Reducing de re to de dicto modality

In my previous post, I gave an initial defense of a theory of qualitative haecceities in terms of qualitative origins: qualitative haecceities encapsulate complete qualitative descriptions of an entity’s initial state and causal history. I noted that among the advantages of the theory is that it can allow for a reduction of de re modality to de dicto modality, without “the mystery of non-qualitative haecceities”.

I want to expand on this, and why qualitative-origin haecceities are superior to non-qualitative haecceities here. A haecceitistic account of de re modality proceeds in something like the following vein. First, introduce the predicate H(Q,x) which says that Q is a haecceity of x. Then we reduce de re claims as follows:

• x is essentially F ↔︎ ∀Q(H(Q,x)→□(∀y(Qy→Fx)))

• x is accidentally F ↔︎ ∃Q(H(Q,x)∧◊(∃y(Qy∧Fx))).

Granted, this involves de re modality for second-order variables like Q. But this de re modality is less problematic because we can suppose the Barcan and converse Barcan formulas to hold as axioms for the second-order quantifiers, and we can treat the second-order entities as necessary beings. De re modality is particularly difficult for contingent beings, so if we can reduce to a modal logic where only necessary beings are subject to de re modal claims, we have made genuine progress.

We will also need some axioms. Here are two that come to mind:

• ∀x∀Q(H(Q,x)→Qx) (things have their haecceities)

• ∀x∃Q(H(Q,x)) (everything has a haecceity).

Now, here is why I think that qualitative-origin haecceities are superior to non-qualitative haecceities. Given qualitative-origin haecceities, we can give an account of what H(Q,x) means without using de re modality. It just means that Qy attributes to y all of the actual qualitative causal origins of x, including x’s initial qualitative state. On the other hand, if we go for non-qualitative haecceities, we seem to have two options. We could take H(Q,x) to be primitive, which always should be a last resort, or we could try to define in some way like:

• H(Q,x) ↔︎ (□(Ex→Qx) ∧ □∀y(Qy→y=x))

where Ex says that x exists (it might be a primitive in a non-free logic, or it might just be an abbreviation for ∃y(y=x)). But this definition uses de re modality with respect to x, so it is not satisfactory in this context, and I can’t think of any way to do it without de re modality with respect to potentially contingent individuals like x.