{"path":"research/strength-calculus-literature/bayesian-and-informal-logic.md","content":"# The link calculus seen from Bayesianism, confirmation theory and informal logic\n\n**Date**: 2026-09-28 · **Type**: literature review, one half of a pair (the sibling report in this\ndirectory covers computational gradual semantics) · **State**: research only, nothing ruled. Written\nfor the calculus choices that ledger Q33 left open, named D1–D8 below.\n\n**The request, verbatim** (founder, 2026-09-28): *\"I want to make conscious choices about such crucial\ndesign and epistemic-calculus considerations and want to deeply consider everything academia has\nproduced that bears on these decisions generally and specifically about these tidbits. Find out much\nmore about all relevant publications, starting with what's documented in the corpus and if anything\nrelevant to this and the relationship to bayesianism etc has been published and how directly it is\nrelevant.\"*\n\n**The decisions graded throughout**: D1 how the two factors combine · D2 raw versus offset (does an\nunexamined supporter add, does a weakened one subtract) · D3 fixed point versus one pass · D4 what an\nunexamined link presumes · D5 sorting critical questions into premise and connection · D6 how answers\nto critical questions weigh · D7 who sets relevance, and how to elicit it · D8 accrual (several\nsupporters, independence, double counting, padding).\n\n**Notation.** C is the claim a link supports or attacks (the target). P is the claim the link starts\nfrom (the supporter). q is the probability of P (its acceptability) and q0 the reference value it is\nmeasured from; the platform uses 0.5. a1 = P(C|P), a0 = P(C|¬P), Δ = a1 − a0. When C explains P (P is\na sign, report or symptom of C): a = P(P|C), b = P(P|¬C).\n\n**How this was checked.** Formulas are either read in the cited source or derived here and checked\nnumerically with a throwaway script (every number below is reproducible from the formulas given).\nSources not read are marked *not read* or UNVERIFIED. Grades: **DIRECT** bears on the decision as\nstated; **ADJACENT** bears after translation or in a neighbouring setting; **BACKGROUND** is context.\nEach entry lists only its DIRECT and ADJACENT grades; every decision not listed is BACKGROUND.\n\n---\n\n## 0. Answer first\n\n1. **The ruled rule is Jeffrey conditioning, provided \"relevance\" means the right quantity.** Suppose\n   learning about P changes C only through P (Jeffrey's *rigidity*: P(C|P) and P(C|¬P) stay fixed).\n   Then P(C) = a0 + Δ·q, so moving P from q0 to q moves C by exactly **Δ·(q − q0)**. With q0 = 0.5 and\n   relevance := Δ = P(C|P) − P(C|¬P), the ruled contribution *relevance × (acceptability − 0.5)* is\n   not an approximation of Bayesian updating; it is the update. Christensen (1999) proposed this Δ\n   (his measure S) and showed it stays fixed when the premise's own probability moves, after\n   describing the ordinary measure as mixing two factors: how strongly the evidence is linked to the\n   hypothesis, and how far the evidence's probability still has to travel. Godden & Zenker\n   (2018) map informal logic's acceptability, relevance and sufficiency onto the same Jeffrey form.\n   Neither paper is in the corpus.\n\n2. **Four determinate answers.**\n   - **Offset, not raw (D2).** A supporter at its reference value contributes nothing.\n   - **A weakened relevant supporter lowers its target, and a refuted objection raises its target\n     (D2).** This is the complementary-evidence principle (if P confirms C, then ¬P disconfirms C) and\n     conservation of expected evidence. The build does the opposite (`WEAK_SUPPORTER_SUBTRACTS = False`\n     in `deliberus/relevance.py`), and that choice has a price the code comment does not name:\n     examining a link can then only help the side that wrote it, so adding links is a free option for\n     both sides. At a presumed relevance of 0.5 the expected gain is +0.0625 of energy per examined\n     supporter; sixteen such links carry an expected energy of 1.0, which QEM maps to a strength of\n     0.75.\n   - **Relevance needs both conditionals (D5, D7).** P(C|P) alone measures sufficiency, not\n     relevance; a premise is a reason only if P(C|P) > P(C|¬P). Useful identity:\n     **Δ = sufficiency + necessity − 1**, with sufficiency = P(C|P) and necessity = P(¬C|¬P). The\n     corpus's earliest statement of the two factors and the machine judge's current prompt both lean\n     toward sufficiency, or toward \"how far would establishing it move C\", which already contains the\n     acceptability factor (§ 5).\n   - **Answers are premises too (D6).** An answer moves the factor it bears on by the same\n     kinematics, weighted by its own acceptability; once the parameter a question probes is settled,\n     further answers about it are screened off. Counting answers is coherent only if answers are\n     exchangeable and certain.\n\n3. **Where Bayes needs inputs the graph does not have.**\n   - **Premise priors q0.** 0.5 for every claim is a uniform prior, and it sets the asymmetry: the most\n     a supporter can add is Δ·(1 − q0), the most its refutation can remove is Δ·q0. The straw-man worry\n     behind the one-sided rule lives here and in presumed relevance, not in the sign rule.\n   - **The combination structure at each target.** Independent contributions (additive), independent\n     sufficient reasons (noisy-OR) and conditionally independent signs (naive Bayes) each make a\n     different number the portable \"relevance\": Δ, Cheng's causal power, or a likelihood ratio.\n   - **Dependence among supporters** (shared sources, common causes).\n   - **Each link's direction**, cause-like or sign-like (Pearl 1988; Wieten et al. 2019), which a\n     SUPPORTS edge does not record.\n   - **Interactions**: explaining away, and jointly necessary premises.\n\n4. **Where Bayes and the gradual semantics disagree.**\n   - **Raw versus offset.** Published semantics mostly aggregate supporters' raw strengths. Potyka's\n     MLP-based semantics has exactly the log-odds form of a naive-Bayes posterior, but with raw\n     strengths, which equals the offset form only when the base score absorbs the baseline.\n   - **Sign.** The gradual semantics let a weak supporter stop supporting; Bayes makes it count against.\n   - **Shape near zero.** QEM's impact function is quadratic at the origin: an energy of 0.05 moves a\n     claim by 0.00125, where a linear probability update would move it 0.05 and a log-odds update\n     0.0125. A lone weak reason comes out 10 to 50 times smaller than Bayes would make it.\n   - **Cycles.** Local one-pass propagation is exact only on singly connected graphs; iterating the same\n     rules to a fixed point on loops is loopy propagation, whose marginals are in general wrong (Weiss\n     2000). A fixed point is not \"the more correct\" computation.\n   - **No equivalence theorem found.** I found no result proving that QEM, DF-QuAD, Euler-based or\n     h-categorizer semantics is a Bayesian update. The closest results are structural (Potyka 2021's\n     logit form; Prakken 2018, where Pollock's weakest link equals the argument's probability only when\n     at most one defeasible step is involved).\n\n5. **How directly each decision is answered.** D1, D2, D5, D6 and D8 get determinate structural\n   answers. D4 and D7 get a framework that needs inputs (priors, measured judge reliability). D3 is\n   reframed: the Bayesian axis is shared ancestry, not iteration.\n\n---\n\n## 1. What the corpus already has\n\n**Already there** (do not re-derive):\n- `docs/research/what-counts-as-evidence/philosophy-of-evidence.md` § 2.3–2.4: firmness versus\n  increase in firmness, the complementary-evidence principle, the measures D, R, L, Kemeny–Oppenheim\n  and Z, Fitelson's measure sensitivity, and Good's weight of evidence, additive only under\n  conditional independence, which the dossier names as both the justification for summed offsets and\n  the condition under which summing is licit. The Christensen cited there is his 2007 and 2010 work on\n  disagreement and higher-order evidence, not the 1999 measure paper used below.\n- `docs/research/what-counts-as-evidence/formal-and-computational-models-of-evidence.md`: Hahn &\n  Oaksford 2007; Thimm 2012, Hunter 2013 and Hunter & Thimm 2017 (0.5 as no commitment versus\n  balanced); Schum's credibility chain and Fenton–Neil–Lagnado idioms; STRING's noisy-OR combination\n  with the prior removed first, s′ = (s − p)/(1 − p), an engineering precedent for the offset form;\n  the Tavtigian naive-Bayes reformulation of ACMG; Oren & Norman's evidential argumentation.\n- `docs/research/what-counts-as-evidence/testimony-experience-and-normative-evidence.md` § 7.3:\n  critical questions as likelihood-ratio parameters (Hahn & Hornikx); Harris, Hahn, Madsen & Hsu 2016\n  (people track a network with separate expertise and trustworthiness nodes).