TYL: So8res is using the word “beliefs” in a slightly idiosyncratic way, to refer to things one simply treats as true and as fit subjects for logical rather than probabilistic inference.
Logical inference is probability-preserving. It does not require that you assign infinite certainty to your axioms.
If your axioms are a1,a2,a3,...,a10 each with probability 0.75, and if they are independent, then a1 & a2 & … & a10 (which is a valid logical inference from a1, …, a10) has probability about 0.06. In the absence of independence, the probability could be anywhere from 0 to 0.75.
Perhaps by “probability-preserving” you mean something like: if you start with a bunch of axioms then anything you infer from them has probability no smaller than Pr(all axioms are correct). I agree, logical inference is probability-preserving in that sense, but note that that’s fully compatible with (e.g.) it being possible to draw very improbable conclusions from axioms each of which on its own has probability very close to 1.
Logical inference is probability-preserving. It does not require that you assign infinite certainty to your axioms.
If your axioms are a1,a2,a3,...,a10 each with probability 0.75, and if they are independent, then a1 & a2 & … & a10 (which is a valid logical inference from a1, …, a10) has probability about 0.06. In the absence of independence, the probability could be anywhere from 0 to 0.75.
Perhaps by “probability-preserving” you mean something like: if you start with a bunch of axioms then anything you infer from them has probability no smaller than Pr(all axioms are correct). I agree, logical inference is probability-preserving in that sense, but note that that’s fully compatible with (e.g.) it being possible to draw very improbable conclusions from axioms each of which on its own has probability very close to 1.