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

[Speakpigeon]

One derives a contradiction.

The right word is indeed "inconsistent". The premises are inconsistent, i.e. one premise implies the negation of the other premise.

Contradictory premises would be p and not p and that's not what we have here.

The premises here are not contradictory.

You need to make sure you know the basics before posting silly arguments.
EB

So, you claim that it is not a problem when "one premise implies the negation of the other premise."

???

I didn't say it wasn't a problem, I said it's not the same thing. And apparently you didn't know.

The crucial point is that there is no justification that any of the definitions used in mathematical logic would be correct. Put all the mathematicians and philosophers alive today together and you still don't have any justification. So, it sure looks impressive but it is vacuous and it is in contradiction with Aristotelian logic and indeed with our logical intuitions as demonstrated by the polls I carried out here and elsewhere.

Your position is pathetic and desperate. You have zero argument. You are dogmatic and impart falsehoods to people.

I'm sure you're a decent guy, but sometimes even those need to be told. Most people are just like you by the way. As soon as they have what they take to be a bit of expertise they go about pretending they know better than you even when in fact they don't. Millions of mathematicians are very much in the same situation and many will "teach".
EB

 

[Angra Mainyu]

I didn't say it wasn't a problem, I said it's not the same thing. And apparently you didn't know.

First, obviously, I knew it literally is not the same thing to have a self-contradictory premise, or a premise that implies the negation of the other. But I know that it makes no relevant difference. In both cases, the argument has an inconsistent set of premises (which could also be called "contradictory", but that is a matter of terminology). In both cases, the conjunction of all of the premises is self-contradictory. In both cases, the premises imply a contradiction, etc.

Second, you actually made it clear that it was not a problem. You said it in this thread, and more precisely in this post, when you misunderstood A Toy Windmill's post, and said

Of course it is valid. Trivially valid. Obviously valid.

Your discussion with ATW also shows you misunderstand this proof for an instance of contradictory premises. I guess ATW explained to you what's the difference between contradictory and inconsistent but I'm not sure you've understood yet.

This kind of proof is called "by contradiction" but this is misleading and you don't understand logic anyway so you can't tell by yourself.

So, yes, you made it clear that there were no problems with arguments with inconsistent premises, provided that the inconsistency is not in one single premise. Of course, your distinction makes no sense, as in any argument with inconsistent premises, one can replace the different premises by the conjunction of them all, which is contradictory, without affecting validity. But no matter, your not making sense is just the usual. The point remains, you made it clear that an inconsistent set of premises with no single contradictory premise was ok, which of course contradicts your other claims, as I have explained in my previous post here.

Your position is pathetic and desperate. You have zero argument. You are dogmatic and impart falsehoods to people.

That is false. Your position is pathetic and desperate. You have zero reasonable argument. You are dogmatic and impart falsehoods to people. I, on the other hand, keep debunking some of your falsehoods.

 

[A Toy Windmill]

Further, there is the question of the use of the theories developed in mathematical logic. Mathematicians are human beings and they reason logically when they want to, and presumably they have to to prove theorems. As human beings, their logic must be the same as that of any other human being. So, either mathematical logic has no role whatsoever in how mathematicians prove theorems or it does. If no role, mathematical logic is literally useless, which would contradict what you say here. So, I will assume that you think it has a role. Now, i asked several times, without result, AM to provide examples of important mathematical theorems whose demonstration relied on mathematical logic. Can you yourself provide examples?
EB

There are now several major mathematical theorems which are held by mathematicians to be theorems because their proofs were formalized and a computer checked that the proofs followed the rules of a mathematical logic. One such theorem is the Kepler Conjecture. There are likely to be many more of these going forward.

As I said before:

[L]ogics are mathematical objects, some of which are accepted by mathematicians as adequate vehicles for formalizing and delimiting their subject.

Mathematicians who know what first-order logic is will invariably accept as proven any mathematical statement whose proof has been formalized and checked to follow the rules of first-order ZFC set theory.

 

[Speakpigeon]

There are now several major mathematical theorems which are held by mathematicians to be theorems because their proofs were formalized and a computer checked that the proofs followed the rules of a mathematical logic. One such theorem is the Kepler Conjecture. There are likely to be many more of these going forward.

Excellent. It's an extremely complex proof. The probability of an error due to the logic used is maximal.

