If you haven’t already, I would suggest you read Carnap’s book, The Logical Foundations of Probability (there’s a PDF of it somewhere online). As I recall, he ran into some issues with universally quantified statements—they end up having zero probability in his system.
As I recall, he ran into some issues with universally quantified statements—they end up having zero probability in his system.
Cox’s probability is essentially probability defined on a Boolean algebra (the Lindenbaum-Tarski algebra of propositional logic). Kolmogorov’s probability is probability defined on a sigma-complete Boolean algebra. If I can show that quantifiers are related to sigma-completeness (quantifiers are adjunctions in the proper pair of categories, but I’ve yet to look into that), then I can probably lift the equivalnce via the Loomis-Sikorski theorem back to the original algebras, and get exactly when a Cox’s probability can be safely extended to predicate logic. That’s the dream, anyway.
I’d be interested in reading what you come up with once you’re ready to share it.
One thing you might consider is whether sigma-completeness is really necessary, or whether a weaker concept will do. One can argue that, from the perspective of constructing a logical system, only computable countable unions are of interest, rather than arbitrary countable unions.
I’m working on extending probability to predicate calculus and your work will be very precious, thanks!
If you haven’t already, I would suggest you read Carnap’s book, The Logical Foundations of Probability (there’s a PDF of it somewhere online). As I recall, he ran into some issues with universally quantified statements—they end up having zero probability in his system.
Cox’s probability is essentially probability defined on a Boolean algebra (the Lindenbaum-Tarski algebra of propositional logic).
Kolmogorov’s probability is probability defined on a sigma-complete Boolean algebra.
If I can show that quantifiers are related to sigma-completeness (quantifiers are adjunctions in the proper pair of categories, but I’ve yet to look into that), then I can probably lift the equivalnce via the Loomis-Sikorski theorem back to the original algebras, and get exactly when a Cox’s probability can be safely extended to predicate logic.
That’s the dream, anyway.
I’d be interested in reading what you come up with once you’re ready to share it.
One thing you might consider is whether sigma-completeness is really necessary, or whether a weaker concept will do. One can argue that, from the perspective of constructing a logical system, only computable countable unions are of interest, rather than arbitrary countable unions.