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When we do a proof by contradiction, we assume a premise, derive a result from it, and if the result is incorrect then we conclude that our initial premise was wrong. However, if we do get a correct result, can we conclude that the premise is true, or was out test just inconclusive?

My guess is that we can go ahead and assume that the promise was correct. If it was possible to derive a true result from a false premise using valid operations, then it would also be possible to work backwards from a correct result to a false premise by performing valid operations. However, something just tells me that this is "too good to be true".

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  • $\begingroup$ 1=0 implies 1=1. We know 1=1 is true. This does not prove that 1=0. $\endgroup$ – DanielWainfleet Dec 30 '15 at 5:56
  • $\begingroup$ See generality of algebra. $\endgroup$ – Lucian Dec 30 '15 at 15:18
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It is possible to begin from false assumptions, perform valid operations, and arrive at a true statement. For instance, assume $0=1$. Multiplying both sides by $0$ yields the true statement $0=0$.

For this reason, you cannot prove that a statement is true by assuming that it is true and then deriving something else that is true. This is a common mistake I've seen among calculus students. Often they just need to write their equations in a different order, and everything works out because the operations they're doing are reversible, but the logical order of statements is important.

You say:

If it was possible to derive a true result from a false premise using valid operations, then it would also be possible to work backwards from a correct result to a false premise by performing valid operations.

This statement is untrue exactly for the reason that not every valid operation is reversible. For instance, in my example we cannot reverse multiplication by $0$ by dividing by $0$. It's not a valid operation. If you're familiar with dot products, then you know that dotting by a vector is not reversible - i.e., if we know $u \cdot w = v\cdot w$, we cannot undo dotting with $w$ to obtain $u=v$. There are tons more examples of operations we do that are not reversible, but this should give you the idea.

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  • $\begingroup$ However, if we restrict ourselves to reversible operations, would this still be a problem? $\endgroup$ – Ovi Dec 30 '15 at 5:58
  • $\begingroup$ In that case we're dealing with operations of the form $\varphi$ where $a=b$ if and only if $\varphi(a)=\varphi(b)$. So every statement in the derivation is equivalent to every other statement in the derivation. In that case, yes, the conclusion is true if and only if the premise is true. However, you will never prove anything other than statements equivalent to the one you started with. $\endgroup$ – kccu Dec 30 '15 at 6:09
  • $\begingroup$ I'd still say it's poor practice to begin with what you want to be true and derive something known to be true, even if done through equivalences. It's better to start from what you know is true and work your way to what you wish to prove. $\endgroup$ – kccu Dec 30 '15 at 6:10
  • $\begingroup$ That seems quite a bit harder to do, a lot of the times I start by assuming what I'm trying to prove is true, deriving a true statement. But when I actually write the proof I start at the true statement and try to derive my way to the thing I'm trying to prove. Would you say this is bad practice/detrimental? $\endgroup$ – Ovi Dec 30 '15 at 6:15
  • $\begingroup$ I should have been clearer - often mathematicians work the "wrong" direction when trying to figure a proof out, but then write up the proof in the "correct" order as you said. That is perfectly fine, and often very useful. I was warning against writing up proofs in which you begin by assuming what you wish to prove. $\endgroup$ – kccu Dec 30 '15 at 6:18
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You can certainly derive true statements from false premises. By the principle of explosion, a false premise implies every statement.

To be concrete, assume $-1 = 1$ (obviously false). Squaring both sides, we get $1 = 1$ (obviously true). Here the problem is that squaring isn't an "operation" that you can undo.

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  • $\begingroup$ If we were to exclude operations which cause the loss of information (such as raising each side to an even power) would this be ok? $\endgroup$ – Ovi Dec 30 '15 at 5:56
  • $\begingroup$ @Ovi you have to make these notions (such as "loss of information") precise first, before you can say anything about them. $\endgroup$ – Artem Mavrin Dec 30 '15 at 5:58

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