Propositional consistency lets us express constraints between sentences (such as ”A and B cannot both be true”) as sentences (such as “¬(A&B)”) in a way the prior understands and correctly enforces.
Any branch contradicting an already-stated constraint is clipped off the tree of possible sequences of sentences.
The probability of any sentence S which is consistent with everything seen so far can’t go below μ(S) or above 1−μ(¬S), since S or ¬S can be drawn next. So, no trolling.
How do I know whether S is consistent with everything seen so far. Doesn’t that presuppose logical omniscience?
Or does consistency here only mean that it doesn’t violate any explicitly stated constraints (such that I don’t have to know all the implications of all the sentences I’ve seen so far and whether they contradict S)?
There’s a difference between “consistency” (it is impossible to derive X and notX for any sentence X, this requires a halting oracle to test, because there’s always more proof paths), and “propositional consistency”, which merely requires that there are no contradictions discoverable by boolean algebra only. So A^B is propositionally inconsistent with notA, and propositionally consistent with A. If there’s some clever way to prove that B implies notA, it wouldn’t affect the propositional consistency of them at all. Propositional consistency of a set of sentences can be verified in exponential time.
Since propositional consistency is weaker than consistency our prior may distribute some probability to cases that are contradictory. I guess that’s considered acceptable because the aim is to make the prior non-trollable, rather than good.
How do I know whether S is consistent with everything seen so far. Doesn’t that presuppose logical omniscience?
Or does consistency here only mean that it doesn’t violate any explicitly stated constraints (such that I don’t have to know all the implications of all the sentences I’ve seen so far and whether they contradict S)?
There’s a difference between “consistency” (it is impossible to derive X and notX for any sentence X, this requires a halting oracle to test, because there’s always more proof paths), and “propositional consistency”, which merely requires that there are no contradictions discoverable by boolean algebra only. So A^B is propositionally inconsistent with notA, and propositionally consistent with A. If there’s some clever way to prove that B implies notA, it wouldn’t affect the propositional consistency of them at all. Propositional consistency of a set of sentences can be verified in exponential time.
Since propositional consistency is weaker than consistency our prior may distribute some probability to cases that are contradictory. I guess that’s considered acceptable because the aim is to make the prior non-trollable, rather than good.