Shifting the burden of proof
Shifting the burden of proof is a kind of special pleading logical fallacy in argumentation whereby the person who would ordinarily have the burden of proof in an argument attempts to switch that burden to the other person, e.g.:
- "If you don't think that the Invisible Pink Unicorn exists, then prove it!"
The burden of proof is a similar concept to "you can't prove a negative". However, the latter is more vaguely defined and potentially misleading.
- "In examining any particular text of argumentative discourse, the question may (and should) be asked: what strength of evidence is required to persuade? "
Establishing the burden of proof
In an argument, the burden of proof is on the person making a positive assertion. That is, if a person says that the moon is made of cheese, then it is up to that person to support this assertion. This is because without evidence, the natural state for humans is uncertainty; arriving at a state of having knowledge generally has a cause that can be communicated to others. Demanding that the other party demonstrate that the moon is not made of cheese would constitute shifting the burden of proof and is generally considered a logical fallacy.
- "when someone positively asserts p, he acquires the obligation to defend his claim "
- "Ei incumbit probatio qui dicit, non qui negat. [The onus of proof lies on the proposition, not on the opposition.] "
It can sometimes be tricky to determine who holds the burden of proof for a given assertion. If a bank's customer sues the bank, claiming that money was illegally withdrawn from his account, the customer is making a positive assertion, and should therefore have the burden of proving it. However, the bank has detailed financial records, and it is therefore easier for the bank to demonstrate that nothing illegal occurred, than it is for the customer to demonstrate that something illegal happened. For this practical reason, the person with access to evidence generally has the burden of proof.
Unfalsifiable claims automatically rest with the claimant because there is no evidence that could possibly decide the question. Bertrand Russell argued for this principle in relation to a celestial teapot.
There is also a reason from practicality. If we accept one claim from a person on faith, why not accept everyone else's claims on faith? However, this leads to contradictory conclusions (i.e. it is a broken compass argument). If we only accept claims from the first person in question, we are being biased, unfair, and committing the fallacy of special pleading.
In addition, just as each person is responsible for their actions, each person is also responsible for their claims. Such responsibility includes not only explaining the claims but also proving the claims. In a market place of ideas, accepting the claim can be viewed as a bargain: the claimant's end of the bargain is to prove the claim and the audience's end of the bargain is to accept the claim if and only if the claimant has upheld their end of the bargain.
Rebuttal using another positive claim
A rebuttal to a positive claim can also be a positive claim, which also must be supported by evidence. Imagine the following conversation:
- A: The moon is made of green cheese.
- B: That's not true: astronauts have gone to the moon and found that it's made of rock.
Here, the statement "the moon is made of green cheese" is a positive assertion, and person A has the burden of proving it. The second statement is a rebuttal of the first, but the statement "astronauts have gone to the moon" is a positive assertion. Since B is making a positive assertion (about space travel), B also has a burden of proof.
Burden on the advocate
- "Ultimately, anyone who is trying to convince another person of his position must shoulder the burden of proof. If someone who believes in God wants to convince someone who doesn’t, then he must offer evidence for his case. If a person who does not believe in God wishes to convince a believer, then the burden of proof is on him. "
This is not a reasonable standard for the burden of proof. If someone is arguing against belief in Santa Claus, she should not be required to produce specific evidence for his non-existence. It is unreasonable to expect a search of the North Pole to be conducted! It is sufficient to point out there is no reliable evidence for the existence of Santa Claus. This is similar to most other skeptical arguments which rely more on analysis of the quality of evidence supporting a claim, rather that providing evidence the claim is false.
Skepticism and the burden of proof
Claiming a lack of belief is the most reasonable position generally does not have a burden of proof because it argues that the justification for a positive claim is too weak to be accepted. Without any evidence at all, doubt is generally the case for all a posteriori propositions. For this reason, skepticism generally does not require specific evidence to conclude "I don't know". However, extreme skepticism or ignoring available evidence is unreasonable.
