While probability extends basic logic it doesn’t extended advanced logic (predicate calculus) as David Chapman argues in Probability theory does not extend logic.
I’m not convonced that probability cannot be made to extend to predicate calculus. You need to interpret “for every” and “exists” as transfinite “and” and “or”, but they are not some other abstruse ingredients impossible to fit.
As far as Chapman describes the situations various mathematicians have put a lot of effort into trying to made a system that extends probability from predicate calculus but no one succeeded in creating a coherent system.
There are two ways to disagree with that:
1) Point to a mathematician who actually successfully modeled the extension.
2) Say that no mathematician really tried to do that.
Say that no mathematician really tried to do that.
I tend to lean on this. There has been work to fix and strenghten Cox’s theorem, as also to extend probability to arbitrary preorders or other categories. I’ve yet to see someone try to extend probability to, say, intuitionistic or modal logic.
I’m not convonced that probability cannot be made to extend to predicate calculus. You need to interpret “for every” and “exists” as transfinite “and” and “or”, but they are not some other abstruse ingredients impossible to fit.
As far as Chapman describes the situations various mathematicians have put a lot of effort into trying to made a system that extends probability from predicate calculus but no one succeeded in creating a coherent system.
There are two ways to disagree with that: 1) Point to a mathematician who actually successfully modeled the extension. 2) Say that no mathematician really tried to do that.
I tend to lean on this. There has been work to fix and strenghten Cox’s theorem, as also to extend probability to arbitrary preorders or other categories. I’ve yet to see someone try to extend probability to, say, intuitionistic or modal logic.