\n- `docs/research/what-counts-as-evidence/law-intelligence-history-journalism-forensics.md`: Lempert\n  1977 (relevant evidence has a likelihood ratio different from 1); Godden 2014 on corroboration.\n- `docs/research/weakest-link-arithmetic-and-the-merge-hunch.md`: minimum, sum and maximum families,\n  Fréchet bounds, Good's log-odds, Pollock's weakest link and his rejection of accrual, Prakken 2005 on\n  accrual, similarity-adjusted semantics, variety of evidence (Bovens & Hartmann).\n- `docs/research/prediction-markets-argumentation.md` lines 402–403: the two factors stated in March,\n  followed by the right object, the conditional pair P(C|P) and P(C|not-P).\n- `docs/inductive-reasoning-research.md`: BIRD's abduction, entailment and deduction pipeline.\n- `docs/research/relevance-in-the-corpus-sweep.md`: six senses of relevance; Johnson & Blair, Govier\n  and Walton 2004 flagged as missing.\n- `deliberus/graph/qbaf.py` and `deliberus/relevance.py`: the offset form, QEM, and the named choices\n  as built on 2026-09-28 (not deployed).\n\n**New here**: Jeffrey kinematics and its equivalence with the ruled form; Christensen's measure and its\nstability under Jeffrey updating; Godden & Zenker 2018; conservation of expected evidence as a design\nargument; virtual evidence, Chan & Darwiche and Wagner; Pfeifer's probabilistic modus ponens interval;\nPrakken 2018; Hunter 2021 and 2022; Cheng's causal power and the portability result of § 2.2; Pearl's\ncausal/evidential distinction and Wieten et al.'s labelled arguments; how people and language models\njudge impact versus posterior probability; Polberg & Hunter 2018; Weiss 2000.\n\n---\n\n## 2. The key derivations, done explicitly and checked\n\n### 2.1 The Jeffrey form and the two factors\n\nBefore the supporter is examined, P(C) = a1·q0 + a0·(1 − q0). Examining it moves P's probability to q.\nUnder rigidity:\n\n    P_new(C) = a1·q + a0·(1 − q) = a0 + Δ·q\n    change    = P_new(C) − P_old(C) = Δ·(q − q0)\n\nSo a link transmits the fraction Δ of its supporter's increment. With relevance := Δ and q0 = 0.5, the\nruled contribution relevance × (acceptability − 0.5) is this change exactly. The corpus's philosophy\ndossier (§ 2.3) already recommends reading a claim's strength as an increment over 0.5 rather than a\nprobability of truth; this is the reading under which the ruled product is Bayes-exact.\n\n**The other measures mix the two factors.** If P is learned for certain:\n\n| Measure | Value in this setting | Depends on the supporter's acceptability? |\n|---|---|---|\n| difference d = P(C\\|P) − P(C) | Δ·(1 − q0) | yes: shrinks to 0 as the supporter becomes accepted |\n| Christensen's S = d / P(¬P) = P(C\\|P) − P(C\\|¬P) | Δ | no |\n| Carnap's P(P∧C) − P(P)·P(C) | Δ·q0·(1 − q0) | yes |\n| ratio, Z | depend on P(C) | yes, and on the target's prior |\n\nOnly Δ, or a monotone function of the pair (a1, a0), is a property of the link alone. Christensen shows\nS stays the same under Jeffrey conditioning on {P, ¬P}; checked numerically over 2,000 random cases.\nThe table also shows the old-evidence problem Christensen analyses: measured after the fact, a premise\neveryone already accepts can confirm nothing. The platform sidesteps it by computing every strength\nfrom a fixed reference (0.5) rather than from the current state.\n\nA notation warning: the corpus's evidence dossier summarizes the Bayesian argumentation programme's\nstrength as \"Δ in posterior, or the likelihood ratio\"\n(`docs/research/what-counts-as-evidence/formal-and-computational-models-of-evidence.md` § 1.8). That\n\"Δ in posterior\" is the difference measure d in the table above, not the Δ of this report.\n\n**Identity.** Δ = P(C|P) + P(¬C|¬P) − 1 = sufficiency + necessity − 1 (the form of Youden's index).\nHunter (2022) builds argument strength from exactly these two probabilities. The corpus's *dependence*\nscore (\"how much the whole needs the part\") is a necessity-type reading and its *lift* question is a\nsufficiency-type reading; bearing is their sum minus one, not a third quantity independent of both.\n\n### 2.2 Which number is portable depends on how reasons combine\n\n| How the target's reasons combine | Probabilistic model | Stable per-link parameter (\"relevance\") | How several links add |\n|---|---|---|---|\n| independent contributions, no interaction | linear: P(C\\|x) = α + Σβᵢxᵢ | Δᵢ = βᵢ | P(C) = α + Σβᵢqᵢ; holds by linearity of expectation even when the premises are correlated (checked) |\n| independent sufficient reasons | noisy-OR with leak b: P(C\\|x) = 1 − (1 − b)·Π(1 − pᵢ)^xᵢ | causal power pᵢ = Δᵢ / (1 − P(C\\|¬Pᵢ)) (Cheng 1997) | P(C) = 1 − (1 − b)·Π(1 − pᵢqᵢ) under independence; one link's change is (1 − b)·R·pᵢ·(qᵢ − qᵢ0), with R = Π over the other links of (1 − pⱼqⱼ) |\n| conditionally independent signs (C explains each Pᵢ) | naive Bayes | the likelihoods aᵢ, bᵢ, i.e. the likelihood ratio | posterior log-odds = prior log-odds + Σ log LRᵢ (Good; modularity) |\n| jointly necessary parts | AND-type | not separable | covered in `weakest-link-arithmetic-and-the-merge-hunch.md` |\n\nWorked check for the noisy-OR row (leak 0.1, powers 0.6, 0.5 and 0.4): the first link's Δ is 0.324 while\nthe other two supporters stand at 0.5, and 0.176 once they stand at 0.95. Its causal power stays 0.6.\nSo in a target whose reasons are independent and each sufficient, a stored Δ goes stale whenever a\nsibling is examined, while a stored causal power does not. **A single stored \"relevance\" number means\nΔ in the first structure, causal power in the second and a likelihood ratio in the third.** The judge\nand the people arguing relevance have to be asked for the number that matches the link's structure,\nand the structure has to be recorded somewhere.\n\n### 2.3 Offset versus raw, and the sign of a weakened supporter\n\n**Unexamined contributes nothing.** q = q0 gives zero change. **Refuted counts against.** By the\ncomplementary-evidence principle (Crupi's SEP entry), if P confirms C then ¬P disconfirms C; a\nsupporter found weaker than its reference lowers C by Δ·(q0 − q).\n\n**Conservation of expected evidence.** Before examination, the expected final acceptability equals q0,\nso the expected change in C is zero. For an examination that settles P either way:\nq0·Δ·(1 − q0) + (1 − q0)·Δ·(0 − q0) = 0.\n\n**What the one-sided rule does.** With contribution r·max(0, acceptability − 0.5), an examination that\nspreads acceptability evenly over [0, 1] has expected contribution r/8: +0.0625 at r = 0.5 (simulated\n0.0626). Sixteen such supporters carry an expected energy of 1.0, a QEM strength of 0.75. The mirror\nholds for objections, which in the build also stop acting when refuted. The one-sided rule therefore\nrewards quantity on both sides; the symmetric rule makes examination zero-sum in expectation, so only\nevidence that is actually found moves a claim.\n\n**The four rules side by side** (energies, with relevance r; supporter acceptability 0, 0.5 or 1):\n\n| Rule | no link | link, supporter unexamined | supporter refuted | supporter confirmed |\n|---|---|---|---|---|\n| Bayes (Jeffrey), Δ = r, q0 = 0.5 | 0 | 0 | −0.5r | +0.5r |\n| raw (the usual gradual-semantics form) | 0 | +0.5r | 0 | +r |\n| offset, one-sided (as built) | 0 | 0 | 0 | +0.5r |\n| offset, symmetric | 0 | 0 | −0.5r | +0.5r |\n\nRaw semantics make \"no link\" weaker than \"unexamined link\", so silence manufactures strength (three\nunexamined supporters at relevance 0.8 give energy 1.2 and strength 0.795). The build makes \"refuted\"\nequal to \"no link\". Only the symmetric offset rule reproduces the Bayesian row.\n\n**The straw-man case, quantified.** A supporter at relevance 0.8 refuted to 0.05 contributes −0.36\n(strength 0.443); at the presumed relevance of 0.5 it contributes −0.225 (strength 0.476). In Bayes the\ndamage a refuted supporter can do is at most Δ·q0: large only if the link genuinely bears on the target\nand the supporter was genuinely plausible beforehand. The attack works on the platform because both\nare presumed: relevance 0.5 for any link a person writes, and q0 = 0.5 for any claim however it was\ninvented. **The sign rule and the presumptions are coupled**: symmetric subtraction is safe to the\ndegree that unargued links carry low presumed relevance and invented claims are not presumed 50/50.\n*(Annotation 2026-09-30: an honest prior for the reason does not by itself close this, because an attacker\npicks a plausible reason; what the attack exploits is selection, a reason chosen because it fails, so its\nrefutation carries little information. The loss is warranted where the conclusion relied on the reason:\n[../starting-values-and-the-problem-of-the-prior.md](../starting-values-and-the-problem-of-the-prior.md) § 6.8.)*\n\n**Raw versus offset in log-odds.** Potyka's MLP-based semantics computes\nlogit(σ_C) = logit(β_C) + Σ wᵢσᵢ = [logit(β_C) + Σ wᵢ·0.5] + Σ wᵢ(σᵢ − 0.5). Raw is offset with the base\nscore shifted by the supporters' baseline. A trained network's learned bias absorbs that shift; a\ngraph whose base score is set to 0.5 for every claim does not.