 Some philosophers spell out “involvement” as containment: A state of affairs is a complex that contains properties and particulars as parts. In order to assess this proposal, let us first sharpen our understanding of the part-of relation. It is controversial weather the relation we aim to pick out with “part-of” is reflexive (everything is part of itself) or not. But it is not controversial that the relation is transitive (if x is a part of y, and y is a part of z, x is a part of z) and antisymmetrical (if x is part of y, and y is a part of x, x is identical to y) (see the entry on  mereology,  sect. 2 and 3). A further plausible principle governing our part-of relation is the supplementation principle: if x is a part of y, there is an object z that is part of y and z does overlap, that is, have a part in common, with x. The supplementation principle implies the intuitively plausible view that nothing can have a single proper part. The supplementation principle together with the uncontroversial axioms for the (proper) part-of relation also implies the more controversial thesis that complex objects with the same parts are identical (see entry on  mereology,  3.2).