Source
Hartmann & Trpin (2026), LIT-165 (Philosophical Studies 183:1817–1843), read in full in NOTE-168, with the Appendix proofs. The reader recomputed the chain and collider examples of §5.1 and the OLI counterexample, and all three reproduce.
What was actually shown
The measure (Definition 1, p. 1829). For a set H of explanantia and an explanandum E, E_Coh(E; H) = [Coh(H ∪ {E}) − Coh(H ∪ {¬E})] / [Coh(H ∪ {E}) + Coh(H ∪ {¬E})]. The base coherence measure CohOG+ is Olsson–Glass overlap divided by its value under the independence distribution with the same marginals (Eq. 3).
Proposition 1. With Shogenji coherence the measure is the Schupbach–Sprenger measure generalised to n explanantia. It is proved in the Appendix; the printed proof has a sign typo in its penultimate line.
Proposition 2. With CohOG+, if P(E|H1) > P(E|H2) and P(E|¬H1) < P(E|¬H2), then {H1} has more explanatory power than {H2}. This is TLI, for single-hypothesis sets only. The Appendix also gives a counterexample to OLI: h = .5, p = .603, q = .103 gives .59; h = .01, p = .65, q = .35 gives .55, with P(E) = .353 in both.
The chain H1 → H2 → E (P(H2|H1) = .9, P(H2|¬H1) = .1, P(E|H2) = .5, P(E|¬H2) = .1). {H1, H2} beats {H2} for P(H1) < .7 (Fig. 1): .768 against .748 at P(H1) = .05. The order reverses at high P(H1): .179 against .207 at .9. The distal cause alone never beats the proximal one (fn. 13).
The collider H1 → E ← H2 (P(H1) = P(H2), P(E|H1,H2) = .7, P(E|H1,¬H2) = P(E|¬H1,H2) = .71, P(E|¬H1,¬H2) = .01). {H1, H2} beats {H1} for P(H1) < .75 (Fig. 2): .611 against .585 at .4.
What could have come out otherwise. The derivative argument for TLI could have failed, and the richer set could have lost at every prior. The reader’s recomputation is what makes the two examples part of the finding rather than the paper’s report of it.
What this does not say
- That Roche and Sober’s theorems are wrong. They still hold for conjunctions, as the authors grant (p. 1834). The measure escapes them by scoring {H1, H2} differently from H1 ∧ H2. Whether to use a set or a conjunction is a modelling choice the paper leaves to judgment: a set when each explanans “retain[s] explanatory relevance” (p. 1828).
- That the measure is a TLI measure in Roche and Sober’s sense. The authors call it one “in the technical sense, but not in the structurally constrained sense that underwrites Roche and Sober’s theorems” (p. 1832).
- When a richer set wins. Two hand-picked parameterisations are existence results. “It is not clear under which conditions” the interacting cause wins (p. 1836), and the advantage is not monotone in the prior.
- That coherence-based measures in general behave so. Other base measures (Douven–Meijs, Fitelson) are said to work on “preliminary computational evidence” not shown. The class of “structurally proper” coherence measures is uncharacterised, and CohOG+ is defended in a separate paper.
- That Lange’s entailment problem is solved. The reply, that auxiliaries should not be counted in the explanans, applies to probabilistic measures generally, rests on a stipulated example, and the authors “do not claim that this resolves Lange’s challenge completely” (p. 1821).
- All of Roche and Sober’s problems. Two of four are set aside by fiat (fn. 4), and only a binary explanandum is treated.
- Anything about weighing explanations of training phenomena. The measure needs a joint prior over the explanantia, which such comparisons do not have (NOTE-168).
Connections
The record holds nothing else on explanatory power or coherence measures. LIT-156 (reciprocal causation) is the nearest philosophy-of-science neighbour and is not engaged.