What is a "sphere model"?

Question

I'm reading the SEP entry on multi-modal logic and there's this passage:

Is this related to what the SEP entry on infinity says about probability, here?:

Kolmogorov notes that if the original probability space is the uniform distribution of points on a sphere, and if B ranges over the set of longitudes (great circles through the poles), then probability conditional on a line of longitude will not be uniform, but instead will be concentrated near the equator. (This fact is known as the “Borel paradox”, because Emile Borel investigated it even before Kolmogorov’s work.) Since every great circle on a sphere can be viewed as a line of longitude with an appropriate choice of pole, this makes the probability conditional on an event depend not only on which event was chosen, but also which family of alternatives it is contrasted with. (We can view each great circle as a longitudinal line through multiple different poles, each of which disagrees about where the equator is.)

Is the second quote in part about the same kind of thing, a "sphere model," as the first quote? Do sphere models have any significant relation to logical lattices, like as alternatives or something? I'm finding this kind of talk to be both intuitive and counterintuitive, oddly. It reminds me of my question about a truth-value sphere.

Answer 1

In context, the article is talking about epistemic or evidence logics that are intended to express the relation between belief and evidence. The beliefs that a rational agent may form when presented with some fresh evidence are underdetermined by the evidence itself. But it may be possible to organise beliefs into those that are more or less plausible depending on the epistemic state of the agent.

Now suppose we use the language of possible worlds and say that a possible world corresponds to a particular (coherent) set of beliefs of an agent. The actual world represents the current epistemic state of the agent. Then to say some beliefs are more plausible than others can be represented as saying that some possible worlds are closer to the actual world than others.

There is no special reason to believe that a particular set of beliefs is uniquely plausible, so beliefs may be organised into collections that have equal degrees of plausibility. If we picture the actual world as surrounded by a universe of possible worlds, we can think of these collections of equally plausibile worlds as concentric spheres that surround the actual world. The closer ones are more plausible and so are in some way easier to reach when we perform belief revision.

The idea of spheres of possible worlds is at least as old as David Lewis' account of counterfactuals (1973) in which he allows that there may be collections of possible worlds that are equally close to the actual world for the purpose of assessing the truth value of counterfactual conditionals.

I don't see these spheres as related to the probability space passage you quoted.