/sci/ - Science & Math

[Anonymous No.2883534]

P -> Q.
¬Q.
Therefore ¬P.

Are there really logicians who think proofs by contradiction are inferior to other proofs? Are they serious or just trolling? If a premise creates a false conclusion, then the premise can't be true.

[Anonymous No.2883591]

I'll take your silence as a no.

[Anonymous No.2883612]

>I'll take your silence as a no.

Lack of evidence is not evidence of lacking

[Anonymous No.2883613]

Why would there be? What gave you that impression? It's self-evidently valid.

[Anonymous No.2883629]

>>2883613
My math professor said so (he said he couldn't elaborate since he wasn't a logician) and A Mathematician's Apology by G. H. Hardy says some logicians disfavor reductio ad absurdum-type proofs.

[Anonymous No.2883642]

P -> Q.
¬Q.
Therefore ¬P.

That's modus tolens AKA proof by counterpositive.

Proof by contradiction goes: Suppose A
.
.
.
X
¬X
Therefore ¬A

[Anonymous No.2883650]

That's not even a proof by contradiction, it's modus tollens. Proof by contradiction would be

(hyp) P
---
Q&~Q (deduction)
Therefore, ~P.

The problem with this kind of proof is that there is no reason given for ~P, because there may be no meaningful connection between P and Q.

Consider the case where P is a universal claim. Then ~P is an existence claim. But no example has been given, and for all we know it could be impossible to find one. In this case, to infer ~P is tantamount to assuming a proof-independent truth about P. But what could be the basis for that?

[Anonymous No.2883659]

>>2883642
So it's the same thing, except with some vertical ellipses thrown in?

[Anonymous No.2883666]

>>2883642
Except you can replace that ellipsis with logical implication since that's what it is.

so A -> X; ~X; :. ~A

Which is Modus Tollens, aka proof by contrapositive.

And to address the OP's concern, there is as far as I know a whole school of mathematics that doesn't take the impossibility of non-existence as proof of existence.

http://en.wikipedia.org/wiki/Constructivism_(mathematics)

However, I don't think it's that they so much as reject the contrapositive law as say that it cannot be used to deduce the existence of new entities.

[Anonymous No.2883684]

Another way of looking at the problem is to question the law

~~P
---
P

The problem with this law is that it assumes that all meaningful statements must necessarily have exactly one truth value, true or false, and that therefore if you prove that ~P is false, therefore P must be true. But why should we assume that there are already truth values for mathematical statements before we have found a proof for them? Do you think there is some independent mathematical reality out there already made?

[Anonymous No.2883693]

>>2883666
All schools accept modus tollens. If you have PROVED
A -> X;
then if you have PROVED
~X;
you can certainly deduce
~A

[Anonymous No.2883725]

>>2883650
>>2883666
But... but... if it's impossible for something to not exist, doesn't that mean it must exist? It's not like there's some middle ground between existence and non-existence.

[Anonymous No.2883760]

>>2883684
>Do you think there is some independent mathematical reality out there already made?

Maybe mathematical reality already exists and we only observe it. Maybe mathematics only exists inasmuch as we construct it. That conundrum falls under the purview of philosophy, a subject that will get you laughed off /sci/.

>The problem with this law is that it assumes that all meaningful statements must necessarily have exactly one truth value

So if I assume P, and I deduce from P that OP is straight (contradiction, OP is a fag), then I can't conclude ~P because OP might be bisexual?

[Anonymous No.2883763]

>>2883725
Suppose in your system you prove that by assuming that there is no object of a certain description, a contradiction follows. Then it would seem to follow from your system that there must be such an object. But for all you know it is impossible to prove in your system that any particular object meets that description. So for all you know your mathematical system may have a sort of hole in it--there is no proof for a particular object that it meets that description, but if you assume that no object meets the description, then you can derive a contradiction. Constructivist mathematics has no such holes.

[Anonymous No.2883777]

>>2883760
You're wrong about the purview. Goedel proved the Incompleteness Theorem IN ORDER TO SHOW that mathematical truth was not just proof in a finitely axiomatizable system, that there was more to mathematical truth than was within human powers to determine.

And regarding OP, the statement "OP is a fag" and the statement "OP is straight" certainly don't exhaust the possibilities, nor would it do so to add "OP is bisexual." Because OP could be a computer. Or there could be no OP.

[Anonymous No.2883792]

OP, that's proof by contrapositive. It is logically equivalent to a conditional proof.

Proof by contradiction shows how a given theorem leads up to a violation of basic principles.

Completely different things, similar in name only.

[Anonymous No.2883796]

>>2883777
>mfw Goedel proved the Incompleteness Theorem within a system founded on unprovable axioms

[Anonymous No.2883809]

>>2883796
The theorem applies equally well with any system. That's the beauty of it. Plug in a system, and out comes the incomplete statement.

[Anonymous No.2883812]

>>2883763
And in trading off those holes constructivist mathematics also loses most of its power and scope. Ho hum.

[Anonymous No.2883835]

>>2883812
Yeah, well, that's the thing.