 (Here, ‘Sizφw’ may be read: ‘z is a sumi of every w such that φw’ and, again, ‘z’ and ‘v’ are assumed not to occur free in φ; similar restrictions will apply below.) Thus, each (P.15ψ,i) says that if there are some φ-ers, and if every φ-er satisfies condition ψ, then the φ-ers have a sum of the relevant type. It can be checked that each variant of (P.15ψ,i) includes the corresponding finitely principle (P.12ψ,i) as a special case, taking ‘φw’ to be the formula ‘w=x ∨ w=y’ and ‘ψw’ the condition ‘(w=x → ξwy) ∧ (w=y → ξxw)’. And, again, it turns out that in the presence of Strong Supplementation, (P.15ψ,2) and (P.15ψ,3) are equivalent. 