In propositional modal logic, we have a clear distinction between models and frames. A frame is a pair of a set W and a binary relation R on W. A model is a frame plus a valuation function that maps atomic sentence letters to subsets of W. Given this distinction, we can define truth at a point in a model, truth on a model (i.e. truth at all points on a model) and truth on a frame (i.e. truth at all points on all models on the frame.)

In the previous post I was considering logics which, as well as modal operators, had actuality operators and quantifiers. Given that extra expressive capacity, where should we draw the frame/model distinction? There might well be a well established precedent for this, but I wasn’t sure what it was. And it might be that different distinctions are useful for different purposes.

Here are two more specific questions.

First, if we have an actuality operator, we need a way of designating one world as actual in each model. Is this something we do at the level of the frame, or is it something done by something like the valuation function?

Second, is the number of individuals in each world set by the frame, or do different models on the same frame have different individuals in the worlds?

As I said, I imagine there are conventional answers to these questions, and the conventions are probably well motivated. But I wasn’t sure what they were.