\n\n### 2.4 The evidential direction: virtual evidence\n\nIf C explains P, and new information e about P bears on C only through P, then the effective\nlikelihood ratio of e for C is\n\n    LR_eff = (a·λ + 1 − a) / (b·λ + 1 − b),     λ = O_new(P) / O_old(P)\n\nwhere λ is the Bayes factor of e for P (Pearl's virtual evidence; Chan & Darwiche show it is the same\nrevision as Jeffrey's rule, specified by a factor instead of a posterior). λ = 1, nothing learned about\nP, gives LR_eff = 1: offset again. λ → ∞ gives a/b. λ → 0 gives (1 − a)/(1 − b), below 1 when a > b:\nrefuting a sign counts against C. To first order, ln LR_eff ≈ (a − b)·(λ − 1): the product of the\nlink's relevance, a − b = sensitivity + specificity − 1 (Nozick's measure), and the supporter's shift\nin odds form. The exact effect saturates as λ grows.\n\nCheck: P(C) = 0.3, a = 0.8, b = 0.3, so q0 = 0.45, P(C|P) = 0.533 and P(C|¬P) = 0.109. Moving q to 0.7,\nJeffrey's rule gives 0.40606 and virtual evidence with λ = 2.852 gives 0.40606.\n\n### 2.5 Chaining: product, minimum and Jeffrey\n\nGiven q = P(P) and r = P(C|P), coherence only fixes an interval (Pfeifer 2013, probabilistic modus\nponens): **rq ≤ P(C) ≤ rq + 1 − q**.\n\n- **Product rq** is Prakken's (2018) internal strength, the probability that the premise and the\n  conclusion both hold. It equals P(C) only if P(C|¬P) = 0, i.e. only if the premise is necessary.\n- **Adams' bound** q + r − 1 is weaker than rq, since rq − (q + r − 1) = (1 − q)(1 − r) ≥ 0.\n- **Minimum** always lies inside the interval (checked on a grid of 9,900 pairs), but consider what\n  it implies. If r ≤ q, min(q, r) = r is the value P(C) takes when P(C|¬P) = r = P(C|P), i.e. when\n  Δ = 0: the weakest-link rule reports the conclusion's probability as if the premise did not matter.\n  If q < r, it corresponds to P(C|¬P) = q(1 − r)/(1 − q), a value that moves with q, so no stable link\n  parameter lies behind it.\n- **Jeffrey** needs P(C|¬P) explicitly. With P(C|¬P) = 0.5 it lands on the interval's midpoint.\n\nPfeifer's own example (q = 0.9, r = 0.8): interval [0.72, 0.82], Pfeifer's strength (precision ×\nlocation) 0.69. For comparison: midpoint 0.77, minimum 0.80, Adams' bound 0.70.\n\n**There are two products, and they answer different questions.** Prakken's rq is the probability that\na chain holds. The ruled product Δ·(q − q0) is the change a link transmits. Pollock's argument against\nprobabilistic strength (that many independent premises quickly make a conclusion improbable) targets\nthe first, and Hahn & Hornikx (2016) answer that it presupposes a conjunction of premises rather than\nconditional probability. Neither objection touches the second.\n\n### 2.6 Many supporters\n\n- **Additive**: Jeffrey shifts add; for the expected value, independence is not needed (§ 2.2).\n- **Log-odds**: log likelihood ratios add only under conditional independence given C and given ¬C.\n  The SEP entry states it as a theorem: under its background principles, modularity for conditionally\n  independent data holds exactly for likelihood-ratio measures.\n- **Double counting**: one report with likelihood ratio 3 at prior 0.5 gives 0.75; the same report\n  counted twice as if independent gives 0.90.\n- **Weak reasons accrue**: Godden & Zenker's Example 1 has four conditionally independent reasons, each\n  with likelihood ratio 5/3, and prior 0.17. Sequential updating gives 0.2545, 0.3626, 0.4867 and 0.6125.\n  Their appendix prints 0.3625, 0.4833 and 0.6093, a small arithmetic slip at the third update (the\n  reason's recomputed probability is 0.18625, printed 0.1875); the conclusion is unchanged. No reason\n  alone clears a 0.5 threshold; together they do.\n\n### 2.7 QEM near zero, against the Bayesian shapes\n\nThe same energy E read four ways (strength moved away from 0.5):\n\n| E | QEM | log-odds (E as a change in log-odds) | linear (E as a change in probability) |\n|---|---|---|---|\n| 0.01 | 0.00005 | 0.0025 | 0.01 |\n| 0.05 | 0.00125 | 0.0125 | 0.05 |\n| 0.1 | 0.00495 | 0.0250 | 0.10 |\n| 0.2 | 0.0192 | 0.0498 | 0.20 |\n\nQEM's impact is E²/(1 + E²), so near zero the log-odds shape exceeds it by about 1/(2E). Both\nBayesian shapes are first-order in small evidence; QEM is second-order, and the gap compounds through\nlevels because a supporter's own strength is itself a QEM output. Part of the measurement in\n`docs/research/relevance-and-the-two-factors.md` § 5 (one claim of 722 changes; none of the 12 links\nwhose supporter has standing moves its target by more than 0.001) is this shape, not only how little\nhas been examined. The choice of influence function is the\nsibling report's subject; the point here is only that it is not Bayes-shaped near zero.\n\n### 2.8 Cycles and shared ancestry\n\nLocal propagation is guaranteed exact only on singly connected graphs; on graphs with loops, iterating\nthe same local rules to convergence gives marginals that are in general wrong (Weiss 2000 derives the\nrelation for a single loop, and shows cases where the most probable assignment is right while the\nmarginals are not). \"Loops\" here include undirected ones: two routes from one premise to one\nconclusion. A one-pass walk and a fixed point both count a shared ancestor once per route. So the\nBayesian axis for D3 is not one pass versus fixed point but whether shared ancestry is modelled.\nA directed cycle in an argument graph reads better as a dependence to model (a common source) or as\ncircular support than as something to iterate through; Wieten et al. (2019) break the cycles that\nevidential shortcuts create when they turn argument graphs into networks.\n\n### 2.9 Presumed relevance and a judge's reliability\n\nIf a judge's reading rₘ is right with probability ρ and otherwise uninformative (drawn from the same\nprior, with mean r0), then **E[relevance | reading] = ρ·rₘ + (1 − ρ)·r0** (derived here in the style of\nBovens & Hartmann's reliability models; simulated 0.777 against the formula's 0.780 at ρ = 0.7,\nrₘ = 0.9, r0 = 0.5). The build's confident/unconfident switch is the special case ρ ∈ {0, 1}. Measured\nagreement between the judge and argued relevance claims would supply ρ per kind of link; ledger Q21\nalready proposes that a pass's measured error rate discount its unexamined outputs.\n\n---\n\n## 3. Per-publication entries\n\n### 3.1 Informal logic: acceptability, relevance, sufficiency\n\n**Johnson, R. H. & Blair, J. A. (1977). *Logical Self-Defense*. McGraw-Hill Ryerson (later editions).**\nNot read. That the relevance–sufficiency–acceptability criteria were introduced here in 1977 is confirmed\nby Godden & Zenker (2018, note 3). Defines a good argument as one with acceptable premises that are\nrelevant to the conclusion and together sufficient for it.\n- D1 ADJACENT: the ruled split (acceptability times relevance, sufficiency as a separate whole-claim\n  question) is theirs.\n- D5 ADJACENT: sorting critical questions into premise and connection applies their split to Walton.\n\n**Govier, T. (2010). *A Practical Study of Argument*, 7th ed. Wadsworth.** Godden & Zenker (note 3)\nconfirm she calls sufficiency \"good grounds\" (the ARG conditions). Not otherwise read.\n- D5 ADJACENT: the same split, in textbook form.\n\n**Blair, J. A. (1992). Premissary relevance. *Argumentation* 6, 203–217. doi:10.1007/BF00154326.**\nAbstract read. Premissary relevance is a property of an argument: premises are relevant when they belong\nto a set that authoritatively warrants the inference, and the warrant carries a conditional proposition\nthat can be justified; argument schemes help identify warrants.\n- D5 DIRECT: relevance lives in the warrant, the connection side, not in the premise.\n- D7 DIRECT: an argued relevance claim is the warrant's conditional made explicit and justifiable,\n  the informal-logic form of the ruled \"argued\" tier.\n\n**Hitchcock, D. (1992). Relevance. *Argumentation* 6(2), 251–270.** Bibliographic only; not read.\nBACKGROUND until read.\n\n**Walton, D. (2004). *Relevance in Argumentation*. Lawrence Erlbaum.** Publisher description and\ncontents read. A method for judging whether a move is relevant, built on a dialectical theory of\nrelevance (a chapter of that name), case studies of irrelevance, and a critique of relevance logic.\n- D7 ADJACENT: relevance judged per move in its dialogue context supports arguing it per link.\n- D5 ADJACENT.\n\n**Freeman, J. B. (2005). *Acceptable Premises: An Epistemic Approach to an Informal Logic Problem*.\nCambridge UP; and (2006). Argument strength, the Toulmin model, and ampliative probability. *Informal\nLogic* 26(1), 25–40.** Hitchcock's review of the book read; the article verified bibliographically. A\npremise is acceptable without further argument under stated conditions, the first a presumption that\nthe mechanism producing belief in the premise is reliable.\n- D4 DIRECT: a theory of what an unexamined premise presumes, keyed to its source. The build presumes\n  nothing for an unexamined supporter (it contributes zero), which is Freeman's no-presumption case\n  applied everywhere.