# Are there "generalized Barcan formulas" in combining logic?

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 := KrxLxr ∧ ¬∃xKrLxr shows that KrxLxr → ∃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):

1. ☂∃xFx → ∃xFx
2. ABF → ☂BAF

For example, let k be "it is known that":

1. Does ◊kF = kF?

At first I thought I had a counterexample using OB ("it is obligatory that") and k, i.e. it seemed "wrong" to say:

1. 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?

• I can't offer much help here other than to say the Barcan formulas do not seem to generalise to modalities other than necessity and are controversial even there. Another one you might like to consider is probabilistic: there is a difference between "it is probable that there exists a lottery ticket that will win" and "there exists a lottery that will probably win". There has been some work done in combining probability theory with quantifer logic, but it is still in its infancy. Jun 12 at 23:43
• I do not think "combining logic" is a thing, SEP is talking about combining different types of modal operators in a single system. Are you asking whether (one way) commutation relations hold for them in such systems? Is it specifically concerning fusions, or any logics, with meaningful semantics, that include two types of operators (deontic and epistemic, say)? Either way, I would not expect people to call commutation relations "Barcan formulas" in general. And for the latter, it should depend on intended semantics of modalities, so is the question which modalities commute (at least one way)? Jun 13 at 19:39
• @Conifold yeah, commutativity of modal operators must be what I'm asking about. IDK why I thought "generalized Barcan formula" had the intended meaning except that many years back I had thought "possible justification = justified possibility" was an example of her scheme (but that was because I was writing an essay and wanted to sound "informed" by dropping her name and my logic education was limited, so I thought commutativity was only a mathematical relation). Jun 13 at 19:47
• Goller-Jung, Complexity of Decomposing Modal and First-Order Theories, p.43:"An axiomatization for K × K can be obtained from the axiomatization for the fusion by adding axioms postulating that the two modal operators commute and have the Church-Rosser property. Thus, combinations between fusion and product can be obtained by dropping either Church-Rosser or half of commutativity." This sounds like formal games, I am more curious which interpreted modalities can be expected to commute, and what it means semantically. Jun 14 at 1:10