Good example. I assign B = 0.5 in all three cases, but I expect the (unknown) value of F to be very similar (and close to 0 or 1) for all three, unlike in the case of three coin flips.
The above probability assessments are only coherent if you judge that proposition 1 has the same truth value as proposition 2; I don’t know how that could be justified.
I might bet on B or C against A or D at odds of epsilon to 1, to be settled when we have thoroughly explored Mars, assuming that if there is life, we will find it. This of course depends on the actual value of epsilon.
Good example. I assign B = 0.5 in all three cases, but I expect the (unknown) value of F to be very similar (and close to 0 or 1) for all three, unlike in the case of three coin flips.
The above probability assessments are only coherent if you judge that proposition 1 has the same truth value as proposition 2; I don’t know how that could be justified.
Now, without using probabilities of 0 or 1, can you coherently assign probabilities to
Sure. B(A) = B(D) = 0.5, B(B) = B(C) = epsilon. (The 0.5 is only good to one significant figure, and even that’s a stretch.)
Just how small is this epsilon? I might want to propose a bet.
If I had a number, I would’ve given the number instead of saying “epsilon” :) What’s your proposed bet?
I might bet on B or C against A or D at odds of epsilon to 1, to be settled when we have thoroughly explored Mars, assuming that if there is life, we will find it. This of course depends on the actual value of epsilon.
So basically you’re saying that the probability of there being life on only one of the hemispheres is arbitrarily small?
Mathematically nonzero, but small enough that we can treat it as zero for practical purposes, yes.