\n- D2 ADJACENT.\n\n**Godden, D. & Zenker, F. (2018). A probabilistic analysis of argument cogency. *Synthese* 195(4),\n1715–1740. doi:10.1007/s11229-016-1299-2.** Abstract, all 25 notes, references and appendix read; the\nmain text is paywalled and was not read. Treats an argument as a reason R offered for a claim C and\nstates acceptability, relevance and sufficiency probabilistically. Updating on a partially accepted\nreason is by Jeffrey conditionalization (note 17), with a likelihood form of the same update (note 18).\nA reason is\nirrelevant when the impact term P(C|R)/P(C) (Carnap's relevance quotient) equals 1, equivalently when\nthe likelihood ratio P(R|C)/P(R|¬C) equals 1 (note 20, citing Korb 2004 and Hahn & Hornikx 2016).\nSufficiency is a contextual threshold on P(C|R), which must also exceed P(C) (note 21). The abstract\nnames the four features cogency depends on: the change in commitment to the reason, the reason's\nsensitivity and selectivity to the claim, prior commitment to the claim, and contextual thresholds. Their\n\"specificity\" is P(R|¬C), a deliberately non-standard use (note 16). Note 24 shows individually\ninsufficient reasons accruing to sufficiency, says this contradicts the weakest-link principle, and\nproposes that a premise which fails to raise P(C) is not a reason at all; the linked/convergent\nanalysis is left to future work. Appendix: see § 2.6 for a small arithmetic slip.\n- D1 DIRECT: the closest published statement of the ruled design; their first two features, the\n  change in commitment to the reason and its sensitivity and selectivity to the claim, are the two\n  factors.\n- D2 DIRECT: Jeffrey's form gives the offset.\n- D5 DIRECT: relevance is a likelihood property of the reason–claim pair; acceptability is commitment\n  to the reason.\n- D8 DIRECT: accrual of weak independent reasons, against the weakest link.\n- D4 ADJACENT: thresholds as context parameters, not constants.\n\n### 3.2 Bayesian argumentation\n\n**Hahn, U. & Oaksford, M. (2006a). A Bayesian approach to informal argument fallacies. *Synthese* 152(2),\n207–236. doi:10.1007/s11229-005-5233-2. — (2006b). A normative theory of argument strength. *Informal\nLogic* 26, 1–24. — (2007a). The rationality of informal argumentation: a Bayesian approach to reasoning\nfallacies. *Psychological Review* 114(3), 704–732. doi:10.1037/0033-295x.114.3.704.** Bibliographic\ndetails verified from reference lists; content as summarized in the corpus dossier: classic fallacies are\nweak versions of inductive arguments whose strength depends on content through likelihoods, so an\nargument from ignorance can be strong when a search would have found the thing.\n- D1 ADJACENT: strength as a posterior or likelihood quantity.\n- D2 ADJACENT: the argument-from-ignorance analysis is complementary evidence at work.\n\n**Hahn, U. & Oaksford, M. (2007b). The burden of proof and its role in argumentation. *Argumentation* 21,\n39–61.** Known here through Hahn & Hornikx (2016, note 11), not read directly: burden of proof is tied to\naction; where a decision is required, the utilities of the possible actions set it, and where none is\nrequired the notion is forced, with no normatively compelling way to set either the level of proof or\nwho carries it.\n- D4 DIRECT: presumptions and burdens are decision parameters, not epistemic quantities. The badge makes\n  no decision, so a \"burden\" on unexamined links is a display choice, and a presumption is a prior.\n\n**Hahn, U. & Hornikx, J. (2016). A normative framework for argument quality: argumentation schemes with a\nBayesian foundation. *Synthese* 193(6), 1833–1873. doi:10.1007/s11229-015-0815-0.** Read in part (open\nrepository copy). For argument from sign, expert opinion and popular opinion, the critical questions\ndetermine the likelihoods; the likelihood ratio is sensitivity over false-positive rate. Note 6: a\ngeneralization \"if A then generally B\" makes P(B|A) high, but A is a reason for B only if\nP(B|A) > P(B|¬A). Answers Pollock's argument that multiple premises quickly yield improbable conclusions\n(it assumes a conjunction of premises rather than conditional probability). Describes Hahn et al.'s\n(2013) network for expert opinion: expertise and trustworthiness are separate nodes; consistency with\nother experts enters as further reports; sub-questions such as qualifications establish the parent node\nand add nothing once it is established. Contrasts Carneades' split of the questions into assumptions and\nexceptions, a burden-allocation device.\n- D5 DIRECT: expertise, field and trustworthiness sit on the likelihood (connection) side; \"did the\n  expert assert it\" is the premise; consistency is further evidence on the target.\n- D6 DIRECT: screening off, the Bayesian reason answers about one parameter do not accumulate.\n- D1 ADJACENT: the likelihood parameterization of a link.\n- D4 ADJACENT: the Carneades burden rules against the Bayesian treatment.\n\n**Harris, A. J. L., Hahn, U., Madsen, J. K. & Hsu, A. S. (2016). The appeal to expert opinion:\nquantitative support for a Bayesian network approach. *Cognitive Science* 40(6), 1496–1533.\ndoi:10.1111/cogs.12276.** Already in the corpus; bibliographic details verified. People's judgments of\nexpert testimony track a network with separate expertise and trustworthiness nodes.\n- D5 DIRECT: empirical support for putting those questions on the connection side.\n- D7 ADJACENT.\n\n**Hahn, U., Oaksford, M. & Harris, A. J. L. (2013). Testimony and argument: a Bayesian perspective. In\nZenker (ed.), *Bayesian Argumentation*, 15–38.** Bibliographic only (one reference list gives 15–28;\nthe book's pages are 15–38). The expert-testimony network referred to above.\n- D5 ADJACENT.\n\n**Korb, K. (2004). Bayesian informal logic and fallacy. *Informal Logic* 24(1), 41–70.** Bibliographic;\ncited by Godden & Zenker for irrelevance as a likelihood ratio of 1 (p. 44).\n- D5 ADJACENT.\n\n**Zenker, F. (ed.) (2013). *Bayesian Argumentation: The Practical Side of Probability*. Springer\n(Synthese Library 362).** Contents checked. The collection holding Hahn, Oaksford & Harris and Pfeifer.\nBACKGROUND.\n\n**Pfeifer, N. (2013). On argument strength. In Zenker (ed.), 185–193.** Read. Coherence-based\nprobability logic; the probabilistic modus ponens interval [xy, xy + 1 − y] from P(F|B) = x and\nP(B) = y (after Pfeifer & Kleiter 2006a); argument strength as precision times location of the tightest\ncoherent interval, s = [1 − (z″ − z′)]·(z′ + z″)/2; argues against confirmation measures when premises\ncontain conditionals.\n- D1 DIRECT: the product rq is only the interval's lower bound; the interval's width is what an unknown\n  P(C|¬P) costs; its midpoint is Jeffrey with P(C|¬P) = 0.5.\n- D4 DIRECT: an unexamined alternative route is imprecision, not zero.\n\n**Corner, A. & Hahn, U. (2013). Normative theories of argumentation: are some norms better than others?\n*Synthese* 190, 3579–3610.** Bibliographic only; not read. BACKGROUND.\n\n**Hahn, U. (2020). Argument quality in real world argumentation. *Trends in Cognitive Sciences*.\ndoi:10.1016/j.tics.2020.01.004.** Abstract read (volume and pages not checked). A review of measures of\nargument quality beyond persuasiveness. BACKGROUND; a good entry point.\n\n**Hahn, U. (2011). The problem of circularity in evidence, argument, and explanation. *Perspectives on\nPsychological Science* 6(2), 172–182. doi:10.1177/1745691611400240.** Abstract read. Reviews the\nliterature on circularity and asks, case by case, whether circularity is really the problem.\n- D3 ADJACENT: whether a support cycle is vicious is a question about content, which iterating to a\n  fixed point cannot settle.\n\n**Bovens, L. & Hartmann, S. (2003). *Bayesian Epistemology*. OUP. — Olsson, E. J. (2005). *Against\nCoherence*. OUP.** In the corpus. Source reliability as a hidden variable; coherence and variety of\nevidence.\n- D8 ADJACENT: the independence and variety conditions behind accrual.\n- D4 ADJACENT: the reliability model behind § 2.9.\n\n### 3.3 Confirmation measures and probabilistic relevance\n\n**Keynes, J. M. (1921). *A Treatise on Probability*.** In the corpus (weight as the amount of relevant\nevidence, distinct from probability). Not read here.\n- D8 ADJACENT: weight is a separate readout (ledger Q5), not a term in the sum.\n\n**Carnap, R. (1950; 2nd ed. 1962). *Logical Foundations of Probability*.** Firmness versus increase in\nfirmness, as presented in Crupi's SEP entry and Sprenger & Hartmann's Variation 1.\n- D2 ADJACENT: the offset rule is the increment notion; the badge should not be read as firmness.\n\n**Good, I. J. (1950). *Probability and the Weighing of Evidence*. Griffin. — (1983). *Good Thinking*.\nUniversity of Minnesota Press.** Bibliographic via Chan & Darwiche's references; content via the SEP.\nWeight of evidence is the log likelihood ratio, additive for conditionally independent items.\n- D8 DIRECT: the condition under which summing is licit.\n- D1 ADJACENT.