Thank you for being clear and explicit about that. This is becoming a rarity these days.

And I think I already have it somewhere, the one from the theorem prover.

It's definitely beyond my means to check exactly what logic and what proof rules underpin that kind of proof, so thanks.

I thought it had to be mathematical logic since there's no other method available to run on a computer but I wasn't going to work on that before being certain.

The only drawback to this example unfortunately is that Kepler's conjecture is intuitively correct and therefore the thing is very likely a theorem, which perhaps makes any mistake less likely.

Also, as I understand it, theorem provers don't entirely work on their own. There is still some input required from mathematicians and it is therefore not a clear that the proof is really entirely based on mathematical logic. At least, there is a doubt.

Also, disproving the thing would have less value since the conjecture is probably true anyway. But I'm sure it would concentrate the minds of not a few people around the world.

I estimate the number worldwide of mathematicians alive and practising today to be in the millions. Do you have an idea about that?

As I said before:

[L]ogics are mathematical objects, some of which are accepted by mathematicians as adequate vehicles for formalizing and delimiting their subject.

Mathematicians who know what first-order logic is will invariably accept as proven any mathematical statement whose proof has been formalized and checked to follow the rules of first-order ZFC set theory.

Yes, a truly frightening situation.

Again, thanks for being forthright. It will save time.

I'm definitely not anywhere near that yet but, in the event, what would be do you think the best venue for publishing a disproof of it? Coming from a maths university drop-out...
EB

 

[Speakpigeon]

Second, you actually made it clear that it was not a problem. You said it in this thread, and more precisely in this post, when you misunderstood A Toy Windmill's post, and said

Of course it is valid. Trivially valid. Obviously valid.

Your discussion with ATW also shows you misunderstand this proof for an instance of contradictory premises. I guess ATW explained to you what's the difference between contradictory and inconsistent but I'm not sure you've understood yet.

This kind of proof is called "by contradiction" but this is misleading and you don't understand logic anyway so you can't tell by yourself.

So, yes, you made it clear that there were no problems with arguments with inconsistent premises, provided that the inconsistency is not in one single premise. Of course, your distinction makes no sense (...)[/url]

LOL. Yes, I'm sure my comment here makes no sense to you. Still, I read what I wrote and I can still sign up to it. Of course it is valid. Trivially valid. Obviously valid.

The clue is given, intentionally, in my comment, but you sure don't want to see it.


Anyway, I guess the best for the two of us is probably for me to just stop even mentioning your name. I don't expect you to understand and I don't expect any information from you since you've... Oops! OK, let's stop here, doesn't matter.

Again, no hard feelings, I'm much more sentimental than I make it out to be and we sort of know each other since quite a while like many people around here. I would have no problem having a drink and laugh with you at the pub. As long as you don't pretend to have any expertise on logic, you'd be safe.
EB

 

[A Toy Windmill]

It's definitely beyond my means to check exactly what logic and what proof rules underpin that kind of proof,

Then how did you decide that the probability of error is maximal?

Also, as I understand it, theorem provers don't entirely work on their own. There is still some input required from mathematicians and it is therefore not a clear that the proof is really entirely based on mathematical logic. At least, there is a doubt.

No human input is needed to recheck the formalized proof. You can download the source code, run the Makefile, and wait for the check to complete.

I'm definitely not anywhere near that yet but, in the event, what would be do you think the best venue for publishing a disproof of it? Coming from a maths university drop-out...
EB

The same place that a university drop-out publishes squaring-the-circle proofs via compass and straightedge, that π is a rational number, that 0.999... < 1 and that Cantor's diagonal argument is false. Remember to include in your abstract that mathematicians are idiots and that you expect your research to be dismissed because of a cabal of academics who just fear the truth.

 

[Speakpigeon]

I'm definitely not anywhere near that yet but, in the event, what would be do you think the best venue for publishing a disproof of it? Coming from a maths university drop-out...
EB

The same place that a university drop-out publishes squaring-the-circle proofs via compass and straightedge, that π is a rational number, that 0.999... < 1 and that Cantor's diagonal argument is false. Remember to include in your abstract that mathematicians are idiots and that you expect your research to be dismissed because of a cabal of academics who just fear the truth.

LOL. It didn't take you long to drop the mask, you evil academic.

Why not just go along with my fancy and give me an honest and considerate answer?

