{"path":"research/how-independent-reasons-add.md","content":"# How independent reasons add: the arithmetic of several reasons for one claim\n\n*2026-10-22, late. His question, after the canon pass's cluster review (`the-canon-distillation-pass.md`\n§ 4e). **Ruled 2026-10-22**: ledger Q46 and all six recommendations of § 6, in his words *\"I'll go ahead and trust your judgment on this and rule all of this blessed - and we'll then review and make sure I learn about the underlying reasons and theory/math/philosophy afterwards. I do want to understand this fully.\"* So the rulings stand now and the learning follows (§ 7).\nThe two literature reports it rests on are kept verbatim beside it in `how-independent-reasons-add/`\n(`literature-pooling-and-dependence.md`, `real-cases-with-known-outcomes.md`), with the scripts that\ncompute every number here (`rules.py`, `cases.py`; run with `uv run python`).*\n\n## 0. Answer first\n\n1. **One reason: the ruled two factors are already the right arithmetic; the squash after them is not.** A\n   reason's contribution, relevance × (the reason's strength − 0.5) (ledger Q33), is Jeffrey's rule for\n   updating on an uncertain reason (`strength-calculus-literature/bayesian-and-informal-logic.md` § 0 item\n   1). Taken alone, a reason that fully decides its claim passes on its own strength: 0.5 + 1 × (0.9 − 0.5)\n   = 0.9. What turns 0.9 into 0.57 is what the code does afterwards: it feeds Potyka's Quadratic Energy\n   Model (QEM) the reason's offset from 0.5 (0.4, not 0.9; Deliberus's convention, `computational-argumentation.md` D2), and\n   QEM's impact E²/(1 + E²) is quadratic near 0, so the small offset all but silences one reason. ⚠ *23 October:\n   this sentence first blamed QEM alone; Potyka's QEM on raw strengths gives one 0.9 supporter 0.72 and two 0.88\n   (§ 9), so the near-silence is the offset and the square together*\n   (`strength-calculus-literature/computational-argumentation.md` § 0: a settling link from a fully\n   accepted claim reaches 0.60 at best; four are needed for 0.9). So yes: under the ruled default a claim\n   with one strong, fully relevant reason inherits that reason's strength, and relevance below 1 passes a\n   share of it (relevance 0.8 gives 0.82).\n2. **Two reasons: what they are worth together depends on one thing, whether they share evidence.** If\n   they are the same evidence twice (one study reported twice, three wet surfaces one dew explains), they\n   are worth one reason: 0.9. If they are independent (different routes from the world to the claim),\n   their odds multiply: 9:1 × 9:1 = 81:1, a strength of 0.988. In between, the literature has a dial: *n*\n   reasons whose errors correlate at ρ are worth *n* / (1 + (*n* − 1)ρ) independent ones, never more than\n   1/ρ (Clemen and Winkler 1985; the corpus has the same formula from Kish, `retrofitting-the-reasons-channel.md`\n   § 9k point 4). Two 0.9 reasons at ρ = 0.5 give 0.949.\n3. **\"45% each\" is averaging, the rule for sources that share everything.** Averaging two 0.9s gives 0.9;\n   a second agreeing reason then adds nothing, which is right only for a restatement. For calibrated\n   sources with different evidence an average is provably too timid (Ranjan and Gneiting 2010), and\n   forecasting tournaments had to push their averages outwards to be accurate (Baron et al. 2014). Measured\n   in evidence, each independent 0.9 reason contributes the same, a ninefold change in the odds; measured in\n   probability points the first appears to add 40 and the second 8.8, only because probability saturates\n   near 1. So the fair answer to *\"how much does each contribute?\"* is half the evidence each, and a display\n   should say it in those units.\n4. **Real cases agree, and say where each rule fails** (§ 5). On five cases with a published outcome the\n   shipped code was the worst rule every time (mean error 0.24 on the four clinical cells, against 0.10 for\n   the best reason and 0.04 for log-odds addition). Reasons with different mechanisms (two coin flips, a\n   clinical score and a blood assay, hearing and sight) added almost exactly in log-odds; signs of one\n   process (one sore throat's symptoms, two tests of one obstruction) kept about four fifths of their sum;\n   and every disaster on record (seven small magnesium trials, two cot deaths, DNA matches between\n   relatives) came from adding reasons that one story outside the claim explains together.\n5. **So the ruled order is right, and the open half has an answer.** The best argument counts until\n   independence is read (ledger Q38); independent arguments then add in log-odds, which is the rule Q38 left\n   open (*\"log-odds, or the summed energy as shipped\"*). Reading independence is one closed question per\n   pair of a claim's reasons (ledger Q40, the keeper's second job), seconds per pair once the pass is\n   trusted (T1), so his expectation about speed holds in principle. It is unbuilt.