"You can't prove a negative !"
"You can't prove a negative !". This is one of the most absurd and obtuse beliefs anyone could have about epistemology. Basic rationality, and basic science, goes completely against it. And yet throngs of skeptics and believers alike (two groups of people I have no love for) spout this nonsense as if it was a fait accompli.
Let me first state the obvious. Science progresses by proving negatives. More specifically, it is by constructing possible models of a phenomena (a hypothesis), and then testing them (falsification), that we advance and build on the knowledge that we already have. By doing so, we prove many negatives along the way. We came to the conclusion that oxygen is the necessary gas in burning because we were first able to disprove the existence of phlogiston, which was the reigning scientific position at the time. More exactly, we now say that oxygen is a better explanation of burning than phlogiston, because the first fits all the facts while the second does not.
In fact, it is considerably easier to prove a negative (or even a universal negative) then it is to prove a positive. To prove a positive requires extensive testing and decades or centuries of confirmation. To prove a negative can take only one experiment ! As soon as a piece of data disproves the proposition, the negative is proven. Of course, if a model fits most facts perfectly and only disagrees with a few facts, it may very well be that the model does not need to be trashed, but rather modified.
The idea of proving negatives by testing is also part of a basic epistemic principle, Occam's Razor. If you have two models explaining the same data, and one is simpler than the other, then the simpler one is true and the complex one is false.
The "can't prove a negative" contingent then tries to rationalize that away by saying that Occam's Razor is just a preference or a probability. This is nonsense. Occam's Razor is nothing more than a restatement of the fundamental rational standard that "to assert something you must have objective evidence". If you have two competing models, and one is more complex than another, then the complex ones has additional entities or processes which have no evidence to validate their existence.
Going over this, the "can't prove a negative" crowd will then try to remove the object of inquiry outside of human cognition. Here is a famous example, Carl Sagan's "The Dragon In My Garage" :
"A fire-breathing dragon lives in my garage"
Suppose (I'm following a group therapy approach by the psychologist Richard Franklin) I seriously make such an assertion to you. Surely you'd want to check it out, see for yourself. There have been innumerable stories of dragons over the centuries, but no real evidence. What an opportunity!
"Show me," you say. I lead you to my garage. You look inside and see a ladder, empty paint cans, an old tricycle -- but no dragon.
"Where's the dragon?" you ask.
"Oh, she's right here," I reply, waving vaguely. "I neglected to mention that she's an invisible dragon."
You propose spreading flour on the floor of the garage to capture the dragon's footprints.
"Good idea," I say, "but this dragon floats in the air."
Then you'll use an infrared sensor to detect the invisible fire.
"Good idea, but the invisible fire is also heatless."
You'll spray-paint the dragon and make her visible.
"Good idea, but she's an incorporeal dragon and the paint won't stick." And so on. I counter every physical test you propose with a special explanation of why it won't work.
Now, what's the difference between an invisible, incorporeal, floating dragon who spits heatless fire and no dragon at all? If there's no way to disprove my contention, no conceivable experiment that would count against it, what does it mean to say that my dragon exists? Your inability to invalidate my hypothesis is not at all the same thing as proving it true. Claims that cannot be tested, assertions immune to disproof are veridically worthless...
If a belief cannot be falsified in any way whatsoever, then what does it mean ? The meaning of a concept is a set of properties that we can use to differentiate between instance of that concept and other objects around us. The meaning of a table can be subsumed into these properties : furniture - flat horizontal surface - one or more vertical legs - used to eat on or support objects. I can use this meaning to look at different things and figure out whether they are tables or not.
Now if you present me a concept that is not falsifiable, such as the dragon, we cannot possibly extract any meaning from that concept. There is absolutely no way to distinguish it from anythnig else - in fact, that is its selling point ! So such talk is, like religious talk, complete gibberish that can only have significance for the individual within a specific inter-subjective context.
The case of the word "god" is very similar. "Weak atheists", agnostics and believers all try to push "god" into unfalsifiability. But as I said before, that makes it meaningless, and therefore disproven because of its inability to fulfill the smallest burden of proof. The other problem is that this unfalsifiability is plainly false, as the Problem of Evil and other atheistic arguments prove.
So where does this silly belief "you can't prove a negative" come from ? I think part of it is the impulse from American skeptics and atheists, who live in a country hostile to rationality and don't want to make more enemies than they need to, are very quick in deflecting possible conflict with believers. So they will adopt such a "tolerent" attitude even though it is completely irrational and destroys their own epistemology. And I doubt it actually helps much, since believers will always be convinced that we think they are wrong... and who can blame them ? They are obviously right in thinking that we do.