Skepticism concerns itself with extraordinary claims, but hardly just the obviously weird ones…
Unlike topic areas such as overt pseudoscience or the paranormal, more conventional claims count as extraordinary merely by being important and contestible — and there are few claims which cannot be important or contested in the right context. There is also the matter of how much a claim requires us to overturn what we already know if we accept it.
Note that the mere illusion of knowledge, in the form of pre-determined conclusions, fed by ideology, emotion, confirmation bias and rationalizations doesn’t count here.
What we need is a particular degree of evidence. Abstract philosophical argument alone with counterfactual ‘maybes’ and ‘possiblys’ are of no help to us here. This is why debates over the existence of a God or Gods are still raging even after thousands of years, strongly argued by both theists and religious skeptics.
To switch gears a bit, since this is not an antitheism blog…
Take a murder trial. This is hardly trivial given the usual penalties to the defendant if convicted, and even without ‘weird things’ involved it’s every bit as extraordinary as the most bizarre alleged haunting, alien abduction claim, or cryptid sighting. We know murders happen. What is contested here is the innocence or guilt of the defendant.
Rarely does this involve claims of improbable phenomena unknown to science…a body, a weapon, opportunity,a plausible motive and compelling case by the prosecution are usually all that’s needed.
The penalty for murder needs high standards of evidence to get a conviction…to back it up in court and make the conviction stick, after all. A conviction could mean the defendant’s very life in some jurisdictions. We’re not talking about being caught nicking candy from the convenience store.
Even a seemingly innocent claim such as ‘the sky is blue’ could require some evidence, especially when told this only seconds after just hearing of a local tornado warning — if the sky’s actually turning green, credulity could be dangerous as the wind picks up before getting to shelter, but one need only a look at the current weather alerts from safety using whatever device is at hand.
So nearly any claim can be extraordinary depending on the right context, and this depends on two major factors, sometimes in combination:
 The importance of the claim: What is at stake if we uncritically accept it, jumping to an unwarranted conclusion with serious consequences? If the claim is true, then there must be evidence strong enough to support it and once that’s in, we’d be wise to heed it. Reality’s a bastitch when spurned, and doesn’t care about your politics, your religion, or what you had for breakfast before updating your Facebook status. If the claim is false, on the other hand, we need to know so that no one wastes time, money, or risks their health or lives pursuing the claim as though it were true. Properly evaluating the evidence will show this. Again, reality doesn’t give a hoot what you believe, silly relativist arguments about the impossibility of objective reality or truth aside.
 The unusual or strange nature of the claim: This mostly applies to claims of ‘weird things’ but also for rare mundane phenomena, which can seem weird to those unfamiliar with them. Does there seem to be no explanation for the claim immediately on hand? Does the claim require that to accept it, we overturn much or even all of modern science to explain it? Given that, let’s face it, “Science, it works, bitches,”* we use science as the gold standard for factual knowledge even while publicly denouncing it. We cannot just accept or reject a claim unless we’ve given it a fair hearing, and that’s where science steps in. We examine the reasoning and the initial evidence, especially the scientific evidence for it. If it doesn’t pass muster, we are right to reject it as improbable or baseless in fact until shown otherwise, or sometimes as impossible. The latter, though, should be done with care.
*attributed to Richard Dawkins
And both of the above can apply to claims in which there is a legitimate controversy, not one manufactured by the media, zealots, or ideologues intent on undermining the science they don’t like. In any case, importance, consequences, and the plausibility of a claim at the time can make even seemingly trivial ones extraordinary…
…context and the stakes raised by the claim are deciding factors here.
“What may be asserted without evidence may be dismissed without evidence.”
~ Christopher Hitchens