You see, your attitude here definitely suggests there isn't any such a place. That the reason a paper is published in academic papers is not on its own merit but because of who you are. Like a No-Dogs-and-Negroes signpost.

It doesn't matter. I'll go through my imaginary friends with pointed ears on Sirius B.

I also notice that in your post here you implicitly ascribe to me the claim that "mathematicians are idiots". Please remember these are your words, not mine.

So, how many mathematicians worldwide?

And, come on, seriously, where could I consider publishing. Suppose you realise a friend of your, a drop-out like me, made a real discovery. Where could he publish?

I recently had an exchange with a guy who falsified Tarski's theorem. He is reduced to publish on line. Shame.
EB

 

[A Toy Windmill]

There you go then. Your results belong in the same place as "falsifications" of Tarski's Theorem.

If you send such a result to any journal they'll reject it immediately, since any mathematical logician knows that a person claiming to have falsified Tarski's Theorem is badly confused.

 

[Speakpigeon]

There you go then. Your results belong in the same place as "falsifications" of Tarski's Theorem.

If you send such a result to any journal they'll reject it immediately, since any mathematical logician knows that a person claiming to have falsified Tarski's Theorem is badly confused.

Thanks, you're making my point for me, Sir.

Let's take a short moment to commemorate the contributions of such people as Copernicus, Galileo, Einstein. Today, they would be rejected "immediately", because, well, sane people "know" these dudes are just "confused".

LOL. You are such a caricature of the evil academic! It's spooky.
EB

 

[A Toy Windmill]

And you've scored a few more points on the crackpot index.

 

[Angra Mainyu]

LOL. Yes, I'm sure my comment here makes no sense to you. Still, I read what I wrote and I can still sign up to it. Of course it is valid. Trivially valid. Obviously valid.

The clue is given, intentionally, in my comment, but you sure don't want to see it.

The fact that you consider your deliberately obscure talk morally acceptable behavior is just a failure on your part. No matter, you said it is valid.

Anyway, I guess the best for the two of us is probably for me to just stop even mentioning your name. I don't expect you to understand and I don't expect any information from you since you've... Oops! OK, let's stop here, doesn't matter.

No. I will not stop, as long as you attack me, even if you do so while claiming not to attack me.

Again, no hard feelings, I'm much more sentimental than I make it out to be and we sort of know each other since quite a while like many people around here. I would have no problem having a drink and laugh with you at the pub. As long as you don't pretend to have any expertise on logic, you'd be safe.

So, if I were to point out (what you call "pretend", with reckless disregard for the truth) that I do have a reasonable understanding of logic, whereas your position is nonsensical, I would not be safe?

Regardless, I definitely not have a drink with you (not merely because I do not drink alcohol). It's not a symmetrical thing. Your behavior toward me is unacceptable, whereas I'm only defending myself (and others) from you.

 

[Angra Mainyu]

There you go then. Your results belong in the same place as "falsifications" of Tarski's Theorem.

If you send such a result to any journal they'll reject it immediately, since any mathematical logician knows that a person claiming to have falsified Tarski's Theorem is badly confused.

Thanks, you're making my point for me, Sir.

Let's take a short moment to commemorate the contributions of such people as Copernicus, Galileo, Einstein. Today, they would be rejected "immediately", because, well, sane people "know" these dudes are just "confused".

LOL. You are such a caricature of the evil academic! It's spooky.
EB

It's like someone sending a paper to a serious geology journal claiming they have conclusive evidence the Bible is right and the Earth is less than 15000 years old. It would be rejected without even reading it, and with pretty good reason. People have to decide how to allocate their time. Allocating time to such papers would be a waste of time (except, say, for the purpose of debunking Young Earth Creationism before some audience, or whatever, but I'm saying that it would be so from the perspective of advancing science).

But okay, let's be fair. Your paper would not be like one of those. That would be unfair to the Young Earth Creationist, who might not be making any transparent logical errors - they're just assessing probability in a dismal manner -, whereas you have made plenty of such logical errors (even in the threads on this forum), and someone who gets a claim like yours is warranted in assessing that you made such errors without reading your paper, even more than the geologist is warranted in reckoning that the Young Earth Creationist is mistaken even without reading the YEC's paper, and even if the YEC claims that he has written a paper based on new arguments or pieces of information that no other YEC has used before.

