Proof by negation requires a person to start with a certain number of assumptions, proof a necessary consequence of those assumptions, and then show that the consequence is false so therefore one of the assumptions must be false. It sounds like it could actually work but there is a small problem. You have no clue what assumptions you are making any time you attempt to relate it to the real world as opposed to using a closed system where you define everything with your limited assumptions. For instance, suppose a guy bumps into me on the subway and I say "I am assuming that guy did not pickpocket me, so my wallet must still be in my pocket" and then I look in my wallet and it is not there. Proof by negation would have you reason that the person stole your wallet because your assumption was false. But in reality your assumptions were infinite and not all of them are known to you. You are assuming: 1) The guy in the green didn't pickpocket you 2) The guy in the red didn't pickpocket you 3) An unknown person didn't pickpocket you (Someone you didn't even see) 4) Your wallet didn't fall out at the station 5) Your wallet didn't fall out in the car 6) Your wallet didn't fall out in an unknown location 7) An unknown yet plausible thing did not occur to cause your wallet to not be there (infinite variations) 8) The wallets subatomic particles behaved in a way that caused the wallet to cease to exist 9) A space alien warped the wallet out of your pocket to get insight into human nature 10) A less plausible unknown thing (that we still can't rule out) occurred to cause your wallet to not be there. (infinite variations) As you can see I have separated the plausible and less plausible to show that there are still infinite assumptions that sound possible to us once we hear it but that we still couldn't think of on our own. However regarding the implausible, by the nature of uncertainty they are things we can never rule out completely we just usually do not hear of such things happen. This distinction is also meaningless in a realm where most of our experiences do not relate or we simply have little information, for example in the realm of subatomic particles. In other words, we have no clue what is plausible or not plausible when dealing with sub atomic particles.
You're also implicitly assuming that you wallet was there before the person bumped into you. You haven't covered all of your bases.
Of course proof by negation is valid. But whether you're doing a direct proof or an indirect proof, if you run a false premise through a valid logical form, you will get a false conclusion. Logic works by GIGO.
Proof by negation works better at the abstract theoretical level, as in mathematics. It's not exactly invalid when applied to the natural universe, it's just difficult to do. You are always a little bit unsure of what exactly you've disproved. You might have inadvertently disproved your own premise: that you've identified a necessary consequence!
It usually works pretty well. I want to know if my keys are on my coffee table. I look on my coffee table, and my keys aren't there. I conclude that my keys are not, in fact, on the coffee table. Hey, I just proved by negation that my keys aren't on the table! My premise is that if they were on the table, I would see them when I looked. Since I didn't see them, they must not be there. Yeah, yeah - that isn't really "proof," since it's possible that my key really were on the table but were simply invisible or something. But you can get as close to certain as you can be in the real world.
Funny you should mention that. I swear that my keys, my cellphone, and my eyeglasses do become invisible or something at times.
