 and ‘ψw’ expresses a condition at least as strong as ‘∀x(φx → Owx)’ (again, unless one assumes the Bottom principle (P.10)). This principle includes the finitary version (P.13ξ) as a special case, taking ‘φw’ and ‘ψw’ as above, so the remarks we made in connection with the latter apply here. An additional remark, however, is in order. For there is a sense in which (P.16ψ) might be thought to be redundant in the presence of the infinitary sum principles in (P.15ψ,i). Intuitively, a maximal common overlapper (i.e., a product) of a set of overlapping entities is simply a minimal underlapper (a sum) of their common parts; that is precisely the sense in which a product principle qualifies as a composition principle. Thus, intuitively, each of the infinitary sum principles above should have a substitution instance that yields (P.16ψ) as a theorem, at least when ‘ψw’ is as strong as indicated. However, it turns out that this is not generally the case unless one assumes extensionality. In particular, it is easy to see that (P.15ψ,3) does not generally imply (P.16ψ), for it may not even imply the binary version (P.13ξ). This can be verified by taking ‘ξxy’ and ‘ψw’ to express just the requirement of overlap, i.e., the conditions ‘Oxy’ and ‘∀x(φx → Owx)’, respectively, and considering again the non-extensional model diagrammed in Figure 4. In that model, x and y do not have a product, since neither is part of the other and neither z nor w includes the other as a part. Thus, (P.13ξ) fails, which is to say that (P.16ψ) fails when ‘φ’ picks out the set {x, y}; yet (P.15ψ,3) holds, for both z and w are things that overlap exactly those things that overlap some common part of the φ-ers, i.e., of x and y. 