While the skeptic can proceed by simply critiquing evidence in most cases, they may be challenged to support their acceptance of skepticism itself. In this case, skeptics do have a burden of proof.
There is no burden of proof
- "Yet I do not ever hear any proof or evidence that they [the burden of proof itself] are true claims. Is this yet another case of a self-defeating claim? It seems so. "
- "Now, if Bigfoot or fairies existed I would expect we would have better evidence of them. They are presumably material things that can be seen, captured, photographed, video recorded etc. It is in part because I would expect that we would have better evidence than we do, that I do not believe they exist. [...] Notice though I am stating my reasons why I do not think fairies or Bigfoot exists. [...] I feel no need to resort to claims someone else needs to shoulder some imaginary burden of proof. "
The above quote is critiquing the supporting arguments for Bigfoot but not mounting a separate case for its non-existence. Being skeptical does not incur a burden of proof or demonstrate its invalidity.
Many philosophers consider the initial plausibility of a claim to be a factor in determining the burden of proof.  This makes little differences as to who has the burden of proof but does affect the standard required for the justification to be found convincing. An extraordinary claim would require an extremely good justification. This concept is often stated as "extraordinary claims require extraordinary evidence".
Burden of proof by claim
Theism has the burden of proof because in apologetics they claim to have knowledge, not just belief, of God.
- "the object of the exercise is, presumably, to discover whether it is possible to establish that the word 'God' does in fact have application. Now to establish must here be either to show that you know or to come to know. But knowledge is crucially different from mere true belief. "
When asked to support the claim that a god exists, it's not unusual for an apologist to respond with "You can't prove God doesn't exist", or similar statements. Essentially, this is an attempt to shift the burden of proof (a logical fallacy).
A rephrasing of "I believe that god exists" into "I don't believe that god does not exist" is also an attempt to dodge the burden of proof - the proposed equivocation seeks to convert a claim acceptance to a claim rejection, but the two are not logically equivalent. "I don't believe that god does not exist" can be (and often is) a position held by atheists, and says nothing about how the person stands on the claim "god exists".
Therefore, apologists are claiming knowledge about a fact. This claim is unfalsifiable. Gods may exist in an inaccessible spiritual world or be immune from testing. "It is said: 'Do not put the Lord your God to the test." Luke 4:12 For this reason, the burden of proof for the existence of God lays on believers and not with skeptics.
Some apologists fully shoulder the burden of proof, such as Norman Geisler with his proposed adequate truth test. However, other apologists are more modest in their claims and only seek to make their view appear as plausible, rather than true beyond doubt. 
In apologetics, a theist is someone who claims that there is a god. A weak atheist is someone who claims belief in God is not justified. This position is very similar to skepticism and some forms of agnosticism. Since the theist is the one making a positive claim, it is the theist's job to demonstrate that a god exists.
"[Like a legal defence,] The presumption of atheism is similarly defeasible. It lays it down that thorough and systematic inquiry must start from a position of negative atheism, and that the burden of proof lies on the theist proposition."
"You ever seen one of those things with a jar on a counter and its full of gumballs? And they have a contest to see who can guess the closest to the actual number of gumballs? As long as there are gumballs in that jar, you and I would agree that the number is either even or odd? What is the default position on the number of gumballs [being odd or even]? [Caller responds "The default position is neutral"] Yes. But if someone asserts 'the number of gumballs in this jar, I believe, is even.', if I'm in the default position, I disbelieve that assertion, I do not accept that assertion, I reject that assertion. That doesn't mean that I think that the number is odd because we only address a single prong of a dilemma at a time. Either a God exists or it doesn't. A theist is offering the proposition that a God exists (that the number of gumballs is even) and I am rejecting their assertion. Theism is the acceptance of that position and atheism is the rejection of that position. It's not the assertion that there are no Gods. I don't believe their assertion. I also don't claim that I know that their assertion was false."
