The first way to think about probability is as a measure of the frequency of an event: if I say that the probability of a coin to land heads up is 50% I may mean that, if I flip the coin say 100 times, on average I will get heads 50 times. This is not going to get us out of Hume’s problem, because probabilities interpreted as frequencies of events are, again, a form of induction
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Secondly, we can think of probabilities as reflecting subjective judgment. If I say that it is probable that the coin will land heads up, I might simply be trying to express my feeling that this will be the case. You might have a different feeling, and respond that you don’t think it's probable that the coin will lend heads up. This is certainly not a viable solution to the problem of induction, because subjective probabilities are, well, subjective, and hence reflect opinions, not degrees of truth.
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Lastly, one can adopt what Okasha calls the logical interpretation of probabilities, according to which there is a probability X that an event will occur means that we have objective reasons to believe (or not) that X will occur (for instance, because we understand the physics of the solar system, the mechanics of cars, or the physics of coin flipping). This doesn’t mean that we will always be correct, but it does offer a promising way out of Hume’s dilemma, since it seems to ground our judgments on a more solid foundation. Indeed, this is the option adopted by many philosophers, and would be the one probably preferred by scientists, if they ever gave this sort of thing a moment’s thought.
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short summary of some probabilistic responses to the problem of induction
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