# Do possible worlds partition the set of all possible states of affairs?

Let S be the set of all ( logically) possible states of affairs ( I could have said " events" or " propositions" maybe).

Let R be the relation : state of affairs x is compossible/ compatible with state of affairs y.

Is this relation an equivalence relation ( maybe transitivity is questionable).

In case it would be an equivalence relation, would the partition it would operate on the set S " give" the set of possible worlds?

Briefly: is a possible world an equivalence class of the set of all possibilities?

## 1 Answer

No, they do not partition states of affairs, and not only because compossibility isn't transitive. It is not a binary relation at all: any pair from x, y, z may be compossible, but not the three of them together. For example, take x "being a right triangle", y "being an isosceles triangle", and z " triangle having a 60° angle". Right or isosceles triangles can have 60° angles, but right isosceles triangles only have angles of 90° and 45°. This is why consistency of systems is so tricky, if it was enough to check that there are no contradicting pairs of assumptions it'd be much easier.

Even intuitively, it would not make sense for possible worlds to partition states of affairs because the same states of affairs may obtain in different worlds. Nor would it make sense to partition sets of states of affairs (i.e. their powerset), because not all of them form worlds, only maximal consistent sets do. The analogy to a partition probably comes up because equivalence classes are also maximal sets of equivalent elements. But for them to partition a set the relation has to be binary, and an equivalence, which compossibility is not.