\n\n**Christensen, D. (1999). Measuring confirmation. *Journal of Philosophy* 96(9), 437–461.\ndoi:10.2307/2564707.** Read in part. Defines S(E, H) = [P(H|E) − P(H)]/P(¬E), which equals\nP(H|E) − P(H|¬E); argues that the standard difference measure conflates how strongly E is linked to H\nwith how far E's probability can still rise; shows S-support is stable under Jeffrey conditioning on\n{E, ¬E}; analyses the old-evidence problem; argues that even this measure reveals a further problem\nunlikely to have a formal solution.\n- D1 DIRECT: the two factors, named and separated in 1999.\n- D2 DIRECT: the difference measure vanishes for accepted premises; S does not.\n- D7 DIRECT: tells a judge what to measure.\n\n**Joyce, J. (1999). *The Foundations of Causal Decision Theory*. Cambridge UP.** Not read; Eells &\nFitelson (2000, note 15) point to Eells's review of it for further properties of the same measure.\nBACKGROUND.\n\n**Fitelson, B. (1999). The plurality of Bayesian measures of confirmation and the problem of measure\nsensitivity. *Philosophy of Science* 66 (Proceedings), S362–S378. doi:10.1086/392738.** In the corpus;\nbibliographic details verified from citing papers. Many results hold under one measure and fail under\nanother.\n- D7 ADJACENT: which measure is stored changes conclusions.\n\n**Eells, E. & Fitelson, B. (2000). Measuring confirmation and evidence. *Journal of Philosophy* 97(12),\n663–672. — (2002). Symmetries and asymmetries in evidential support. *Philosophical Studies* 107(2),\n129–142.** The 2000 note read in part: normalizing (Christensen's S) is argued to be no improvement as an\nall-purpose measure of support; among the desiderata Christensen proposed, \"support should not vanish\nas P(E) approaches 1\" is violated by the difference and ratio measures and satisfied by S and the\nlikelihood measure. The 2002 paper shows several symmetries fail in general.\n- D7 ADJACENT: S is contested as a confirmation measure. Its use here is narrower: it is the Jeffrey\n  slope, an identity rather than a contested choice.\n\n**Crupi, V., Tentori, K. & Gonzalez, M. (2007). On Bayesian measures of evidential support: theoretical\nand empirical issues. *Philosophy of Science* 74(2), 229–252. doi:10.1086/520779.** Via the SEP and the\ncorpus. The Z (relative distance) measure.\n- D7 ADJACENT: Z depends on the target's prior, so it cannot be a link-only number.\n\n**Tentori, K., Crupi, V., Bonini, N. & Osherson, D. (2007). Comparison of confirmation measures.\n*Cognition* 103(1), 107–119. doi:10.1016/j.cognition.2005.09.006.** Abstract read. In an experiment on\njudged evidential impact, the Kemeny–Oppenheim measure was among the best predictors, its only rival the\nordinally equivalent likelihood measure, and it was the only candidate meeting their normative\nconditions.\n- D7 DIRECT: what people track when asked about impact.\n\n**Tentori, K., Chater, N. & Crupi, V. (2016). Judging the probability of hypotheses versus the impact of\nevidence: which form of inductive inference is more accurate and time-consistent? *Cognitive Science*\n40(3), 758–778.** Abstract read. Impact judgments were more accurate and more consistent over time than\nposterior probability judgments, and predicted the direction of errors in the probability judgments.\n- D7 DIRECT: ask for impact (the link), not for the conclusion's probability.\n\n**Mastropasqua, T., Crupi, V. & Tentori, K. (2010). Broadening the study of inductive reasoning:\nconfirmation judgments with uncertain evidence. *Memory & Cognition* 38(7), 941–950.\ndoi:10.3758/mc.38.7.941.** Abstract read. With the evidence's uncertainty set numerically or by ambiguous\npictures, people's judgments of impact correlated highly with normative Bayesian impact measures; the\npaper frames the update by Jeffrey's rule.\n- D7 DIRECT: people can judge the impact of an uncertain premise.\n- D1 ADJACENT: people are sensitive to the uncertainty factor, i.e. to the product.\n\n**Crupi, V. \"Confirmation\". *Stanford Encyclopedia of Philosophy* (first 2013; revised 2025).** Read in\npart: the complementary-evidence principle; the final-probability principle; Theorem 2 sorting relevance\nmeasures into difference, ratio and likelihood-ratio families, with modularity for conditionally\nindependent data holding exactly for the likelihood-ratio family.\n- D2 DIRECT: complementary evidence.\n- D8 DIRECT: modularity.\n\n**Sprenger, J. & Hartmann, S. (2019). *Bayesian Philosophy of Science*. OUP.** Contents read, text not\nread: Variation 1 (firmness, increase in firmness, the plurality of measures), 4 (learning conditional\nevidence), 5 (old evidence), 6 (probabilistic measures of causal strength).\n- D1 ADJACENT; D7 ADJACENT: causal-strength measures are the portable-relevance question of § 2.2.\n\n**Cheng, P. W. (1997). From covariation to causation: a causal power theory. *Psychological Review*\n104(2), 367–405. doi:10.1037/0033-295X.104.2.367.** Read in part. The contrast ΔPᵢ = P(e|i) − P(e|¬i)\n(her Equation 1, the same quantity as Christensen's S) and generative causal power\npᵢ = ΔPᵢ/(1 − P(e|¬i)) (Equation 8); the contrast underestimates power when other causes already\nproduce the effect.\n- D1 DIRECT: which relevance is portable when reasons are independent and each sufficient.\n- D7 DIRECT: what to ask for in that structure.\n- D8 DIRECT: saturation.\n\n**Lempert, R. (1977). Modeling relevance. *Michigan Law Review* 75, 1021–1057.** In the corpus: evidence\nis legally relevant when its likelihood ratio differs from 1.\n- D5 ADJACENT; D7 ADJACENT.\n\n### 3.4 Uncertain evidence\n\n**Jeffrey, R. C. (1965; 2nd ed. 1983). *The Logic of Decision*. McGraw-Hill; University of Chicago\nPress.** Probability kinematics (chapter 11, as cited by Mastropasqua et al.).\n- D1 DIRECT; D2 DIRECT.\n\n**Pearl, J. (1988). *Probabilistic Reasoning in Intelligent Systems*. Morgan Kaufmann.** Contents read.\nVirtual evidence (as analysed by Chan & Darwiche); exact local propagation on singly connected networks\n(as restated by Weiss 2000); § 10.3 reprints the paper below with the C-E system.\n- D3 DIRECT: exactness only without undirected loops.\n- D1 ADJACENT.\n\n**Pearl, J. (1988). Embracing causality in default reasoning. *Artificial Intelligence* 35(2),\n259–271.** Abstract read. Nearly every default rule is either expectation-evoking (fire suggests smoke)\nor explanation-evoking (smoke suggests fire); formal systems that ignore the distinction misreason, and\nthe C-E system controls which rules may chain.\n- D5 ADJACENT: undercutters of explanation-type links are alternative explanations.\n- D1 ADJACENT: chaining a support through links of mixed direction can manufacture support.\n\n**Diaconis, P. & Zabell, S. L. (1982). Updating subjective probability. *Journal of the American\nStatistical Association* 77, 822–830.** Via Chan & Darwiche: Jeffrey's rule does not commute.\nBACKGROUND (the platform recomputes, it does not accumulate).\n\n**Wagner, C. (2002). Probability kinematics and commutativity. *Philosophy of Science* 69(2),\n266–278.** Evidence specified by Bayes factors makes revisions commute.\n- D7 ADJACENT: store a link's evidential strength as a factor, not as the posterior it produced.\n\n**Chan, H. & Darwiche, A. (2005). On the revision of probabilistic beliefs using uncertain evidence.\n*Artificial Intelligence* 163(1) (pages UNVERIFIED). doi:10.1016/j.artint.2004.09.005. Shorter version\nIJCAI-03, 99–105.** Abstract and introduction read. Jeffrey's rule and virtual evidence both obey\nprobability kinematics and differ only in how evidence is specified (a posterior versus a likelihood\nratio); the two translate into each other; virtual evidence specifies evidence by Bayes factor, as Wagner\nasked, and so commutes; bounds on how much a revision can change beliefs.\n- D1 DIRECT: both forms are the same revision.\n- D7 DIRECT: a person's \"I'm 80% sure\" can be read either way, and the two readings give different\n  answers.\n\n### 3.5 Chaining and combining\n\n**Pollock, J. L. (2001). Defeasible reasoning with variable degrees of justification. *Artificial\nIntelligence* 133(1–2), 233–282.** Abstract and parts read. The weakest-link principle: an argument\nconfers the minimum of its premises' degrees of justification and its reasons' strengths. Rejects the\naccrual of reasons; defeaters too weak to defeat may still diminish; a limited form of collaborative\ndefeat; argues degrees of justification do not follow the probability calculus. His two-witness example:\nconcurring testimony rests on a different probability, not a function of the two single ones.\n- D1 DIRECT: the source of the minimum alternative.\n- D8 DIRECT: the case against accrual, and for dependence mattering.\n- D2 ADJACENT: diminishers.\n\n**Pollock, J. L. (2010). Defeasible reasoning and degrees of justification. *Argument & Computation*\n1(1), 7–22.** Bibliographic only.\n- D1 ADJACENT.\n\n**Prakken, H. (2018). Probabilistic strength of arguments with structure. *KR 2018* (pages\nUNVERIFIED).