Complex numbers don’t have an ordering, which seems counterintuitive as usually you can talk about things being more or less plausible. (Though they do get used in quantum mechanics for something that could be said to resemble plausibility, but that introduces a whole bunch of complicating factors that are not relevant here.)
When it comes to number systems with infinitesimals, you could often in principle use them, but in practice they aren’t relevant because infinitesimal values will almost always be outweighted by non-infinitesimal values.
I think so. Complex numbers and infinitesimals are IIRC the only possible alternatives to the reals, but complex numbers only apply to certain limited quantum contexts (roughly speaking, complex numbers apply when information is perfectly preserved, while real numbers apply in contexts where there’s information leakage into the environment), while infinitesimals can be approximated perfectly by real numbers. So in everyday contexts (which is presumably where “common sense” applies), plausibility is captured by real numbers.
Complex numbers don’t have an ordering, which seems counterintuitive as usually you can talk about things being more or less plausible. (Though they do get used in quantum mechanics for something that could be said to resemble plausibility, but that introduces a whole bunch of complicating factors that are not relevant here.)
When it comes to number systems with infinitesimals, you could often in principle use them, but in practice they aren’t relevant because infinitesimal values will almost always be outweighted by non-infinitesimal values.
(See amplitude if you want to look at the quantum generalisation of probability from R to C)
If it is not necessary to use reals, can you still say that probability theory as extended logic follows from common sense?
I think so. Complex numbers and infinitesimals are IIRC the only possible alternatives to the reals, but complex numbers only apply to certain limited quantum contexts (roughly speaking, complex numbers apply when information is perfectly preserved, while real numbers apply in contexts where there’s information leakage into the environment), while infinitesimals can be approximated perfectly by real numbers. So in everyday contexts (which is presumably where “common sense” applies), plausibility is captured by real numbers.