What exactly is an “event set” in this context? I don’t think a hypothesis would necessarily correspond to a particular set of events that it permitted, but rather its own probability distribution over which events you would be more likely to see under that hypothesis. In that sense, an event set with no probabilities attached would not be enough to specify which hypothesis you were talking about, because multiple hypotheses could correspond to the same set of permitted events despite assigning very different probabilities to each of those events occurring.
A hypothesis corresponds to an “event” in Kolmogorov probability theory. Such an event is the set of mutually exclusive and exhaustive “outcomes” under which the hypothesis is true. An outcome is basically a possible world. So a hypothesis can be thought of as the set of (or the disjunction of) possible worlds in which the hypothesis is true. So the probability of an event/hypothesis is exactly the sum of the probabilities of its outcomes.
That being said, this way of seeing things is problematic, since most hypotheses are true in infinitely many “possible worlds”. It is not clear how we could sum infinitely many probabilities. And if we don’t use possible worlds as outcomes, but use arbitrary DNFs, the decomposition (partition) becomes non-unique, because outcomes are no longer distinguishable from normal events. Possible worlds at least have the special feature of being “atomic” in the sense that they can’t be decomposed into mutually exclusive and exhaustive disjuncts.
Ideally we would want finitely many non-arbitrary outcomes for each hypothesis. Then we could apply the principle of indifference, assign equal probabilities, and create the sum. But that doesn’t seem to work.
So the whole disjunctive approach seems not exactly promising. An alternative is Wittgenstein-style logical atomism, which supposes there are something like “atomic propositions”, presumably about sense data, which are all independent of each other. Hypotheses/events are then supposed to be logical combinations of these. This approach is riddled with technical difficulties as well, it is unclear how it could be made precise.
What exactly is an “event set” in this context? I don’t think a hypothesis would necessarily correspond to a particular set of events that it permitted, but rather its own probability distribution over which events you would be more likely to see under that hypothesis. In that sense, an event set with no probabilities attached would not be enough to specify which hypothesis you were talking about, because multiple hypotheses could correspond to the same set of permitted events despite assigning very different probabilities to each of those events occurring.
A hypothesis corresponds to an “event” in Kolmogorov probability theory. Such an event is the set of mutually exclusive and exhaustive “outcomes” under which the hypothesis is true. An outcome is basically a possible world. So a hypothesis can be thought of as the set of (or the disjunction of) possible worlds in which the hypothesis is true. So the probability of an event/hypothesis is exactly the sum of the probabilities of its outcomes.
That being said, this way of seeing things is problematic, since most hypotheses are true in infinitely many “possible worlds”. It is not clear how we could sum infinitely many probabilities. And if we don’t use possible worlds as outcomes, but use arbitrary DNFs, the decomposition (partition) becomes non-unique, because outcomes are no longer distinguishable from normal events. Possible worlds at least have the special feature of being “atomic” in the sense that they can’t be decomposed into mutually exclusive and exhaustive disjuncts.
Ideally we would want finitely many non-arbitrary outcomes for each hypothesis. Then we could apply the principle of indifference, assign equal probabilities, and create the sum. But that doesn’t seem to work.
So the whole disjunctive approach seems not exactly promising. An alternative is Wittgenstein-style logical atomism, which supposes there are something like “atomic propositions”, presumably about sense data, which are all independent of each other. Hypotheses/events are then supposed to be logical combinations of these. This approach is riddled with technical difficulties as well, it is unclear how it could be made precise.