Burden of proof
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Burden of proof (or onus probandi in Latin) is the obligation that somebody presenting a new or remarkable idea has to provide evidence to support it when asked. In a scientific context evidence is experimental or empirical data (although in some branches, well thought out mathematics may suffice). Once some evidence has been presented, it is up to the opposing "side" to disprove the evidence presented or explain why it may not be adequate. For example, in identifying a chemical compound, an analyst may present a spectrum to support their hypothesis but a reviewer may point out that it is insufficient, explain why by offering an alternative interpretation and state more data is needed, usually suggesting specific data that would be required. This sort of procedure happens constantly in the scientific method, repeating until everyone is happy that the data and explanation match.
Denialists of evolution and global warming have a habit of merely ignoring evidence and claiming that the burden of proof still rests with the proponents of those concepts. Debates frequently descend into arguments about which side the burden of proof lies with.
Often, someone will present a new idea and say that it must be accepted because it cannot be disproved. This is insufficient because without evidence there is no reason to accept an idea, even if there is no contrary evidence. One example is that of a simulated reality, which proposes that the human race does indeed live in The Matrix and we are a computer simulation. There is no evidence against this idea, in fact, it may be impossible to fully disprove, but as there is no real evidence for it there is no reason to accept the idea as real. Another famous example is the teapot proposed by Bertrand Russell, the existence of which cannot be disproved.
Indeed, if one is making an exceptionally bold claim (such as with alternative medicine that claims miraculous cures) then exceptional evidence is expected in its support. Or as Carl Sagan said, "extraordinary claims require extraordinary evidence".
 Null hypothesis
If someone has presented you with an idea and says that the burden of proof is on you to disprove the idea, work out what the null hypothesis is and then put their evidence for the idea against it.
The person claiming something is possible or has happened needs to produce evidence to refute the null hypothesis.
If they have considerable and well-tested evidence, the burden of proof may reasonably be considered to be on the person claiming that the evidence does not hold.
Some theists maintain that unless atheists can disprove the existence of a god, or gods, their position is untenable. It does, however, depend on which of the many thousands of mankind's gods one has in mind; strangely, such theists cheerfully accept the arguments against every god except their own preferred one(s).
On like lines, some anti-theists maintain that because theists cannot prove the existence of their chosen god or gods, their position is untenable. A similar fallacy, from the opposite direction.
The burden of proof lies with whoever is making the assertion. If there's no evidence either way, it's a matter of faith. Expressing a personal belief / disbelief in the existence of a certain god / goddess is sound enough, but claiming their opinion is factual or denouncing the opponent's claim as false without any proof supporting one idea or disproving the other is fallacious. While an Unfalsifiable claim / hypothesis can be proven neither right nor wrong, it is reasonable to dismiss it as non-factual if it lacks logical supporting evidence.
 See also
- Flying Spaghetti Monster
- Russell's Teapot
- Presumption of innocence
- Negative evidence
- Logical, Objective, Verifiable Evidence (L.O.V.E.)