I'd first like to make the following point: there is definitely a point at which evidence does make it reasonable to believe extraordinary claims. For example, if all of Europe claims to have seen the stars spell out a clear sentence for the past 5 nights, we should believe that the stars did spell out a clear sentence for the past 5 nights over Europe, despite the extraordinary nature of this claim.
Now, consider the following scenario.
Person A clearly believes that he has witnessed an extraordinary event in such a way that we are faced with only two possibilities: either he really did witness the extraordinary event, or he hallucinated. The extraordinary event is something which we have no evidence against or evidence in favour of other than A's claim.
For example, imagine that someone bolts out of the Amazon rainforest, very clearly terrified, claiming to have seen a monster very clearly for some time (a time that hallucinations can certainly often last for). His description makes it clear that we would not have expected to have discovered this monster if it really did exist. (Perhaps he describes it as able to materialise and dematerialise at will, extremely fast and vicious, so it's usually very hard to see or escape from if seen and it doesn't leave tracks.) This means that there is no evidence against the existence of this monster, and there is no evidence in favour of the existence of this monster prior to this person's claim.
I think that it's clear that we should believe that the person hallucinated.
However, why should believe that the person hallucinated? The probability that a given sight is a hallucination is surely something like 1 in 1,000,000. Why do we choose to believe something so improbable instead of believing the man's claim?
I have the following suggestion to answer my question above, but I'm not sure if it's correct, for a reason that I will adduce at the end of this section.
It is true that the probability that a particular sight is due to a hallucination is something like 1 in 1,000,000. However, the probability that someone will hallucinate on a randomly chosen day is assumedly quite high. The fact that this second possibility is quite high makes the proposition that this person hallucinated more reasonable to believe. (I admit that I haven't explained why the second probability being high has this effect. At the moment I can't explain it clearly.)
This reasoning is apparently very similar to the reasoning we use to see why it's reasonable to believe that the particular person who won the lottery won it fairly. One might have said the following: "The probability of a particular person will fairly win the lottery is about 1 in 14,000,000. That is a tiny probability. Therefore, it is more reasonable to believe that the particular person who won the lottery won it unfairly." Using the paragraph above, we may explain why this is incorrect. It is true that it is very unlikely that a particular person will win the lottery fairly, but it is almost certain that someone will win the lottery fairly if nobody interferes.
What makes me unsure about my reasoning is the following. Imagine that a mass extinction event had previously occurred, and that person A and the person who hears A's claim are the only humans left alive in the universe. In this case, the probability that someone will hallucinate on a randomly chosen day becomes extremely small due to the tiny remaining human population. If so, according to my reasoning above, it would become more reasonable to believe that A saw what he claims to have seen than to believe that he hallucinated. This might be correct, but I am not sure.
Is my suggestion correct?
EDIT
I am just adding this edit to clarify and emphasise a few points and to avoid confusion and misinterpretation.
- In my Europe case, I specifically say that the case I am making up is when all of Europe actually claims to have seen the stars spell out a clear message.
- The scenario that my question is about is only when the person 'clearly believes that he has witnessed an extraordinary event'. (This precludes the possibility that the person is lying.)
- The scenario that my question is about is only when 'we are faced with only two possibilities: either he really did witness the extraordinary event, or he hallucinated'. (This precludes the possibility that anything else is responsible for his belief that he witnessed the event.)
- The scenario that my question is about is only when the 'extraordinary event is something which we have no evidence against or evidence in favour of other than A's claim'. (This precludes scenarios where we would have expected to be aware of what A claims to have seen if it really existed, for example.)
- I did not say that the probability that a person will hallucinate is roughly 1 in 1,000,000. I said that the probability that a given sight is a hallucination is roughly 1,000,000.
- In the lottery case that I mention, I am specifically talking about 'the particular person who won the lottery'. I am not talking about someone claiming that he won the lottery with no evidence to his claim.
- The mass extinction event that I mention is nothing to do with the essence of the story I talk about. I only intended to provide a quick explanation for why in my story there are only 2 people left alive in the universe. If it's confusing, please ignore it: the point is that in my story, there are only 2 people left alive in the world.
Apologies if I didn't make these points clear enough in my question.