What exactly makes modal logic intensional? In what follows, for illustration, I will focus on propositional modal logic (MPL).
I know that the modal operators in MPL are intensional since the truth values of modal formulas depends not only on the actual world but on all relevant possible worlds. Formulas whose main operator is "necessity" are true iff they are true in all possible worlds (accessible from our world). Formulas whose main operator is "possibility" are true iff they are true in at least one possible world (accessible from our world).
Moreover, the truth values of non-modal formulas in MPL are assigned per possible world. Each non-modal formula gets an interpretation and a valuation per each possible world in the model.
Now, my question is: is MPL intensional in virtue of (i) its modal operators or (ii) both its modal operators and non-modal formulas? In other words: would MPL still be intensional if we removed the modal operators from it but otherwise keep the semantics unchanged?
Another way to put my point is: Consider a logic that is exactly like MPL, except without modal operators (call this logic pseudo-modal). Pseudo-modal logic would still be evaluated using a model M = <worlds, accessibility relation, interpretation function>; however, its vocabulary would be the vocabulary of plain propositional logic. Would pseudo-modal logic be intensional? If so, in virtue of what?
Pseudo-modal logic would evaluate formulas per possible world (just like MPL). However, it would not have any formulas that are evaluated across all accessible possible worlds (i.e., formulas whose main operator is modal). Thus, would it not be the case that, in pseudo-modal logic, the extensions of the atoms evaluated per possible world fully determine the truth values of all other formulas? If so, wouldn't pseudo-modal logic be extensional instead of intensional?