** Read in part. Internal strength is the probability of the conjunction of an argument's\npremises and conclusions, computed by the chain rule, with defeasible rules given strengths above 0.5.\nIt coincides with Pollock's weakest link when at most one defeasible rule is involved; with more than\none, the weakest link departs from it. Rejects defining strength as P(conclusion | premises). Also a\nresult under an accrual assumption, and dialectical strength from a single distribution.\n- D1 DIRECT: when the minimum is and is not the probability.\n- D8 ADJACENT.\n\n**Hunter, A. (2021). Argument strength in probabilistic argumentation using confirmation theory.\n*ECSQARU 2021*, LNCS 12897, 74–88.** Read in part. Argument and rule strength from the confirmation\nmeasures Cᵈ (difference), Cˢ (Christensen's S) and Cᵏ (Kemeny–Oppenheim), applied to an argument's\nevidence and claim; compositional strength.\n- D1 DIRECT: a published use of S as argument strength.\n- D7 ADJACENT.\n\n**Hunter, A. (2022). Argument strength in probabilistic argumentation based on defeasible rules.\n*International Journal of Approximate Reasoning* 146, 79–105. doi:10.1016/j.ijar.2022.04.003.**\nAbstract read. Strength from the probabilistic necessity and sufficiency of premises for the claim, the\nsufficiency of competing premises, and the necessity of the premises for competing claims; axioms; four\nmeasures.\n- D1 DIRECT; D5 ADJACENT: necessity and sufficiency are the two terms of Δ (§ 2.1).\n\n**Adams, E. W. (1975). *The Logic of Conditionals*. Reidel.** Via Pfeifer 2013: a valid argument's\nconclusion uncertainty is bounded by the sum of its premises' uncertainties.\n- D1 ADJACENT: the bound is weaker than the product (§ 2.5).\n\n### 3.6 Probabilistic argumentation\n\n**Dung, P. M. & Thang, P. M. (2010). Towards (probabilistic) argumentation for jury-based dispute\nresolution. *COMMA 2010*. — Li, H., Oren, N. & Norman, T. J. (2011). Probabilistic argumentation\nframeworks. *TAFA 2011*.** Venues verified; content not read. BACKGROUND.\n\n**Thimm, M. (2012). A probabilistic semantics for abstract argumentation. *ECAI 2012*, 750–755. —\nHunter, A. (2013). A probabilistic approach to modelling uncertain logical arguments. *IJAR* 54(1),\n47–81. — Hunter, A. & Thimm, M. (2017). Probabilistic reasoning with abstract argumentation frameworks.\n*JAIR* 59, 565–611.** In the corpus. The epistemic reading, with 0.5 as no commitment.\n- D4 ADJACENT: 0.5 as \"no commitment\" versus \"balanced\".\n\n**Polberg, S. & Hunter, A. (2018). Empirical evaluation of abstract argumentation: supporting the need\nfor bipolar and probabilistic approaches. *IJAR* 93, 487–543.** Observations read. People interpret\nstatements and relations differently from the intended reading; they may declare two statements\nconnected while unsure of the relation's nature; they assign agreement beyond three values; the\nperceived strength of a relation need not be tightly connected to agreement with its source; support\nis needed alongside attack.\n- D1 DIRECT: empirical evidence that people keep the two factors apart.\n- D5 ADJACENT.\n- D7 ADJACENT: uncertainty about a relation's type argues for recording the type.\n\n**Hunter, A., Polberg, S. & Thimm, M. (2020). Epistemic graphs for representing and reasoning with\npositive and negative influences of arguments. *Artificial Intelligence* 281, 103236.** Verified\nbibliographically; title-level content. BACKGROUND, possibly D2/D6 ADJACENT after reading.\n\n**Potyka, N. (2021). Interpreting neural networks as quantitative argumentation frameworks. *AAAI-21*.**\nRead. The MLP-based semantics sets a claim's log-odds to its base log-odds plus a weighted sum of its\nsupporters' and attackers' strengths; acyclic graphs are computed in one linear-time pass; cyclic ones\nare guaranteed to converge when edge weight times in-degree stays below 4; a continuous variant\nconverges more widely. Its\nrelation to Bayes, derived here: it has the form of a naive-Bayes (logistic) posterior with the\nsupporters' strengths in place of their truth values, exact when every supporter is settled and an\napproximation otherwise, and the constants that would make an unsettled supporter neutral sit in the\nbase score (§ 2.3).\n- D1 ADJACENT; D2 ADJACENT; D3 ADJACENT: convergence conditions say when a fixed point exists, not\n  whether it is right.\n\n**Negative finding.** No paper found shows that QEM, DF-QuAD, Euler-based or h-categorizer semantics is a\nBayesian update under stated assumptions. Structural correspondences exist: the one above; Prakken 2018's\nweakest-link coincidence; and the corpus's own formula for DF-QuAD's aggregation, 1 − Π(1 − sₘ), which is\nthe noisy-OR combination for independent sufficient reasons applied to raw strengths\n(`docs/research/qbaf-gradual-semantics-research.md` § 1). The search was not exhaustive.\n\n### 3.7 Bayesian networks and argument graphs\n\n**Vreeswijk, G. A. W. (2005). Argumentation in Bayesian belief networks. *ArgMAS 2004*, LNCS 3366.**\nAbstract read. Arguments and defeat inside Bayesian networks; an algorithm for the top-k strongest\nadmissible arguments as a tractable alternative to k most probable explanations. BACKGROUND.\n\n**Grabmair, M., Gordon, T. F. & Walton, D. (2010). Probabilistic semantics for the Carneades argument\nmodel using Bayesian networks. *COMMA 2010*, 255–266.** Abstract read. Carneades argument structures\ntranslated into networks, with argument applicability and statement acceptability propagated through\nconditional probability tables.\n- D5 ADJACENT: assumptions and exceptions become table structure.\n- D1 ADJACENT.\n\n**Keppens, J. (2012). Argument diagram extraction from evidential Bayesian networks. *Artificial\nIntelligence and Law* 20(2), 109–143.** Bibliographic only. BACKGROUND.\n\n**Fenton, N., Neil, M. & Lagnado, D. A. (2013). A general structure for legal arguments about evidence\nusing Bayesian networks. *Cognitive Science* 37(1), 61–102.** In the corpus: reusable fragments,\nincluding evidence accuracy, dependence between witnesses, and explaining away.\n- D5 DIRECT: the accuracy idiom separates a report from its accuracy, premise from connection.\n- D8 DIRECT: the dependence idiom.\n\n**Timmer, S. T., Meyer, J.-J. Ch., Prakken, H., Renooij, S. & Verheij, B. (2017). A two-phase method for\nextracting explanatory arguments from Bayesian networks. *IJAR* 80, 475–494.\ndoi:10.1016/j.ijar.2016.09.002.** Read in part. From a network, a support graph; from the support graph\nand observations, arguments. Strength is measured incrementally (likelihood ratio) or absolutely\n(posterior odds), and arguments collect all pro and con reasons, as the network weighs them.\n- D1 ADJACENT: the incremental versus absolute distinction applied to arguments.\n- D8 ADJACENT.\n\n**Verheij, B. (2017). Proof with and without probabilities. *Artificial Intelligence and Law* 25,\n127–154. doi:10.1007/s10506-017-9199-4.** Details verified (the corpus marks them unverified). Case models\nconnecting arguments, scenarios and probabilities. BACKGROUND.\n\n**Wieten, R., Bex, F., Prakken, H. & Renooij, S. (2019). Constructing Bayesian network graphs from\nlabeled arguments. *ECSQARU 2019*.** Read in part. Arguments whose inferences are labelled causal,\nevidential or other become a network whose arrows follow the causal direction. An undercutter of an\nevidential inference is an alternative cause, captured in the table of a head-to-head node. Cycles\ncreated by evidential shortcuts are broken. Predecessor: \"Exploiting causality in constructing Bayesian\nnetwork graphs from legal arguments\" (2018).\n- D5 DIRECT: undercutting questions are interactions, not scalings of one link.\n- D1 ADJACENT: direction decides what is portable.\n- D3 ADJACENT: cycles are broken, not iterated.\n\n**Weiss, Y. (2000). Correctness of local probability propagation in graphical models with loops. *Neural\nComputation* 12(1), 1–41.** Abstract read. Exact on singly connected graphs; for a single loop, an\nanalytic relation between propagated and correct marginals; cases where the most probable assignment is\nright though the marginals are wrong.\n- D3 DIRECT.\n\n**Noisy-OR (Kim & Pearl 1983; noisy-MAX, Henrion 1989).** Cited in\n`docs/research/weakest-link-arithmetic-and-the-merge-hunch.md`.\n- D8 ADJACENT: the combination rule for independent sufficient reasons (§ 2.2).\n\n### 3.8 Eliciting relevance from people and from models\n\n**Renooij, S. (2001). Probability elicitation for belief networks: issues to consider. *Knowledge\nEngineering Review* 16(3), 255–269.** Abstract read. Issues and methods when experts are the only\nsource of a network's probabilities.\n- D7 ADJACENT.\n\n**van der Gaag, L. C., Renooij, S., Witteman, C. L. M., Aleman, B. M. P. & Taal, B. G. (1999). How to\nelicit many probabilities. *UAI 1999*, 647–654 (arXiv:1301.6745).