematicians the world over and that probably at least 95% of them would end up calling you stupid if you dared insist to their face such arguments are not valid. I think this is a remarkable fact of our modern world. 165 years of mathematical logic to get to this point that most mathematicians would say that most non-mathematicians are logically inept.

EB

 

[Speakpigeon]

To the extent that there is a single correct notion of validity, and the poll representative, the results show that two thirds of people are confused.

So, why would the poll shows people are confused?!

People all vote broadly the same and they broadly give the same kind of justification for their vote. The poll's results are no indication that people are confused.

In fact, given that most people don't have any training in formal logic, the poll is an indication that people get to agree on the argument being not valid by following their intuition. So, in effect, people don't even have to think to lean in favour of non-validity. Please anyone who voted not valid and had to think long to arrive at a decision correct me if I am wrong on this.
EB

 

[A Toy Windmill]

So, why would the poll shows people are confused?!

There are four poll options, with votes split evenly on three. If you constructed the poll to give exclusive and exhaustive options, that means that two-thirds of people voted wrong whatever answer is right.

I'm not going to nitpick this further. Your poll is scientifically as worthless as your thread in the science forum asking why there is no science of logic.

165 years of mathematical logic to get to this point that most mathematicians would say that most non-mathematicians are logically inept.

People who are expected to work in subjects that require impeccable logical rigour will need training in logical rigour. This is not peculiar to mathematics. Training is needed, for instance, to judge when a poll is scientifically worthless, such as the ones you have conducted on this forum.

 

[steve_bank]

EB, good I was? The OP is one in a long stream of basically the same syllogism. Do you engage with people outside of your bubble?

He who makes the poll biases the results. If you had a 'logical fallacy' category I would have voted.

Your interpretation of invalid and valid is a bit muddy.

I do not know if mathematicians think classical logic practitioner' are inept. Classical logic has a lot of problems in usage. It is imprecise.

In terms of contract law there is a clear difference between the words will and shall in terms of legal meaning. Lawyers are immersed in classical logic. Same with politicians but they would probably not realize it as such.

People do not use fprmalogic or Bolean ASlgbra in everyday reality.

I read that an old complicated treaty when reduced to formulaic showed the treaty actually did nothing.

Back in the 80s I had to analyze a military contract and specification and make a cost schedule bid.

To be sure I had all the conditions and requirements covered I reduced all to symbolic logic so I could see it all visually.

Logic is about using the right tool for the problem at hand. For clarity I used written syllogisms a time or two. Made it easier to grasp the problem..

 

[Speakpigeon]

There is always one "guest" or two, beyond members.
Maybe often the same ones but probably not always.

Hey, we have one member (me) and 26 guests on the Logic and Epistemology forum!

Unless it's members "lurking" incognito? Nah.

26 guests! Are we getting an audience beyond FTF?

26 sounds a lot. But is it a reliable indication?
EB

 

[Speakpigeon]

So, why would the poll shows people are confused?!

There are four poll options, with votes split evenly on three. If you constructed the poll to give exclusive and exhaustive options, that means that two-thirds of people voted wrong whatever answer is right.

Nobody voted "I don't know", so there's not objective reason to assume any voter to be confused.

None of the four options could be dispensed with.

The option "doesn't make sense" was meant to set appart "confused" people.It just happens that some people prefer to vote this than to vote "not valid". However, given the justification they all give for voting "doesn't make sense", it is clear that they don't think the argument is valid, and therefore that it is not valid.

I would assume that there are two kind of voters in this category. First, people who think the argument is a non-starter and that the answer is really too bloody obvious for anyone asking for them to vote. Their vote means "not valid" coming with an expression of utter dismay. Second, some people may be prepared to change their vote if they could be convinced. However, clearly, nobody would be prepared to change their vote.

So, there is nothing "confused" about that and we have six posters who don't think the argument is valid and only two who think it is.

Fast is in a category of his own since his reply is based on the false premise that by "valid" I meant valid as defined in the context of mathematical logic.

I'm not going to nitpick this further. Your poll is scientifically as worthless as your thread in the science forum asking why there is no science of logic.

You're welcome to prove my question on the science of logic would be worthless. For now, your comment is obviously without merit.

165 years of mathematical logic to get to this point that most mathematicians would say that most non-mathematicians are logically inept.

People who are expected to work in subjects that require impeccable logical rigour will need training in logical rigour. This is not peculiar to mathematics. Training is needed, for instance, to judge when a poll is scientifically worthless, such as the ones you have conducted on this forum.

False premise. You would need to prove that the logic used by mathematicians for the last 165 years is correct. There's no proof that it is correct and my poll on the Squid argument shows it isn't.
EB

 

[fast]

Fast is in a category of his own since his reply is based on the false premise that by "valid" I meant valid as defined in the context of mathematical logic”

Ordinarily, I would not care what a person means by what they say (as far as the interpretation of what they say goes) except to understand what they mean when it diverges from what they say. In other words, there is sometimes a difference between what is said and what is meant. The referent of the word “zebra” (and not the referent of the word “horse”) continues to have a zebra as a referent of the word even after the child at the zoo pointing at a zebra says, “ooh mommy, look at the pretty horse.” The child is referencing the zebra and doing so with the word “horse,” but the referent of the term “zebra” is not altered because of the child’s misuse; indeed, that’s why it’s not just a use but a misuse of the word “horse.”

If you call a tail a leg, how many legs does a dog have? I’m going with four because what you call a tail doesn’t make it a leg, and though you might mean something by ‘leg’ that is different than what is ordinarily meant by “leg,” that doesn’t change anything, except the answer to what is meant is five whereas the answer to what is asked is four.

Although I’m watchful not to conflate the lexical usages of words with the stipulative usages of words, I almost always put the stipulative usages in single quotes, but I think now maybe the single quotes should be better reserved for the individualistic element of stipulative usages; after all, why in the world would I (why should I?) single quote wide spread usages as unusual and alternative—when they are merely alternative and not unusual.

By the way, lately I’ve been trying my damndest to figure out if “validity” is a particular variety of a homonym. There’s so much darn imprecision in the English language that I still haven’t come to grips with whether I’m dealing with the very same word (with different meanings) or two different words (with the same spelling). Nothing ever seems to be easy.

 

[steve_bank]

Valid simply means conclusion follows from premises. In logic a valid argument is about form not content. In practice in response to an argument one might say your argument is valid but your premise is incorrect.

In EBs syllogism conclusion does not follow from premise, there is no connection between premise and conclusion. It is invalid

A valid argument with meaningless content.
p1 a wog is zog
p2 a boz is zog
c both a wog and a boz are zog

or
p1 a is c
p2 b is c
c both a and b are c


An average person presented with the OP syllogism may not have the technical jargon to describe it but would dismiss it as nonsense.

It remands me of the kids logic puzzles that were passed around when I was growing up. A plane crashes herd on the border between two countries and all die. Which side of the border are the survivors buried?

The OP represents a common political tactic. Present a series of true statement's seemingly related to a tropic like an election, then declare a conclusion related to the election but unrelated to the premise.

Out in the real world people use logic even if they have never studied it formally. We all do.

In engineering we'd call it overwhelming someone with technical terms and theory and then declaring a conclusion unrelated to a premise. It works when trying to deal with non technical people.

Or arm waving followed by a conclusion.