Why is proof by contradiction even accepted professionally? It's so retarded...

Why is proof by contradiction even accepted professionally? It's so retarded. I'm gonna prove this dog is a reptile by assuming it's a fish. Does it have gills? No. Then I can assume it can be anything I want as long as it's not a fish, so it's a reptile. And if your work is extensive and boring enough no one will bother to prove otherwise. Proof by contradiction is a scam.

Other urls found in this thread:

math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/
en.wikipedia.org/wiki/Law_of_excluded_middle
en.wikipedia.org/wiki/Double_negation
twitter.com/NSFWRedditVideo

Except in your example, if you're trying to prove a dog is a reptile by contradiction then you have to assume that it's not a reptile.

I fucking HATE you OP just DIE no one LIKES YOUUUUUU

I'm assuming the dog is not a reptile by assuming it is a fish, though.

WHOOPS you got ASS-WHOOPED OP WOW!

You actually brought more attention to his moronic thread instead of letting it slide

Next time go to the Options field, below the Name field, and write "sage" before posting. Also, don't forget to report this inane trolling garbage.

naaa no one LIKES YOUUUUU you're dum :))

>what are categories and sets

"Suppose there is a largest integer N. N+1 does not exist." - N.J. Wildberger.

DUMB idiot

math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/

logicians BTFO

Proof by contradiction does rely upon the Law of the Excluded Middle, which is a problem.

However, your representation of such a proof demonstrates a woeful lack of understanding regarding how proof by contradiction works.

Essentially, the idea is that one must first accept a system of logic in which something cannot be simultaneously true and false (this is the Law of the Excluded Middle). If assuming something is true leads to a situation where a contradiction is a *direct* consequence thereof, then you can safely derive that it must be false. Conversely, the same process may be used to prove something as true if its falseness yields a contradiction.

The problem here is that the LEM represents a binary system, and precludes the possibility as truth and falseness being relative to another dimension, such as time. A notable analogy to this problem is the Copenhagen interpretation of the dual-slit experiment in quantum physics, where a photon was shown to be simultaneously wave and particle. More directly applicable may be the fact that we cannot divide by zero in maths because it leads to a situation where 0/0 must be both 1 and 0, which would basically undo the entire system. Much of our entire understanding of the universe is built on the LEM.

But you are correct to instinctually doubt it, OP. It's a good thing that you do not accept it by mere blind faith, and that you question it. Although it is extremely useful, and necessary to develop a framework by which we can logically discuss a fantastically large amount, it is incomplete, and represents a limit of knowledge.

Keep thinking. Question everything.

>Proof by contradiction does rely upon the Law of the Excluded Middle, which is a problem.
If your logic does not have excluded middle, then YOU are the problem

Sick burn, bruh.

But actually, yeah - non-binary processing is the future. Bivalence places limitations on information that prevent us from seeing patterns of acausality. Implementing a system that allows for a simultaneous valuation allows for a more accurate modeling of real-world scenarios. If that's a problem for you, then I'm totally okay with being a part of that problem.

>Essentially, the idea is that one must first accept a system of logic in which something cannot be simultaneously true and false (this is the Law of the Excluded Middle).

No, this is the law of non-contradiction. LEM says that something *must* be either true or false.

Actually there are many reasonable logics in which LEM doesn't hold, like intuitionistic logic.

Falsifiability.
"I claim that A has the properties of B. A does not have the properties of B. A is not B."

precisely why there is no largest number

...

It's equivalent in logic to the well ordering principle therefore it's valid.

You're wrong.

Proof by contradiction does not depend on LEM.

The following is proof by contradiction.
>>Suppose P
>>... (reach a contradiction)
>not P

This is another form of proof by contradiction
>>Suppose not P
>>... (reach a contradiction)
>not not P

Again, no LEM.

However, LEM is equivalent to double negation elimination, which is as follows.
>not not P
>P

If you apply double negation after the second example of proof by contradiction above then you get the thing you are mistaking for proof by contradiction.
>>Suppose not P
>>... (reach a contradiction)
>not not P
>P

In pleb tier informal English one would phrase this:
>Suppose P is false. Reach a contradiction. Conclude P is true.
There are many problems here, not the least of which is using the nonsense term "P is false" when we never actually prove a statement is false (protip: we only prove the negation is true).

If you refuse to leave your pleb tier ways behind then you may find it helpful to think of them as two forms of contradiction. Positive contradiction and negative contradiction.

You idiot, "not a reptile" and "fish" aren't equal sets. infact, "fish" is a proper subset of "not a reptile".

ah, a baby high-schooler who doesn't understand existential qualifiers.

in proof by contradiction, you take a logical statement:
A implies B
An equivalent wording of this statement is:
(Not B) implies (Not A)
In proof by contradiction, we show that that assuming B is false tells us that A is false. This is equivalent to saying that B follows from A. Thus, we have a logical tool.

If you don't understand this, you're brainlet garbage. Sorry.

This is also wrong. That equivalence (the contrapositive) is actually derived by using contradiction.

If you remember the contrapositive then it's a nice mental reminder/shortcut but it is no substitute for understanding the underlying proof system itself.

I'm sorry user but you are also a brainlet.

I think some of you may be missing a valid point hat OP raises -- sufficiently dense and unread jargonese can be used to obfuscate when proof by contradiction is being misapplied in the way OP suggests. Defining the mutually exclusive sets can be done improperly, and when it is, the results are in error.

