This is kind of annoying when I hear "You can't prove a negative". When using a stricter mathematical logic it would appear you can. As I looked on wiki, it was also like describing a creepy stalker... If I bake a pie, and I always put the pie on the window sill to cool, if a person sees no pie on the window sill, it means I did not bake a pie. If P=Q, not Q proposes not P. Hence, I can prove a negative.
That is not proving a negative, that is a conclusion based on the original critera. Just to be annoying; not seeing a pie on the window does not prove you did not bake a pie. If you baked a pie and accidentaly dropped it when removing it from the oven it would not be on the window sill cooling. The point is that you cannot prove a negative in relation to a scientific hypothesis or theory
The pie example had obvious flaws. But mathematically, can you prove a negative? Like 1+1 is not 3. I'm not trying to be philosophical (or actually I don't know) and that may appear to be more strict and weird, but... Can you mathematically prove a negative. Mathematics is not science or philosophy.
Sure if I have 3 apples in my hand and I claim there is only 1, you can prove my statement is wrong. \(1\neq 3 \)
It's a widely accepted bit of folk-logic out there on the street, despite the fact that it's totally false. Of course you can. In any logic class it's done routinely. In the words of the Oxford Guide to Philosophy p.793, "The negation of a proposition P is proved by taking P as a premiss and demonstrating that, in conjunction with previously established premisses or axioms, a contradiction follows." That's called a reductio ad absurdum proof and it is one of the most common strategies used in mathematical proofs. What's more, one can sometimes prove a negative by simple enumeration. Suppose I want to establish that it's true that 'There are no snakes in my drawer'. And suppose that there are a finite number of things in the drawer. So I open my drawer and look at object #1, nope, not a snake. Proceeding through each object in turn, I verify that each one isn't a snake. We do this all the time in real life, when we are looking for things like can-openers. Of course this only works when the universe of discourse is fixed and finite. It doesn't work so well when the universe of discourse is open and unbounded. If we don't know how many animals exist in the natural world, checking each animal in turn and verifying it isn't a snake doesn't guarantee that the next animal we check won't be. We can only conclude that there are no snakes in the world when we are sure we've looked everywhere and checked every animal. It's important to recognize that the same problem applies to positive statements too. We can observe that all swans in our experience have been white, and perhaps even rashly conclude that 'all swans are white', before we happen upon our first black swan. That's the famous problem of induction, the problem of explaining how open-ended universal conclusions might be drawn from finite data-sets of particular instances. The relevance of that problem to things like the laws of physics should be obvious. http://departments.bloomu.edu/philosophy/pages/content/hales/articlepdf/proveanegative.pdf
That looks like an example of modus tollens, which is a valid form of inference. If P, then Q ~Q Therefore ~P https://en.wikipedia.org/wiki/Modus_tollens If we assume the truth of both of your initial premises, then we have just proven that they logically imply that P has to be false. Just about any proof that one encounters in mathematics or in theoretical physics is going to derive its conclusion by valid inference from whatever the proof's initial assumptions are. If the goal is to prove a proposition true or false without any initial assumptions apart from principles of logical inference, then the conclusion would have to be either a tautology or a logical self-contradiction.
"You can't prove a negative" has some specific applications. It is by no means universal. One very common application is more like "you can't prove the non-existence of something". Very commonly applied to God, but also to pink unicorns, black swans, orbiting teapots and invisible garage-dwelling dragons.
Just to be even more annoying (to you, though), your critique is flawed: if, as you suggest, the pie is baked and accidentally dropped, then you are merely describing a case in which the first premise is false. The premise was that if the pie is baked then it is always put on the window sill. Introducing an "accidentally dropped" is thus not in line with the premise, and all you are saying is that if the premise is false then the conclusion is false, which is to be expected of a valid argument. Yazata is correct in that this is an example of Modus Tollens, which is a valid form of argument.
I don't really see your point but that is OK, I will assume you are correct. I find the whole area of mathematical logic and proofs way to tedious to hold my attention for very long.
The electron has a negative charge. This is defined by tradition and can be seen in the lab for proof.
In constructive mathematics, to make a negative statement, then you really do have to have a direct proof of it. One cannot simply rely on proof by contradiction.
Interesting, there is a live example on OP. You, Physbang, in one of the threads claimed that you are a PHD (PhD). Since your posts were mostly abusive, lacked merit and you wrote PhD as PHD, I suggested that "you are not a PhD". James R jumped in and he erroneously asked me to prove the negative that you are not a PhD. I cannot. Why I cannot ? Simply because the information (the truth)whether you are a PhD or not is in your possession and you are not sharing it despite my asking. So an adverse inference can be drawn that you are not a PhD but to be fair to you that is not the conclusive proof. So the point is that if A expects B to prove the negative, then B must be aware (or noticed) of the truth with regards to affirmation of the point. Its like you cannot expect a person to prove that he is 'not guilty' even if certain presumptions of guilt are present against him, even in that case affirmative (suggesting his guilt) must be put to him to counter. Apart from this proof of negative in general is no way differnt from proof of positive. Its like this..Prove that God does not exist, it is as good or bad as asking ...Prove that God exists.
Agreed. In my experience the expression "you can't prove a negative" usually arises when someone is advocating an additional hypothesis. The phrase comes up in arguments involving people (often on both sides of the argument) who do not understand the principle of parsimony, a.k.a. our good friend, Ockham's Razor.
Further thought: one also comes across people intoning a related cliché: "absence of evidence is not evidence of absence". This seems to me quite wrong. Absence of evidence is certainly not proof of absence, but sufficient absence of evidence (i.e. after a thorough search) may well be enough to allow a provisional finding that the thing looked for is, indeed, most likely absent. In other words, there is evidence, though not proof, of absence.
Slightly off-topic, but this reminds me of the story of the logic teacher explaining to students that while two negatives cancel each other and make a positive, two positives cannot make a negative. Whereupon a voice from the back of the class murmured, "Yeah. Right."
The proof of anything, including a negative, is a truth, and the truth about anything is only a value, not an absolute, even in mathematics. If there appears to be three apples in your hand, one could be a facsimile of an apple made from wax or plastic. You wouldn't know unless you sampled it for flavor, and if you did that by taking a bite, then you would have less than three apples in your hand, or maybe even less than two if you picked the wrong one to sample. There are branches of mathematics where 1+1 literally equals 3, and not by means of exploiting a division by zero. R.W. Hamming, a famous person in the field of error correcting codes has pointed this out as part of a strange but true feature of his field of endeavor. Galois fields are even stranger, as I recall. For the problem he was working on, it made perfect sense. Liberty and freedom are likewise values, not absolute. The freedom to walk down the sidewalk depends on respecting laws related to pedestrian traffic, and to do so in a manner that respects the rights of the homes a and businesses on the same street, which is to say it is usually a good idea to do so with appropriate apparel. That sidewalk, signs, benches, snow removal and general maintenance were paid for with taxpayer money. Nothing in this scenario is really "free". And nothing you can think of is really an absolute truth or falsehood either. Bertrand Russell's example of the orbiting teapot is the best example I can think of that shows you can't prove a negative. If someone says there is a teapot in orbit, it would be pretty much impossible to prove that there isn't. Of course, we can be fairly certain that the ISS probably does have a teapot, or the equivalent, which kind of wrecks the whole idea unless you specify your orbiting teapot is orbiting somewhere else.
Russell's Teapot is actually * orbiting Sol, not Earth. So, ISS doesn't count. Please Register or Log in to view the hidden image! *not
