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> In fact, logic can do things probability theory can’t. However, despite much hard work, no known formalism completely unifies the two! Even at the mathematica

> In fact, logic can do things probability theory can’t. However, despite much hard work, no known formalism completely unifies the two! Even at the mathematical level, the marriage of rationality and empiricism has never been fully consummated. >Furthermore, probability theory plus logic cannot exhaust rationality—much less add up to a complete epistemology. I’ll end with a very handwavey sketch of how we might make progress toward one. Nevermind the handwavey plan of attack that follows, the preceding premise seems a bit too handwavey to begin with for my needs before I invest in further reading. Statements like "logic can do things that probability can't" beg for a citation or at least a proof by counterexample. I'm interested, but the initial argument needs more supporting evidence - maybe it's just me.

foerbert 5y ago
This is just a blind guess, but the difference may come when talking about impossibilities and/or extreme improbabilities. Empirically there might not be a way to distinguish between actually impossible and merely unobserved/improbable. If an event is so unlikely that it probably has never occurred in the observable universe, how would you empirically differentiate between the two? Meanwhile I would think rationality and logic would face no such obstacle. (Naturally rationality has it's own limitations, but that's a whole new topic.)
cevi 5y ago
Quantifiers are what is missing in probability theory but present in logic - while we could potentially write down mixtures of quantifiers and probabilistic symbols, we don't have great ways to interpret them or to reason about them. That is the point he is trying to make.
- drdeca 5y ago
  I don’t see how that is, at least when quantifying over a countable set. A sigma-algebra is closed under countable unions and countable intersections, and that lets us produce events corresponding to quantification over a countable set. Like, we can talk about the probability that a particular sum of random variables converges, by handling quantifiers in this way. That seems like handling quantifiers and probability together well. Now, nesting “the probability of” type expressions, is maybe another story? But I’m not sure that that’s a problem with “extending logic”

foerbert 5y ago

This is just a blind guess, but the difference may come when talking about impossibilities and/or extreme improbabilities. Empirically there might not be a way to distinguish between actually impossible and merely unobserved/improbable. If an event is so unlikely that it probably has never occurred in the observable universe, how would you empirically differentiate between the two? Meanwhile I would think rationality and logic would face no such obstacle. (Naturally rationality has it's own limitations, but that's a whole new topic.)

cevi 5y ago

Quantifiers are what is missing in probability theory but present in logic - while we could potentially write down mixtures of quantifiers and probabilistic symbols, we don't have great ways to interpret them or to reason about them. That is the point he is trying to make.

drdeca 5y ago
I don’t see how that is, at least when quantifying over a countable set. A sigma-algebra is closed under countable unions and countable intersections, and that lets us produce events corresponding to quantification over a countable set. Like, we can talk about the probability that a particular sum of random variables converges, by handling quantifiers in this way. That seems like handling quantifiers and probability together well. Now, nesting “the probability of” type expressions, is maybe another story? But I’m not sure that that’s a problem with “extending logic”