Source
Luque (2016), LIT-205 (Biology & Philosophy 32(1):97–125), §3, p. 7, read in full in NOTE-101 from the author’s PhilSci-Archive manuscript. The equation is Price’s (1970); Luque states and derives it. The derivation was worked through by the promotion audit (batches 056 and 058).
What was actually shown
Setting. Ancestors i at time t, with frequency q_i, character z_i and fitness w_i, the number of descendants each leaves at t + 1. Each ancestor’s descendants have mean character z’_i, and Δz_i = z’_i − z_i. The population is closed: every entity at t + 1 descends from, or is, an entity at t, so there is no migration. Write w̄ = Σ q_i w_i and z̄ = Σ q_i z_i.
Derivation. Descendant frequencies are q’_i = q_i w_i / w̄, so the descendants’ mean character is z̄’ = Σ q_i w_i z’_i / w̄. Then
w̄ Δz̄ = Σ q_i w_i (z_i + Δz_i) − w̄ Σ q_i z_i = [Σ q_i w_i z_i − w̄ z̄] + Σ q_i w_i Δz_i = Cov(w, z) + E(w Δz),
with covariance and expectation taken over ancestors weighted by q_i. With relative fitness ω = w / w̄ this is Δz̄ = Cov(ω, z) + E(ω Δz). Only closure and the ancestor–descendant mapping are used, and the interval from t to t + 1 is arbitrary.
Equivalent partitions. Luque’s Table 1 lists others, among them Okasha’s Cov(ω, z’) + E(Δz), obtained by writing z’_i = z_i + Δz_i inside the covariance instead of the expectation. They are rearrangements of the same identity.
What could have come out otherwise. The identity could have needed an assumption beyond closure, such as additivity, discrete generations or a fixed population size. It does not.
What this does not say
- It does not say which partition of Δz̄ a given causal model licenses. Different equivalent forms assign the same change differently to “selection” and “transmission”; that question is THEORY-673’s.
- It predicts nothing on its own. Fitnesses and the transmission term must come from a model, so it is not dynamically sufficient.
- It fails without closure. With migration a further term is needed.
- It does not carry Luque’s generalization-sketch reading, the f = ma analogy or the replies to the tautology charge. Those stay in THEORY-673, which is Proposed.