Conflating atheism with other views
Some apologists disagree and say atheists have a burden of proof because they must also have other views that require justification.
- "your belief system is not neutral. Lack a belief in God is only part of a worldview. [...] The atheist has to answer the question, ”Why is there something” according to the atheistic worldview. The atheist has to justify their belief in rationality. The atheist must give reason for the existence of free will. While the word “atheist” may give the impression that it only has to do with a lack of belief in God, the reality is that they are “naturalists” (often materialists) and, as such, must give a positive explanation for the claims of their worldview. "
While atheists generally have many other positive beliefs, it is a non sequitur to say atheism has a same burden of proof. Not all atheists accept naturalism and it is a hasty generalization to claim that they do. If naturalism is thought to need justification, that is a separate issue to the burden of proof of atheism.
- "I am asking whether the skeptic is justified in calling into question the truth of 'God exists.' Why not put the burden on him? Why not insist that he is attempting to convict of irrationality generations of human beings, rational animals like himself, whole cultures for whom belief in the divine and worship are part of what it is to be a human being? Were all those millions, that silent majority, wrong? [...] Is there thus a prima facie argument against atheism drawn from tradition, the common consent of mankind both in the past and in the present time? I think so."
The problem with this reasoning is it employs the majority argument in an attempt to shift the burden of proof to the skeptic; this tacitly admits that the burden of proof originally lays with the apologist. Also, the argument itself fails because general belief in something (or even the general predisposition to believe) does not reliably indicate the truth of the belief (i.e. it is the fallacy of argumentum ad populum).
- "Does not the burden of proof then fall on the shoulders of the skeptic? Yes. And the skeptic is the first to admit this-or at least to exemplify it. I would hazard the view that more attention is paid to theism, religious belief, the existence of God, as a problem to be dealt with, as something that is an intellectual task, by the skeptic than by the believer. I have met many more militant skeptics than I have believers who look as if they were going to toss and turn all night unless they developed an airtight proof for the existence of God."
Apart from having questionably factual validity, it is also an ad hominem since shifts the argument on to the red herring of skeptic's behaviour, which this tells us nothing about the actual burden of proof.
Strong atheism probably has a burden of proof because it is a positive assertion that God does not exist.
Some apologists argue that the usual meaning of atheism is strong atheism. This appears to leave atheists with the burden of proof.
- "The denial of God’s existence (the claim that he does not exist) is what is traditionally called atheism. "
"‘Atheism’ means the negation of theism, the denial of the existence of God."
- — Stanford Encyclopedia of Philosophy 
- "Atheism is not "I don't know," that's agnosticism (again, Greek: A is still without; gnostos is knowledge). Atheists affirmatively say there is not a god "
Despite this, most self professed atheists consider the broader definition of weak atheism to be the usual meaning of the word. 
Well-established scientific theories
Often, in debates over well-established scientific theories, the person arguing against the mainstream view will say that it is not up to him to disprove the theory, but that it is scientists' job to demonstrate it. This is true, in a sense: when a new scientific hypothesis is introduced, its proponents have the onus of demonstrating it. The rest of the scientific establishment has no obligation to disprove the new hypothesis. However, a hypothesis can only rise to the rank of theory by being repeatedly tested, and by accumulating evidence in its favor. Non-specialists may then rely on a valid argument from authority that a theory accepted by the vast majority scientists should be accepted as true. This should now be taken into account in general discussions, as well as non-expert critics of the theory. However, this does not apply to expert scientists who may sometimes disagree with their peers.
Thus, if person A says that relativity is unproven, and person B asks A for evidence, this may be seen as shifting the burden of proof. However, B is really asking A to support their positive assertion that mainstream science is wrong.
In online debates, when a person challenges a well-established scientific theory, it is almost invariably the case that that person does not know or does not understand the evidence for the theory. This is particularly evident when discussing evolution with the majority of creationists.