** Read. Each conditional probability\nis shown as a fragment of text with a scale carrying numerical and verbal anchors; the verbal anchors\nwere placed on the numbers empirically (certain 100, probable about 85, expected about 75, fifty-fifty\n50, uncertain about 25, improbable about 15, impossible 0); probabilities sharing a conditioning context\nare asked together; experts gave 150–200 probabilities an hour, as a first step of stepwise refinement.\n- D7 DIRECT: the relevance judge's five named steps are this move; the literature places verbal steps on\n  numbers by measurement and asks the conditionals of one context side by side.\n\n**Bordley, R. F. (1982). A multiplicative formula for aggregating probability assessments. *Management\nScience* 28(10), 1137–1148. doi:10.1287/mnsc.28.10.1137.** Abstract read: an axiomatic derivation of a\nmultiplicative formula for combining experts' probabilities.\n- D7 ADJACENT: how several readings would pool, if ever; pooling formulas presume the assessments are\n  distinct sources, which is why counting votes is the wrong model for relevance.\n\n**Zhu, J.-Q. & Griffiths, T. L. (2024). Incoherent probability judgments in large language models.\narXiv:2401.16646.** Abstract read. Language models' probability judgments show human-like incoherence,\nbetter described by the Bayesian sampler than by probability theory plus noise.\n- D7 DIRECT: check a judge's readings for coherence (for example both conditionals and their\n  complements).\n\n**Li, S. & Rui, H. (2024). Dual traits in probabilistic reasoning of large language models.\narXiv:2412.11009.** Abstract read. Two modes of posterior judgment, one following Bayes' rule and one\nbased on similarity; models struggle to recall base rates.\n- D7 DIRECT: ask for impact or likelihoods, not posteriors that need base rates.\n\n**Nafar, A., Venable, K. B., Cui, Z. & Kordjamshidi, P. (2025). Extracting probabilistic knowledge from\nlarge language models for Bayesian network parameterization. arXiv:2505.15918.** Abstract read. Across 80\npublic networks, conditional probabilities obtained by querying models beat random, uniform and\ntoken-probability baselines, and work best as priors refined with a little data.\n- D7 DIRECT: a machine's relevance is a prior that arguments then update.\n\n**Feng, Y., Zhou, B., Lin, W. & Roth, D. BIRD: a trustworthy Bayesian inference framework for large\nlanguage models. arXiv:2404.12494 (ICLR 2025).** In the corpus. The model proposes factors; an explicit\nBayesian model computes.\n- D7 ADJACENT.\n\n**Rescala, P., Horta Ribeiro, M., Hu, T. & West, R. (2024). Can language models recognize convincing\narguments? *Findings of EMNLP 2024* (arXiv:2404.00750).** Abstract read. Models match humans at telling\nstrong arguments from weak; combining predictions from different models beats humans.\n- D7 DIRECT: supports a second model family, and pooling across families.\n\n**Let's think var-by-var: large language models enable ad hoc probabilistic reasoning.\narXiv:2412.02081 (authors UNVERIFIED).** Abstract read. The model proposes variables and moment\nconstraints; a probabilistic model is then fitted.\n- D7 ADJACENT.\n\n---\n\n## 4. The design space per decision\n\n**D1 how the two factors combine.**\n- *Product (as built).* In the effect domain this is Jeffrey's rule when relevance means Δ and 0.5 is\n  the supporter's reference: bilinear, zero when either factor is zero (§ 2.1; Christensen 1999;\n  Godden & Zenker 2018). People's impact judgments track it (Mastropasqua et al. 2010), and they keep the\n  two factors apart (Polberg & Hunter 2018).\n- *Minimum (Pollock).* No link-level Bayesian reading: as a transmitted probability it assumes the\n  premise irrelevant whenever the link is weaker than the premise (§ 2.5). Pollock's own case is against\n  probabilistic chains, answered by Hahn & Hornikx; Prakken shows the minimum equals the chain's\n  probability only with one defeasible step. The minimum earns its place where it already lives, among\n  jointly necessary parts, not between one link's two factors.\n- *Explicit Jeffrey.* Needs P(C|¬P), which nobody supplies; with 0.5 it gives Pfeifer's midpoint.\n- *Evidential links.* The exact effect is the virtual-evidence form (§ 2.4), of which the product is\n  the first-order version; it saturates for large shifts.\n- *Noisy-OR.* A product with a third factor, the room other reasons leave (§ 2.2).\n- **Left for the founder**: the product has the Bayesian shape; what remains open is which quantity\n  relevance names (D7) and which structure a target's links have (D8).\n\n**D2 raw versus offset.**\n- *Offset* is determinate (Jeffrey; virtual evidence at λ = 1; STRING's prior removal, in the corpus).\n- *Raw* makes an unexamined link add, manufacturing strength from silence (§ 2.3).\n- *Sign.* Bayes says a refuted relevant supporter lowers its target and a refuted objection raises its\n  target. The one-sided rule (as built) gives both sides a free option (§ 2.3). The straw-man risk is\n  real, but it sits in two presumptions: relevance 0.5 for any person-written link, and q0 = 0.5 for any\n  claim. Options consistent with Bayes: (i) symmetric subtraction with presumed relevance 0 until argued\n  or confidently proposed; (ii) symmetric subtraction with presumed relevance only for links the\n  target's own source asserted; (iii) symmetric subtraction with a reference value per kind of claim\n  instead of 0.5. The build's option, one-sided, trades the straw-man risk for the padding drift.\n\n**D3 fixed point versus one pass.**\n- Local rules are exact only on singly connected graphs; iterating to a fixed point on loops is an\n  approximation (Weiss 2000). Both one pass and a fixed point double-count shared ancestry (§ 2.8).\n- The probabilistic view favours the built one pass on the acyclic part, with directed cycles treated as\n  a finding (a shared dependence or circular support, Hahn 2011; broken, as in Wieten et al. 2019)\n  rather than something to iterate through. Circularity is not always vicious: the corpus's dossier\n  records Hahn & Oaksford's point that self-dependent justification can still be informative in some\n  structures, which is a question about content that iteration cannot answer. The measurement that matters is not yet made: how many\n  claim pairs are joined by more than one route.\n- Convergence conditions for fixed points (Potyka 2021) say when a fixed point exists, not whether it is\n  right.\n\n**D4 what an unexamined link presumes.**\n- *Presumed relevance* is a prior mean of bearing for links of that provenance. It can be measured: the\n  mean judged or argued relevance per provenance, as the corpus already did for dependence (0.17–0.49 by\n  writer).\n- *Presumed acceptability* is none in the build (an unexamined supporter contributes zero), which is\n  Bayes-consistent under offset. Freeman argues some sources carry a presumption; Hunter & Thimm\n  distinguish 0.5 as \"no commitment\" from 0.5 as \"balanced\".\n- *An unknown alternative route* is imprecision (Pfeifer), not zero; 0.5 is its midpoint.\n- *Burden* is defined only relative to a decision (Hahn & Oaksford 2007b); Carneades' assumptions and\n  exceptions (ledger Q9's typed questions) allocate it in dialogue, not in probability.\n- *A judge's reading* can be shrunk by its measured reliability instead of switched on or off (§ 2.9).\n\n**D5 sorting critical questions.**\n- Bayes sorts cleanly: premise questions move q; connection questions move the likelihoods (Δ, or a and\n  b); rebuttals are further evidence on the target (Hahn & Hornikx; Harris et al. 2016; Fenton et al.\n  2013). Blair (1992) puts relevance in the warrant, which is the connection side.\n- Two refinements. Questions about alternative causes, explanations or competing hypotheses are\n  interactions (Wieten et al. 2019; the explaining-away idiom), not scalings of one link; a product rule\n  approximates them only while the alternative's probability stays fixed. Base-rate questions bear on\n  the target's prior when relevance is a likelihood ratio, and on the link only when relevance is Δ.\n- The build (`deliberus/extraction/schemes.py`, `CQ_BEARS_ON`) matches Bayes for expert opinion\n  (credibility, field, trustworthiness and evidence basis on the connection side; assertion as premise;\n  consistency as rebuttal). The design doc does not (§ 5, item 1).\n\n**D6 how answers weigh.**\n- An answer is a premise about a parameter: it moves that parameter by Jeffrey's rule, weighted by its\n  own acceptability, and once the parameter is settled further answers about it are screened off (Hahn\n  & Hornikx on Hahn et al. 2013). The ledger's direction for Q3 (weigh by standing) is the Bayesian one.\n- Counting is coherent only if answers are exchangeable (interchangeable, conditionally independent\n  samples of one parameter) and certain; critical-question answers are neither. The ravens lesson in the\n  corpus's philosophy dossier is the same point.\n- Record answers as factors, not as the posteriors they produced, so order does not matter (Wagner 2002;\n  Chan & Darwiche 2005).\n\n**D7 who sets relevance, and how.**\n- *What to ask.