The burden of proof is a commonly invoked principle, used to end an argument by pointing out that there is an unsatisfied burden of proof that falls on one of the parties, and thus that their position can be justifiably considered false unless they produce the required proof (actually, the required evidence, “proof”, while colloquial, is too strong of a term). Indeed we might even make the case that all arguments ultimately rest on some claim involving a burden of proof. Even the validity of a claim backed by a mathematical proof rests on the assumption that we aren’t all systematically making the same logical error when considering that proof, and must “refute” claims that we are making such a mistake by arguing that the person making them bears the burden of demonstrating that we are making such a mistake; people holding the proof to be accurate are not required to demonstrate that we are not making any such systematic mistake. But, despite its widespread use, there are many confused ideas about who bears the burden of proof and why. Partly, I suspect, this is because the notion is usually invoked only rhetorically, investigations need only produce the best explanation, they are not required to defend that explanation against other possibilities (whether we even should debate things or whether investigations should stand by themselves, to be compared to other investigations but not pitted against them, is something that I will leave aside for the moment, although I lean towards the latter). Because of its informal use people tend to assume that the burden of proof rests with the more outrageous claim, that which we deem the less probable or less rational. But clearly that can’t actually work. Not only is there a problem in establishing what these standards are, but once we have established them there is no need for further proofs; given any two such claims, plus the ability to pass such judgment, taking into consideration the available evidence, clearly warrants us in believing the less outrageous, the more probable and more rational claim by those standards, and there is no need to further bother ourselves with the burden of proof. So, rather than being dependant on such standards, the burden of proof is something we use to decide which alternative is more rational or more probable.
The natural next move is simply to assert that the burden of proof always lies with existence claims, such that the claim that something exists always bears a burden of proof in comparison with the equivalent non-existence claim. I would agree with this principle, in the vast majority of cases, but as presented it isn’t obvious why we should believe it. Furthermore, there are a number of other cases where it would seem that we need to invoke the burden of proof, but in which both alternatives involve existence claims. For example, we would say that the claim that the force of gravity is conveyed by gravity gnomes bears the burden of proof in comparison to the claim that the force of gravity is conveyed by gravitons, even if the gravity gnomes and the gravitons are claimed to have exactly the same observed effects. Now, we could try to cast the difference between these two claims in terms of existence, arguing that the first postulates the existence of gnome-like properties, while the second doesn’t. But, not only does that make the metaphysically questionable move of attributing existence to properties, but it opens up the problem of wondering whether gravitons are postulated to have not-gnome-like properties, thus putting the two on an even footing. Worrying about such negative properties is exactly why it is a bad idea to ascribe existence to properties, and I would rather avoid arguing that the claim something lacks properties does not imply that it has the property of lacking those properties. It is better then to simply look for a better understanding of the burden of proof than worry about such issues.
Let us return then to more fundamental questions before trying to say exactly where the burden of proof lies, and instead consider why the burden of proof even exists in the first case. Simply considering common cases where the burden of proof is invoked reveals that it is usually used to distinguish between a well-confirmed hypothesis and an “implausible”, to speak colloquially, alternative that simply hasn’t been disproved. Note that is not used to distinguish between two alternative hypotheses that make exactly the same predictions, and which are thus indistinguishable even in principle; those cases we can deal with using a principle of epistemic indifference, which states that if two claims make exactly the same predictions then they have the same content, and that any apparent differences are simply a linguistic confusion arising from trying to say “more than we are able”. Consider, for example, the unicorn hypothesis as compared with the no unicorn hypothesis and the god hypothesis compared to the no god hypothesis, U vs, ~U, G vs, ~G. In their natural, original, forms both sides make fairly strong claims and are relatively easy to verify. If we run into a unicorn or god then U or G are confirmed, but if we consistently fail to then ~U and ~G are confirmed instead. Thus in their original forms the U and G hypothesis are relatively quickly “disproved” and we may claim that we know ~U and ~G (even though the possibility of making an observation that confirms U or G can never be completely ruled out). However, those who firmly hold U or G to be true, for whatever reasons, will wish to revise their hypothesis so that they aren’t disconfirmed by the lack of such observations. Thus they may propose new, modified versions, the rare unicorn and rare god hypothesis (RU and RG), which claim that unicorns and god are around, but that observing them is extremely unlikely, as unlikely as needed in order to account for our continual failure to make confirming observations. And it may be claimed that RU and RG fit the evidence just as well as ~U and ~G do. It is here that the burden of proof comes into play. We say that, despite the fact that they both fit the evidence equally well, RU bears a burden of proof that ~U doesn’t, and similarly with RG and ~G.
