This is an addendum to Trope Theory Meets Bradley's Regress. In that paper I touched upon the question whether the compresence relation is dyadic or not, but did not delve into the matter in any depth. Now I will say a little more with the help of George Molnar's excellent discussion in Powers: A Study in Metaphysics (Oxford 2003), pp. 48-51. Molnar draws upon Peter Simons, "Particulars in Particular Clothing: Three Trope Theories of Substance (Philosophy and Phenomenological Research, September 1994, 553-575), which I have also consulted.
On the theory we are considering, ordinary particulars are bundles of properties, where properties are assayed as tropes. The bundling relation, which makes unitary things from collections of tropes, is standardly called 'compresence.' But now we need to consider the 'adicity' of this relation and how it bears upon Bradley's regress. There seem to be three possibilities: (1) Compresence is a dyadic or two-termed relation that holds pair-wise between all the tropes in a trope-bundle. (2) Compresence is a triadic or three-termed relation which holds among a trope, a distinct trope, and a spatiotemporal position. The relation would then have the form: T1 is compresent with T2 at position P. (3) Compresence is an n-adic relation, n being the number of tropes in the bundle. So if there were one thousand tropes in a bundle, the compresence relation for that bundle would be chiliadic! On the third option there is exactly one compresence relation for each bundle.
Ad (1): Compresence as Dyadic. Compresence, no matter what its 'adicity,' is a relation that must satisfy the following two constraints. (A) It must connect every trope in the bundle to every other trope. Suppose a bundle consists of non-relational tropes T1, T2, and T3. Then dyadic compresence connects T1 to T2, T2 to T3, and T1 to T3. The result is that each trope is connected to every other one. (B) The second constraint is that compresence is itself a trope assuming, as we are, that trope theory is a one-category ontology, an ontology that attempts to account for everything with tropes and constructions from tropes. Thus the relation of compresence occurs in reality as one or more particulars (unrepeatables), and not as a multiply exemplifed universal. So in the example just given there are, in addition to the three non-relational tropes T1, T2, and T3, three relational compresence tropes C1, C2, and C3.
But now it seems that dyadic compresence gives rise to a vicious infinite regress. In the minbundle we are considering there are six tropes: three ordinary tropes and three compresence tropes (C-tropes). But now it seems there must be 15 additional C-tropes to connect the six tropes just mentioned. For if the six tropes are a, b, c, d, e, f, then constraint (A) demands that each be connected to every other one. This yields 15 more C-tropes: Cab, Cbc, Ccd, Cde, Cef, Cac, Cad, Cae, Caf, Cbd, Cbe, Cbf, Cce, Ccf, Cdf. (Since compresence is a symmetrical relation, Cxy = Cyx.)
We are thus embarked upon an infinite regress. One might try to stop it before it starts by denying constraint (A) or constraint (B). We cannot drop the first constraint since the job of compresence is to tie together ALL the tropes in a trope-bundle. And we cannot drop the second constraint, according to which compresence is itself a trope, because, first, we need a unifier to connect tropes, and second, in a one-category ontology that connector can only be a trope.
Might one prevent the regress by claiming that, e.g., C1 (which connects T1 to T2) somehow includes within itself its relata T1 and T2, and is individuated by them, so that C1 cannot exist except as connecting T1 and T2? This, roughly, is Anna-Sofia Maurin's solution which I discuss and reject in section 5 of Trope Theory Meets Bradley's Regress.
The infinite regress would thus appear unavoidable. But is it vicious? One might think that dyadic compresence supervenes or is founded upon the natures of its relata. If two tropes were such that they could not exist without being compresent, then their compresence would supervene upon their natures and there would be no need for any additional entities to connect the C-trope to what it connects. But most tropes are merely contingently related to one another, which entails that their mere existence does not suffice to make them compresent. Here is a homey example. I make a meatball which has a particular, quite determinate, spheroid shape. This shape trope must be connected to some size trope or other, but there is no necessity that it be connected to some one determinate shape trope. It is a contingent fact that my meatball has a diameter of one inch. So if it does have a diamater of one inch, then the shape trope and the size trope are contingently connected, which is to say that their compresence cannot be supervenient upon their natures.
Upshot: dyadic compresence succumbs to Bradley's regress!
Ad (2): Compresence as Triadic. The idea here is that the bundling relation is three-termed: T1 is compresent with T2 at spatiotemporal position P. Thus the shape trope and the size trope of my meatball are compresent now on my counter. Is this consistent with the meatball's being in my frying pan a moment from now? Rather than examine this question, I will just point out that the regress objection that spells the doom of dyadic compresence also knocks out triadic compresence.
Ad (3): Compresence as N-Adic. This is is the idea that the bundling relation, compresence, has as many terms or relata as the number of tropes in the bundle. So if an ordinary particular is a bundle of one thousand nonrelational tropes, then that one bundle is held together by one chiliadic compresence trope. Molnar states that "On this account of compresence, no regress arises." I don't understand why he says this. Suppose you have a trope bundle consisting of five nonrelational tropes, and suppose compresence is n-adic. Then the bundling relation for this trope-bundle will be a pentadic or five-termed compresence relation. But by constraint (B) above, compresence is itself a trope. Thus our bundle with contain at least six tropes. And by constraint (A) above, compresence must connect every trope in a bundle to every other one. So the pentadic C-trope must be connected to the tropes it connects. Since this C-trope does not supervene on the natures of the tropes it connects, there is need for a six-termed C-trope to connect it to what it connects. But then the infinite regress is up and running in. So, unless I am missing something, an infinite regress arises whether compresence is dyadic, triadic, or n-adic.
One criticism that Molnar makes (49) echoing Simons (561) is that n-adic compresence "turns all the properties of an object into essential properties, that is, essential members of the bundle." How then is accidental change to be accounted for? A second problem Molnar raises is this. Suppose n-adic compresence unifies the tropes in object O1 while m-adic compresence unifies the tropes in object O2. If O1 and O2 have unequal numbers of properties, as seems possible, then the respective C-tropes witll have different 'adicities.' But then what makes them both compresence tropes? It cannot be their exact resemblance since their 'adicities' are different. How might a one-category trope ontology account for the fact that both C-tropes are compresence tropes?