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.

A cardinality objection to unrestricted modal profiles

The modal profile of an object tells us which worlds the object exists in and what it consists of in those worlds.

The unrestricted modal profiles (UMP) thesis says that for any map f that assigns to some worlds w a concrete object f(w) in w and that assigns nothing to other worlds, there is a possible concrete object O_f such that O_f exists in all and only the worlds w to which f assigns an object and has the property that in w, O_f is wholly composed of f(w) (or of parts of f(w) that compose f(w)).

One can think of UMP as the next step after unrestricted composition (UC) which holds that for any concrete objects there is an object composed of them. UC is not enough to guarantee the existence of ordinary objects like tables and chairs, since there is no guarantee that the modal profile of a UC-guaranteed object composed of the particles in a table will match the modal profile of a table. But UC+UMP, plus a thesis about the physical world being made of temporal parts of particles, will give us what we need here.

However, UMP is false.

1. There is a set of all actual concrete objects.
2. If UMP is true, then for any cardinality K, there are at least K actual concrete objects.
3. So, if UMP is true, there is no set of all actual concrete objects. (By 2)
4. So, UMP is not true. (By 1 and 3)

Now, claim (1) is very plausible. Claim (3) follows from (2) since for any set there is a greater cardinality by Cantor's Theorem and so it's impossible to have a set whose cardinality is at least as large as every cardinality.

That leaves (2). Assume UMP. Suppose K is any infinite cardinality (we don't need to worry about the finite case). For any K, there is a possible world w with at least K concrete objects (say, K photons). Let x be any concrete object in the actual world @. Then there will be at least K maps f with the property that f(@)=x and f(w) is a concrete object in w and f assigns nothing to any other world. To each such map f there corresponds at least one distinct object O_f in the actual world. (Distinct as difference in modal profiles implies non-identity of objects by Leibniz's Law.) So there are at least K concrete objects in the actual world.