I did a search for "epistemic Barcan formula" and got only one result, with the sample being:
An example with the Barcan formula in an epistemic context shows that our intuitions are much clearer in this case. The rejection of the epistemic Barcan formula is straightforward since the possible truth of Romeo knows that someone likes him, but he doesn’t know who := Kr∃xLxr ∧ ¬∃xKrLxr shows that Kr∃xLxr → ∃xKrLxr is not a truth of epistemic logic.
From skimming that essay, I get that one can represent a scheme of Barcan formulas by generalizing over the non-quantifier operator. (Searching for "generalized Barcan formula" delivered a number of results, e.g. this one, but I haven't yet checked to see if the sense of the phrase, there, is exactly what I'm wondering about.) I also read somewhere in the SEP (I don't recall where precisely) that even the ◊-operator can be understood, though, as a quantifier after a fashion, or as being quantifier-ish (I suppose that, since I accept Kant's take on the modal categories, I should have no objection to reading possibility/necessity in this way). But so putting all that together, I want to distinguish between two dimensions of generalization over the Barcan formula (letting ☂ be for non-quantifiers generally):
- ☂∃xFx → ∃x☂Fx
- ☂A☂BF → ☂B☂AF
For example, let k be "it is known that":
- Does ◊kF = k◊F?
At first I thought I had a counterexample using OB ("it is obligatory that") and k, i.e. it seemed "wrong" to say:
- OBkF = kOBF
But on second thought, I wasn't so sure that that's "wrong" on its own terms so much as lacking in nuance (taken strictly, it seems plainly true, but we have outside reasons to ambiguate the purpose of OB).
Question: supposing something like (2) is admissible for ☂A, B, which permutations are admissible for ☂A, B, C or then ☂A, B, C ... X? From skimming the SEP article on combining logics, it looks like I'm asking about what they call "fusion." But I tried "Find: Barcan" on that article and it gave no results, so either I'm misusing the phrase "Barcan formula" in the first place or maybe generalizations over that formula always have to involve quantifiers?