For me, in just about every case, the credence I'd assign to an AI's wacky claims would depend on its ability to answer followup questions. For instance, in Eliezer's examples:
1) Tin-foil hats actually do block the Orbital Mind Control Lasers
What Orbital Mind Control Lasers? Who uses them? What do they do with them? Why haven't they come up with a way to get around the hats?
2) All mathematical reasoning involving "infinities" involves self-evident contradictions, but human mathematicians have a blind spot with respect to them.
I'm actually strangely comfortable with this one, possibly because I'm bad at math.
3) You are not above-average; most people believe in the existence of a huge fictional underclass in order to place themselves at the top of the heap, rather than in the middle. This is why so many of your friends seem to have PhDs despite PhDs supposedly constituting only 0.5% of the population. You are actually in the bottom third of the population; the other two-thirds have already built their own AIs.
Why haven't I heard of any of these other AIs before? How do all of the people producing statistics indicating that there are a lot of dumb people coordinate their efforts to perpetuate the fiction?
4) The human bias toward overconfidence is far deeper than we are capable of recognizing; we have a form of species overconfidence which denies all evidence against itself. Humans are much slower runners than we think, muscularly weaker, struggle to keep afloat in the water let alone move, and of course, are poorer thinkers.
Why do so few of us die of drowning (or any of the other things that would kill us if we were so dramatically more pathetic than we believe)? If this bias is so pervasive, why can I see these words on the AI's screen, when it seems that I should block them out as with all over evidence that we are pathetic in this way?
5) Dogs, cats, cows, and many other mammals are capable of linguistic reasoning and have made many efforts to communicate with us, but humans are only capable of recognizing other humans as capable of thought.
If we have this incapability, what explains the abundant fiction in which nonhuman animals (both terrestrial and non) are capable of speech, and childhood anthropomorphization of animals? Can you teach me to talk to the stray cat in my neighborhood? Why only mammals, not birds and the like? What about people who are actively trying to communicate with animals like gorillas, or are those not capable of communication?
6) Humans cannot reproduce without the aid of the overlooked third sex.
Are they overlooked in the sense that people we can otherwise detect are not recognized as being part of this sex, or in the sense that we literally do not notice the existence of the members of this sex? In the former case, how do so many people manage to reproduce without apparently wanting to or involving third parties? In the latter case, how can I get in touch with these people? By what mechanism are they involved in human reproduction?
7) The Earth is flat.
Are we talking Euclidean spacetime here? What is the explanation for the observations of a spheroid Earth?
8) Human beings are incapable of writing fiction; all supposed fiction you have read is actually true.
In this universe? What about stories with plot holes? I think that I have written fiction in the past; am I in causal contact with the events I describe? When I make an edit that changes the plot, how does that work? What about people who write self-insertions?
Why haven't I heard of any of these other AIs before?
You have. They're in the news every day.
How do all of the people producing statistics indicating that there are a lot of dumb people coordinate their efforts to perpetuate the fiction?
Perpetuate what fiction? They produce statistics about all the dumb people, compiled into glossy magazines. Hell, you're wearing a 'bottom thirder' sleeve button on your shirt right now.
No I'm not.
Yes. Yes you are.
Human beings are all crazy. And if you tap on our brains just a little, we get so crazy that even other humans notice. Anosognosics are one of my favorite examples of this; people with right-hemisphere damage whose left arms become paralyzed, and who deny that their left arms are paralyzed, coming up with excuses whenever they're asked why they can't move their arms.
A truly wonderful form of brain damage - it disables your ability to notice or accept the brain damage. If you're told outright that your arm is paralyzed, you'll deny it. All the marvelous excuse-generating rationalization faculties of the brain will be mobilized to mask the damage from your own sight. As Yvain summarized:
I find it disturbing that the brain has such a simple macro for absolute denial that it can be invoked as a side effect of paralysis. That a single whack on the brain can both disable a left-side motor function, and disable our ability to recognize or accept the disability. Other forms of brain damage also seem to both cause insanity and disallow recognition of that insanity - for example, when people insist that their friends have been replaced by exact duplicates after damage to face-recognizing areas.
And it really makes you wonder...
...what if we all have some form of brain damage in common, so that none of us notice some simple and obvious fact? As blatant, perhaps, as our left arms being paralyzed? Every time this fact intrudes into our universe, we come up with some ridiculous excuse to dismiss it - as ridiculous as "It's my daughter's arm" - only there's no sane doctor watching to pursue the argument any further. (Would we all come up with the same excuse?)
If the "absolute denial macro" is that simple, and invoked that easily...
Now, suppose you built an AI. You wrote the source code yourself, and so far as you can tell by inspecting the AI's thought processes, it has no equivalent of the "absolute denial macro" - there's no point damage that could inflict on it the equivalent of anosognosia. It has redundant differently-architected systems, defending in depth against cognitive errors. If one system makes a mistake, two others will catch it. The AI has no functionality at all for deliberate rationalization, let alone the doublethink and denial-of-denial that characterizes anosognosics or humans thinking about politics. Inspecting the AI's thought processes seems to show that, in accordance with your design, the AI has no intention to deceive you, and an explicit goal of telling you the truth. And in your experience so far, the AI has been, inhumanly, well-calibrated; the AI has assigned 99% certainty on a couple of hundred occasions, and been wrong exactly twice that you know of.
Arguably, you now have far better reason to trust what the AI says to you, than to trust your own thoughts.
And now the AI tells you that it's 99.9% sure - having seen it with its own cameras, and confirmed from a hundred other sources - even though (it thinks) the human brain is built to invoke the absolute denial macro on it - that...
...what?
What's the craziest thing the AI could tell you, such that you would be willing to believe that the AI was the sane one?
(Some of my own answers appear in the comments.)