Fake Proofs by Contradiction

[Kriminal99]

I have noticed that "Proofs" by contradiction seem to lack any strength above and beyond just about any other argument that people can come up with. Whenever a person wishes to "prove" something by contraction, they can simply use a couple of assumptions (preferably unrecognized so as to make it less obvious what is really going on) that contradict each other in a non trivial manner.

 

Then they simply avoid exploring the implications that these two assumptions create by themselves, such that when a contradiction does arise, you are free to attribute it to some other assumption.

 

Imagine a proof by contradiction of the following form.

 

X is a statement we intend to prove false using proof by contradiction.

Y is an intermediary statement that is required in the proof.

 

Proof:

 

X is true.

 

Y is true.

 

If not Y, then not X.

 

Not Y, so not X.

 

Contradiction, therefore X is not true.

 

This is an obviously failed proof by contradiction because it assumes that both Y and not Y.

 

However, we can make it less obviously wrong (and yet still wrong) by making an assumption that can be proven in a non trivial way to necessitate that a later assumption is false.

 

For example:

 

X is true

 

Y/2 + 8 = 10

 

Y = 5

 

If Y = 5, then not X

 

Contradiction.

 

We can extend the complexity of the information that implies Y as much as we want to make it less and less obvious that clause 2 contradicts with clause 3. We can even make it state it in a way that makes it seem like this is not likely to be true, even though it is. It is easy to do this using words instead of math equations were people are used to knowing that the equation implies a specific value for Y.

 

We have come to use the concept of "proof" so liberally that it has ceased to have any meaning.

[sanctus]

Contradiction proof does work though if done properly, one has just to watch out if Y is really implied or not. for example:

if X>1 then Y=1/X<1. Now notX implies notY (i.e. if Y=1/X>1 then X<1).

 

But it is true by contradiction starting with some false or not complete assessements you can prove anything...

[Qfwfq]

Proof by contradiction (also called reductio ad absurdum) is based on application of modus tollens. The assert to be disproved is supposed hypothetically only to the purpose of arguing an implication. The assert A is not supposed outside of arguing A => B so, at that point, given that B is false, so is A.

 

This is an obviously failed proof by contradiction because it assumes that both Y and not Y.

Apparently you are trying to twist the matter into a known but distinct fact: if one assumes an assert as being true and the same as being false, equally as hypotheses, then any other assert may be argued as true in a standard way (also based on modus tollens). This simply shows it makes no sense to allow even one specific assert to be both true and false.

 

Try to avoid confusing the two things...:)

[Kriminal99]

Not really looking for naive regurgitated school book interpretations of logic, just pointing out something generally fallacious about "proof" by contradiction.

 

It helps to read an argument before responding as if you understand what it was saying.

 

The idea here is that a person has a limited amount of ability to recognize all the implications of any statement.

 

If I give assumption Y, and assumption X in a proof by contradiction, and say 20 years from now it will be proven that Y is equivalent to saying not X (but it is in no way obvious this is the case) then up until the point that such a proof is created the "proof" will look legitimate.

 

Then suddenly it is not true. How can it ever be a "proof" if this is the case?

[sanctus]

Because in all the cases where the implication not Y implies not X is satisfied, you can use the logic as proof.

 

It is more like physics if you want, as long as nothing proves the contrary (for example a new experiment) a theory is right, then when something proves it wrong it is only right in the case where experiments prove it. Just like Newtonian mechanics has been proven wrong for high speeds, it is is still right to use to calculate the time it takes you by car to get from A to B...

 

Anyway it depends on how concise you can pin down the (not-)Y-statement, ie. if Y is not greater than 2 implies X=Y/2 is not greater than 1, will be hardly ever proven wrong.

[Kriminal99]

Doing so creates a situation where the concept of "proof" becomes trite and useless. I could prove something today, then disprove it tomorrow. Should anyone really stop trying something because it has been proven impossible in this case?

[sanctus]

Doing so creates a situation where the concept of "proof" becomes trite and useless.

Not really, when you use the proof you have to check that your are in a situation you can use that proof. More mathematically that all the suppositions of the given proof are satisfied.

Should anyone really not trying something because it has been proven impossible in this case?

 

Well, you either accept the proof (and then you will not try it) or you re-derive the proof and see for yourself that it is impossible or you prove the proof wrong.

[Kriminal99]

Not really, when you use the proof you have to check that your are in a situation you can use that proof. More mathematically that all the suppositions of the given proof are satisfied.

 

Well, you either accept the proof (and then you will not try it) or you re-derive the proof and see for yourself that it is impossible or you prove the proof wrong.

 

Then I restate the common issue that such proofs are rife with unrecognized assumptions which are unknown to be true or false.

 

One way people try to get around this is to define everything themselves instead of leaving it connected to the real world. Then it becomes simple to see that the things they assume are true (based on deduction using their oversimplified definitions) but the overall proof has no bearing on the real world in this case.

 

Or I just completely ignore the proof because it is fundamentally flawed...