On the Types of Answers to the Special Composition Question

There are three broad types of answers that can be given to the Special Composition Question. They are divided by their position on the number of composite objects relative to objects in general:

(N) [∀x: x ∈ M] ~∃y(Pxy ^ x =/= y)
(U) [∀x: x ∈ M] [∀y: y ∈ M] ∃z(Pxz ^ Pyz)
(C) ~(U) ^ ~(N), or [∃x: x ∈ M] ∃y(Pxy ^ x =/= y) ^ [∃x: x ∈ M][∃y: y ∈ M] ~∃z(Pxz ^ Pyz)

(N), or nihilism, claims that there are no cases of composition. All material objects that exist have no proper parts (proper parts satisfy Pxy and x=/=y). Thus, no matter how I arrange my bread, meat, and cheese, they will never compose another object. All material objects are simple.

(U), or universalism, claims that for any two distinct objects (x =/= y), they compose an object. Thus, the two slices of bread compose an object. That object and the meat compose an object. And that object and the cheese compose another object. This theory still allows for simple objects. Note, however, that an object cannot be a part of a composite object if another of the objects parts already contains the first as a part. That is, all objects can be parts only once; there is no double dipping of parthood. Consider objects A, B, and 3. Object 3 is the composite of objects A and B. It is impossible for there to be a 4th object that is composed of object 3 and object B. This is because object B is a part of object 3.

(C), compatiblism, claims that composition sometimes occurs. This answer is logically incompatible with either (N) or (U). Nihilism and compatibilism disagree on there being at least one composite object. Universalism and compatibilism disagree on there being an instance of failed composition. All three disagree on the number of material objects (or at least the number of potential material objects).