The contradiction and contrapositive are equivalent. I personally learned contrapositive first in undergrad. You can use boolean logic of:
(A implies B) if and only if ( A or (Not B))
(this is the logical definition of implication) you can use that equivalent to get the contrapositive before the contradiction.
I am not saying you are wrong, I'm saying that you can look at it either way and lose nothing mathematically

This is actually a decent bait.
Sage

slight error:
meant:
((Not A) or B)
looks like i am a brainlet, but not for the reason you stated. kek.

Almost.

It's not (a and not b) which is equivalent to
not a or b

see
>when you do Putnam problems for daily gains but can get your fuckin logical operators right
rest in peace

Your proof is false because "not being a fish" doesn't imply "reptiles".

A better example is the infinitude of primes. If you assume you only had finitely many primes, then you can construct a new number out of those finitely many primes, which none of them divide.

So you assume the existence of something, and show that assumption makes no sense.

Your example is really just stupid!

It's only equivalent in the sense that contradiction + double negation = contrapositive.

In a logic without double negation (e.g. intuitionistic logic) they are not equivalent.

Thanks; that commentary is helpful. I guess I'm wrong. Or maybe it's a problem of language. Here's the wiki on LEM, which literally states that contradiction and LEM are very related, and complementary: en.wikipedia.org/wiki/Law_of_excluded_middle - both stem from the idea that two things cannot be true at the same time, and that in and of itself is a limit placed upon a system of logic that disallows certain things from being expressed.

Do you find it helpful to imagine that you have the "correct" understanding, and other ways of expressing things are a product of "pleb tier" thinking? Confidence is very important, but this seems to make you needlessly sound like a derisive prick, which doesn't seem particularly useful if you're trying to persuade others to see things your way, does it?

That wiki article talks about the old school classical laws of thought which is a really old way of talking about classical logic. Modern logic is performed using a proof system and there's a whole field of research called structural proof theory that studies different calculus for performing proof.

A modern introduction to formal logic does none of that old school stuff. Instead one takes a formal approach by first introducing a formal language and then equipping it with a proof system (then a bunch of discussion about semantics and models completes the logic). While introducing the proof system one is given a list of rules that tell you how to introduce or eliminate each logical operator. For instance,

>P
>Q
>------&I (& introduction)
>P&Q

The negation introduction is what people call "proof by contradiction", or at least the negative version of it. Negation does not have a negation elimination rule but under classical logic it does have a double negation elimination rule. If you combine both rules you can produce a positive proof by contradiction.

If you understand both rules separately however you develop a deeper understanding of the logic and are more capable of working with non-classical logics. In particular, intuitionistic logic does not have double negation but it does have negation introduction. Unfortunately someone who has become too accustomed to informal "proofs" would struggle to understand whether or not proof by contradiction exists in intuitionist logic and worse they may struggle to understand which proofs are valid and which are not.

The idea that you have a large number of self proclaimed proof writers out there who don't understand the underlying proof system is embarrassing and nothing short of pleb tier.

en.wikipedia.org/wiki/Double_negation
>Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,[2] but it is disallowed by intuitionistic logic.[3]

Yeah; I've studied LSL and LMPL. I have worked with proofs at a collegiate level (albeit somewhere around a decade ago). It still seems like there is a binary foundation to the logic, however, even if it goes all the way down to bivalence, which most people would likely see as an immutable bedrock. What you're addressing is important, though, because people too often see ~ introduction as a way of saying "not," when it is actually stating "it is not the case that." It's a subtle difference, but one with large implications. To problematize the situation, I would ask whether double negation doesn't rely on a binary approach, however - to say "it is not the case that it is not the case that x" is equivalent that "it is the case that x" seems to forego the possibility of a case where it could be not the case that it is not the case, yet still not the case that it is the case. Is this a misunderstanding of the double negative?

sounds reasonable to me

i mean he said it right there, suppose there's a largest integer.

>seems to forego the possibility of a case where it could be not the case that it is not the case, yet still not the case that it is the case. Is this a misunderstanding of the double negative?

If I understood your post correctly, yes double negation does forego that possibility. Under intuitionistic logic (where double negation doesn't exist) one may sometimes introduce axioms in order to abuse the possibility that "not not P" is provable but that "P" is not.

An example would be smooth infinitesimal analysis where infinitesimal numbers are introduced by abusing this property. Specifically, given an infinitesimal number i the following two statements are provable:
>it is not the case that i is equal to 0 (not i=0)
>it is not the case that it is not the case that i is equal to 0 (not not i=0)
However, the statement
>i is equal to 0 (i=0)
is unprovable.
Smooth infinitesimal analysis is also interesting in that it uses what are called anti-classical axioms. The idea is that under classical logic these axioms are inconsistent because the proof system of classical logic is stronger (the proof of inconsistency is unprovable under intuitionistic logic).

Because, for all the interesting thing constructive logic is, without excluded middle you can't prove shit and everything you think you know about math ceases to be.

Did you think a b) are the same? TO BAD, YOU JUST USED LEM. Except, for integers is OK. For reals is not.

And, and Cauchy reals and Dedekind reals are not equivalent anymore either.

All of topology goes to the trash.

NO. This is madness.

If you want, use Computable Mathematics: study what can do explicit Turing machines (or whatever) in a classical logic setting.

If constructive logic interest you then that's OK as another mathematical subject, or for the internal languages of topoi, or whatever.

Just don't go full Brouwer. You never go full Brouwer.

>Brouwer
>Spends life developing a philosophy that avoids LEM
>Gets killed while crossing the street from one side to the other
>If he could have excluded the middle of that street, he would have lived
Irony is the only god confirmed