How do we justify our knowledge of the external world?
Knowledge of the external world seems to be fallible in any case if we put the threshold of success at the highest level, namely 100% certainty. But this still raises a question: if we want to avoid complete skepticism, how can we be certain that our knowledge is at least likely to be true? In order to create a probability about the validity of our knowledge of the external world we need to start from perception. The problem is that we can be certain of the existence of perception but not the source of it (the matrix/the real world), and that is essential for the knowledge of the external world.
In order to calculate our probability we then need the number of possible events E and the one favourable event F we're looking for:
E = 2 possible events are external source or non-external source (matrix, hallucination, dream etc.)
F = 1 favourable event i.e. external source
P(F) = F/E = 1/2 = 50%
It seems to me that both possibilities are equally likely....
Setting external world skepticism aside for a moment, suppose I'm about to roll a die. Now there are two possibilities: it will come up 1 or it won't. If I reason as you did, I will conclude that the probability is 1/2 that the die will come up 1. Something has gone wrong here. For one thing, we can't get the answers to probability questions just by counting. There are many ways to slice up the space of possibilities, and if we use your rule, the answer we get will depend on how we do the slicing. This is a well-known problem, and there is no simple fix. But there's another problem: the probabilities here aren't chances. They are degrees of belief. Even if we thought (though we shouldn't) that the right way to slice things up is that our experience has an external source or it doesn't, without adding nothing more fine-grained, we don't have to agree that the two possibilities are equally probable. You say "it seems to me that both possibilities are equally likely." It's worth wondering whether you...