And this is why we need the burden of proof, because such transformations can be made with any claim, we can always qualify our assertions in such a way that it confirming them becomes arbitrarily hard to do, so that they always fit with the observed facts. For example, we could claim that the laws of physics are wildly different from what they are normally supposed to be, but that this difference only manifests itself when everything present is green. And since it is virtually impossible to create this set-up, where even the instruments making the observations are completely green, this hypothesis can’t be ruled out in a strict sense. But to admit that we simply can’t rule such possibilities out and that we must treat them as equally likely, given the lack of evidence distinguishing the two, as some have a tendency to claim, is simply absurd. Because this is not like the cases of epistemic indifference, accepting one of the two possibilities may have serious consequences for our understanding of the world. For example, we might postulate that all boxes that are never opened and never inspected contain dead cats, which would have radical consequences for our other beliefs, because we would have to postulate new theories about how the dead cats got there, and so on.
With that in mind I can cut to the chase, and point out that, while the available evidence may fit two available hypothesis equally well, that doesn’t mean that it confirms both of them, and that thinking it does is a linguistic/logical confusion. The “burden of proof” is a device for pointing this out then, for pointing out that while a particular hypothesis is consistent with the available evidence that it isn’t supported by it, while the other is. This goes back to the problem of confirming “all crows are black”. As it has been pointed out this statement is logically equivalent to “all non-black objects are not-crows”. And thus it would seem that since observing objects of other colors which fail to be crows confirms the second it should confirm the first as well. Maybe that doesn’t seem absurd to you. But consider that such evidence would still confirm the hypothesis that all crows were black even if no crows had even been observed. And in that situation it would equally support the hypothesis that “all crows are blue” (and that all grues are green, for that matter). But clearly that is absurd, the same evidence can’t be legitimately taken to entail two contradictory hypotheses (at least, not if we want to retain the idea that the evidence lends support to the claim). I won’t get into how we are to avoid this dilemma; there are a number of solutions, some of which involve breaking the apparent symmetry by revising the original claim so that it contains an existence clause. It suffices to point out that to resolve the dilemma we end up with the solution that while observing a number of non black, non-crow object is consistent with the claim that crows are black that it doesn’t justify that conclusion. And so we can legitimately use that distinction when we say that some claim bearing the burden of proof means that while that claim is not contradicted by the available evidence neither is it supported by it, while the alternative is, even if we are unwilling to say exactly what being supported by the available evidence consists in.
And we can see how this understanding of the burden of proof justifies the way it is usually employed. For example, the hypothesis that rare unicorns exist bears the burden of proof over the hypothesis that unicorns don’t exist, because while the lack of observed unicorns doesn’t contradict that hypothesis neither is the claim that rare unicorns do exist supported by it. To be actually supported would require an observation of these rare unicorns. Similarly, the hypothesis that gravity gnomes carry gravitational force carries the burden of proof over the hypothesis gravitons carry the gravitational force, because the hypothesis involving gravity gnomes involves some additional claims regarding their gnome-like properties while that with gravitons does not (and I would gently remind the reader that claim that gravitons lack gnome-like properties is a claim only in name only, because it asserts nothing about gravitons over and above their role in conveying the gravitational force; to understand it as a claim is to fall prey to the very linguistic/logical confusion we are trying to avoid). Finally, to use a new example, if we come across 100 boxes and we open 99 of them, only to find that they are empty, the hypothesis that the last one contains a ball bears the burden of proof over the claim that it is empty. The sequence of empty boxes justifies the hypothesis that they are all empty, which justifies the claim that the last box is empty; it does not justify the claim that they are all empty except the last one, even though it does not contradict it.