The thesis of Weak Unrestricted Composition says that every pair of objects has a fusion. This thesis has been argued by Contessa (Analysis 72(3):455–457, 2012) and Smith (Erkenntnis 84(1):41–55, 2019) to be compatible with the world being junky and hence to evade an argument against the necessity of Strong Unrestricted Composition proposed by Bohn (Analysis 69(1):27–31, 2009a, Philos Q 59(235):193–201, 2009b). However, neither Weak Unrestricted Composition alone nor the different variants of it that have been proposed in the literature can provide us with a satisfying answer to the special composition question, or so we will argue. We will then go on to explore an alternative family of purely mereological rules in the vicinity of Weak Unrestricted Composition, Cardinal Composition: A plurality of pairwise non-overlapping objects composes an object iff the objects in the plurality are of cardinality smaller than κ. As we will show, all the instances for infinite κs determine fusion and are compatible with junk, and every instance for a κ>ℵ0 is furthermore compatible with gunk and dense chains of parthood.