 

[Angra Mainyu]

There are now several major mathematical theorems which are held by mathematicians to be theorems because their proofs were formalized and a computer checked that the proofs followed the rules of a mathematical logic. One such theorem is the Kepler Conjecture.

True and nice example, but the bar is much lower if one just wants a theorem that requires what Speakpigeon calls "mathematical logic". There is no need for first-order formalization for that.

 

[A Toy Windmill]

It's like someone sending a paper to a serious geology journal claiming they have conclusive evidence the Bible is right and the Earth is less than 15000 years old.

Far worse for Tarski's Theorem. Anyone claiming to have falsified it is just stating a logical absurdity. It's a theorem. You can't falsify it.

The theorem is a fairly easy result of the machinery needed to prove Gödel's first theorem, and there's a cottage industry of internet people who think they've falsified that. I can tell you out of the gate that all such people are badly confused, and most don't even have the requisite knowledge to understand what they are criticizing. They are not often so candid as Speakpigeon, who admits it is definitely beyond their means to check where the logic used in the Kepler conjecture is wrong, but still judges that the probability of wrongness is maximal.

 

[Speakpigeon]

It's like someone sending a paper to a serious geology journal claiming they have conclusive evidence the Bible is right and the Earth is less than 15000 years old. It would be rejected without even reading it, and with pretty good reason.

Your logic really sucks. I wouldn't have any reason to send a paper about the Bible being right or the Earth being less than 15000 years old.
EB

 

[Speakpigeon]

Far worse for Tarski's Theorem. Anyone claiming to have falsified it is just stating a logical absurdity. It's a theorem. You can't falsify it.

It is possible to falsify theorems whose proof is wrong,

You are assuming the proof is correct only because you believe it is correct. Your belief doesn't make the proof valid..

Speakpigeon, who admits it is definitely beyond their means to check where the logic used in the Kepler conjecture is wrong

This is not what I said. You don't seem to understand much overall.
EB

 

[A Toy Windmill]

It is possible to falsify theorems whose proof is wrong,

Which proof is that? The first proof I read of Tarski's Theorem was back in 2004 and was not Tarski's original. In fact, I daresay the proof was original to my lecture material. If pressed to provide a proof of the theorem today, I would figure one out on my own, since it's one that I find myself running through my head every so often, and I know plenty of formal systems in which to state it.

You still don't falsify theorems. You show that a purported theorem was never a theorem, and falsify the conjecture.

You are assuming the proof is correct only because you believe it is correct. Your belief doesn't make the proof valid..

You are assuming this counterpoint isn't utter crackpottery because you believe it isn't utter crackpottery. Your belief doesn't mean you aren't a complete crackpot.

This is not what I said. You don't seem to understand much overall.
EB

You could always correct me. Why the shyness?

Oh well. Put it on the backlog with all the other points you are too scared to address.

I'll respond to you again once you've address Angra Mainyu's excellent defence of the definition of logical validity.

 

[fast]

P1: P and Q
C: P or Q

Is that valid?
I’ve been wrestling with that for two days

ETA: I want to say yes, but certain examples get tricky.
I need either some apples or some oranges
You have some apples and oranges
Seems easy enough

But
I need either some salt or pepper
You have both, but they’re mixed
Maybe what was really meant was I need either pure salt or pure pepper

Any pitfalls to consider?

 

[Angra Mainyu]

It's like someone sending a paper to a serious geology journal claiming they have conclusive evidence the Bible is right and the Earth is less than 15000 years old. It would be rejected without even reading it, and with pretty good reason.

Your logic really sucks. I wouldn't have any reason to send a paper about the Bible being right or the Earth being less than 15000 years old.
EB

Your failure to understand what others say is a constant annoyance. It's an analogy, of course.

 

[A Toy Windmill]

P1: P and Q
C: P or Q

Is that valid?
I’ve been wrestling with that for two days

ETA: I want to say yes, but certain examples get tricky.
I need either some apples or some oranges
You have some apples and oranges
Seems easy enough

But
I need either some salt or pepper
You have both, but they’re mixed
Maybe what was really meant was I need either pure salt or pure pepper

Any pitfalls to consider?

Does the conclusion need to be an "or" to highlight the issue?

I need some salt. You have salt and pepper, but they're mixed.