* The contrast between the target's standing with and without the supporter (Δ), or for\n  sign-like links the likelihoods. Not \"how far would establishing it move the target\" (that is d, which\n  contains the acceptability factor), and not \"would it settle the target\" (sufficiency).\n- *Who asks well.* People judge impact more accurately and consistently than posteriors (Tentori et al.\n  2016), track normative impact measures with uncertain evidence (Mastropasqua et al. 2010), and the\n  Kemeny–Oppenheim form fits them best (Tentori et al. 2007). Models are incoherent in human-like ways\n  (Zhu & Griffiths 2024) and weak on base rates (Li & Rui 2024), so ask them for impact and check\n  coherence; their conditionals serve best as priors (Nafar et al. 2025); different families pooled beat\n  one (Rescala et al. 2024).\n- *How to elicit.* Verbal steps placed on numbers by measurement, and the two conditionals of one context\n  side by side (van der Gaag et al. 1999).\n- *Never voted.* Pooling formulas presume distinct sources (Bordley 1982); a brigade is one source\n  counted many times, so the ruling against votes has a probabilistic reason as well as a principled one.\n- *Discounting.* ρ·rₘ + (1 − ρ)·r0 with ρ measured (§ 2.9).\n\n**D8 accrual.**\n- Accrual is licit exactly when reasons are independent in the sense their structure requires: no\n  interaction for additive contributions, independence of mechanisms for noisy-OR, conditional\n  independence given C and ¬C for log-odds (§ 2.2, § 2.6). Correlated reasons double-count (0.75 becomes\n  0.90). Weak independent reasons really do accrue (Godden & Zenker), against the weakest link; Pollock's\n  two witnesses show the dependence structure decides by how much.\n- QEM's energy sum is additive then squashed, which is none of the three exactly, and its second-order\n  origin makes a lone weak reason nearly invisible (§ 2.7). DF-QuAD's product aggregation is the noisy-OR\n  shape applied to raw strengths.\n- Padding on support links: under naive Bayes, near-duplicate supporters violate conditional\n  independence. The corpus's fair-share discounting and \"a shared source becomes a node\" are the\n  Bayes-consistent repairs. The one-sided sign rule reopens padding on this channel (§ 2.3).\n\n---\n\n## 5. Where the corpus is wrong or missing something important\n\n1. **`docs/research/relevance-and-the-two-factors.md` § 2 puts \"is the expert credible\" under\n   acceptability.** The build (`schemes.CQ_BEARS_ON`) and the Bayesian models (Hahn & Hornikx 2016;\n   Harris et al. 2016; Schum's credibility chain, already in the corpus) put credibility, field,\n   trustworthiness and evidence basis on the connection side, and only \"did the expert assert it\" on the\n   premise side. The doc should follow the code.\n2. **\"Conditional relevance P(C|P)\" names sufficiency.** The phrase in\n   `docs/research/prediction-markets-argumentation.md` line 402, quoted in the two-factors doc § 1 as\n   the earliest statement of the two factors, describes how likely the target is given the supporter.\n   Relevance is the difference in the pair on the very next line, P(C|P) against P(C|not-P) (Hahn &\n   Hornikx note 6; Godden & Zenker note 20; Lempert, in the corpus).\n3. **The machine judge is asked for the wrong quantity.** `deliberus/relevance_judge.py` asks how far\n   the target would move if the supporter were true beyond doubt, and anchors its top step on \"it would\n   decide the second outright\". The first is d = Δ·(1 − q0), which already contains the supporter's\n   current credibility, the very factor the prompt tells the judge to leave out; the second is\n   sufficiency. The product rule needs Δ, the contrast between the target with and without the supporter.\n   A question that asks for it: *compare a world where the first claim is true with one where it is\n   false; how different is the second claim's standing?* The magnitude table maps \"settles\" to 1.0,\n   which overstates Δ unless the supporter is also necessary (Δ = sufficiency + necessity − 1). The\n   dossier's summary of Bayesian argument strength as \"Δ in posterior\" (§ 2.1's notation warning) is the\n   same slide from Δ to d. *(2026-09-30: under the drop-out ruling relevance means the gain, which the judge's question asks; this objection stands for the starting-value path. `relevance-and-the-two-factors.md` § 7.)*\n4. **Link structure is not recorded, so a stored relevance is portable only by luck.** Whether a link is\n   cause-like, sign-like, an independent sufficient reason or a jointly necessary part decides which\n   number is stable (§ 2.2). Pearl's causal/evidential distinction and Wieten et al.'s labelled arguments\n   are absent from the corpus.\n5. **The one-sided sign rule's cost is not recorded next to its reason.** The comment in\n   `deliberus/relevance.py` names the straw-man risk and that probability theory disagrees, but not the\n   free option it creates on both sides, which is padding reopened on the support channel (§ 2.3). The\n   straw-man risk is located in the two presumptions, which the doc does not say.\n6. **The two-factors doc § 3 says strength flowing along support links \"needs the fixed-point\n   computation the published model uses\".** On the probabilistic view a fixed point on loops is an\n   approximation and one pass is exact only without shared ancestry; the relevant measurement (claim\n   pairs joined by more than one route) has not been made.\n7. **The § 5 measurement in the two-factors doc is explained only by sparse examination.** QEM's\n   second-order origin is also part of why nothing moves (§ 2.7).\n8. **Missing literature, most important first**: Christensen 1999 and Godden & Zenker 2018 (the closest\n   published statements of the ruled design); Jeffrey, virtual evidence and Chan & Darwiche;\n   conservation of expected evidence; Cheng 1997; Pfeifer 2013 and Prakken 2018; Hunter 2021 and 2022;\n   Tentori et al. 2007 and 2016 and Mastropasqua et al. 2010; Polberg & Hunter 2018; Weiss 2000; Pearl's\n   causal/evidential distinction and Wieten et al. 2019; van der Gaag et al. 1999; the 2024–2025 work on\n   eliciting probabilities from models.\n9. **Minor corrections to the dossier's flags.** Verheij 2017 (*Artificial Intelligence and Law* 25,\n   127–154) is now verified; Fitelson 1999's page range S362–S378 is confirmed by citing papers; the\n   Carnap preface detail remains unverified here too.\n\n---\n\n## 6. Open questions only the founder can decide\n\n1. What does \"relevance\" mean on this graph: the contrast between the target with and without the\n   supporter (Δ), how far establishing the supporter would move the target (d), or whether it would\n   settle the target (sufficiency)? *(For now, 2026-09-30: the gain, under the drop-out ruling; the proposed starting-value path would make it the contrast. `relevance-and-the-two-factors.md` § 7.)*\n2. Should a refuted relevant supporter lower its target, and a refuted objection raise its target, as\n   Bayes says, given that the one-sided rule lets both sides gain by adding links? *(Ruled 2026-09-30, for now: no. The reconciling path, honest starting values, is open as ledger Q35; § 7.)*\n3. If subtraction becomes symmetric, which presumption should carry the straw-man defence: no presumed\n   relevance until a link is argued or confidently proposed, presumed relevance only for links the\n   target's own source asserted, or a starting value per kind of claim instead of 0.5 for everything? *(Moot while subtraction stays off; the proposed path answers it with a starting value per claim, § 7.)*\n4. Should links record what kind of connection they are (cause-like, sign-like, an independent\n   sufficient reason, a jointly necessary part), since which relevance number stays true depends on it?\n   *(2026-10-08: the shape of a written-out step, ledger AD23, proposes this record from the\n   philosophy side; the two were joined in `the-squirrel-and-the-strength-calculus.md` § 4.)*\n5. How should several supporters combine: adding probability changes, as independent sufficient reasons\n   (each closing part of the remaining gap), in log-odds as independent evidence, or by QEM's energy sum,\n   which is none of these?\n6. Is it intended that a lone weak reason is nearly invisible until others join it, as QEM's shape near\n   zero makes it?\n7. Should a cycle be computed through, or treated as a sign of a shared dependence or of circular support\n   to be modelled or flagged?\n8. Should a machine's relevance reading be discounted by its measured agreement with argued relevance,\n   rather than used in full whenever the judge says it is confident?\n9. Once the parameter a question probes is settled, should further answers about it stop counting, as\n   Bayes says, rather than keep adding?\n10. What should the platform presume about a claim nobody has examined: 0.5 for every claim, or a\n    starting value that depends on how the claim entered the graph? *(2026-09-30: the sign question turned out to be this question; proposed path, honest starting values, ledger Q35, § 7.)*\n"}