Carl, that is a good point. I’m not quite sure what to say about such cases. One thing that springs to mind though, is that in realistic examples you couldn’t have investigated each of those options to see if it was a real option and even if you could, you couldn’t be sure of all of that at once. You must know it through some more general principle whereby there is, say, an option per natural number up to a trillion. However, how certain can you be of that principle? That is isn’t really up to only a million?
Hmmmm… Maybe I have an example that I can assert with confidence greater than one minus a billionth:
‘The universe does not contain precisely 123,456,678,901,234,567,890 particles.’
I can’t think of a sensible, important claim like Eliezer’s original one though, and I stand by my advice to be very careful about claiming less than a billionth probability of error, even for a claim about the colour of a piece of paper held in front of you.
Carl, that is a good point. I’m not quite sure what to say about such cases. One thing that springs to mind though, is that in realistic examples you couldn’t have investigated each of those options to see if it was a real option and even if you could, you couldn’t be sure of all of that at once. You must know it through some more general principle whereby there is, say, an option per natural number up to a trillion. However, how certain can you be of that principle? That is isn’t really up to only a million?
Hmmmm… Maybe I have an example that I can assert with confidence greater than one minus a billionth:
‘The universe does not contain precisely 123,456,678,901,234,567,890 particles.’
I can’t think of a sensible, important claim like Eliezer’s original one though, and I stand by my advice to be very careful about claiming less than a billionth probability of error, even for a claim about the colour of a piece of paper held in front of you.
Toby.