\n\n## 1. His questions, verbatim\n\n*\"Contrast what effect this has. 1) until the two are shown independent... ONLY the best one powers the\nconclusion/claim. Ok so if there's only one strong reason let's say at 90% strength/truthiness... then the\nclaim is powered by that and so simply kinda inherits that 90% strength? or what? and the other reasons\ndon't do anything before shown independent, but hopefully in the daily course of things in the near future\nwhen we have trusty models running microtasks, that usually won't take more than a few seconds or a minute\nor so for the standing work / machine passes to whir up and finish judging that and indicate it on the\ngraph and thus open this up, right 2) when the two are shown independent, let's say, and they start\n\"adding\"... do they each contribute 45% to the conclusion if they each let's say have 90% strength each? Or\nhow should the math work here? What would common sense say/expect, and what does our\ncurrent/ruled/ledger-unruled-but-pondered math say?\"*\n\nAnd, on the first answer (*today one fully relevant reason at strength 0.9 lifts its claim from 0.5 to only\n0.57 … by probability, one gives at least 0.9 and two give about 0.99*): *\"Read up on all angles on this in\nthe corpus and consult the corpus graph and research what we've not covered on disk online and reflect\ndeeply and test a few examples (not ones we've used before, ideally real-life examples of wildly different\nkinds) and present me with some recommendations.\"*\n\n## 2. What the corpus already holds\n\n| Topic | Where | What it says |\n|---|---|---|\n| the two factors are Jeffrey's rule | `strength-calculus-literature/bayesian-and-informal-logic.md` § 0, § 2.1 | contribution = Δ × (q − q0), exactly the ruled product with every start at 0.5 |\n| three combination structures | same note § 2.2 | independent contributions (linear), independent sufficient reasons (noisy-OR), conditionally independent signs (naive Bayes, log-odds add); each makes a different per-link number portable |\n| QEM is not Bayes-shaped near 0.5 | same note § 2.7; `computational-argumentation.md` § 0, D8, § 5 question 6 | QEM's impact is second-order near neutral; a lone weak reason comes out 10 to 50 times smaller than Bayes; question 6 (*how far may one link move a claim?*) was put and never ruled |\n| weak reasons accrue | `bayesian-and-informal-logic.md` § 2.6 | Godden and Zenker: four reasons each with likelihood ratio 5/3 take a claim from 0.17 to 0.61; one report counted twice overstates |\n| the logistic form, noted for later | `starting-values-and-the-problem-of-the-prior.md` § 6.9 | Potyka's logistic semantics combines a start and independent reasons exactly as naive Bayes; *\"a separate calculus choice for later … honest starts make it bite\"* |\n| best reason, then a lift | `pollock-against-accrual-and-the-overlap-check.md`; ledger Q37 | Pollock and Prakken both refuse to sum; the lift above the best reason is a claim that can be attacked |\n| the ruling | `retrofitting-the-reasons-channel.md` § 9h; ledger Q38 | within an argument the weakest premise; across arguments the best until independence is read, then they add; how they add left open |\n| the rival test | same note § 9k point 2; ledger Q40 (wording ruled 2026-10-22) | *suppose the claim is false; is each reason's most likely explanation the same one?* |\n| effective number | same note § 9k point 4; ledger Q43 | *n*/(1 + (*n* − 1)ρ), ceiling 1/ρ, for display; at ρ = 0.8 the ceiling is 1.25, close to the ruled default |\n| his own view | ledger Q37 (14 October) | *\"reasons pass on their strength in whole to the claim they argue for … Whether there's 1 or 14 reasons\"*; *\"Overlapping roots of evidence obviously shall not be added up.\"* |\n| the target | ledger Q41; `the-number-in-the-world-and-the-repertoire.md` § 1 | probability given the record is the target; its row 4 calls the two factors Jeffrey's rule, which holds for the contribution and not for the strength QEM returns from it |\n\n## 3. What the literature adds (not in the corpus before tonight)\n\nFull reports, verbatim, with every citation and what was read in full: `how-independent-reasons-add/`.\n\n- **Three ways to pool and when each is right** (Dietrich and List 2016; Dietrich 2010). Averaging\n  (linear pooling) has procedural defences and no epistemic one. The geometric pool suits sources that hold\n  the same information and differ in interpretation; it also returns 0.9 for two 0.9s. The multiplicative\n  pool, group odds = product of the members' odds over the shared prior's, is the only rule under which a\n  member learning a private fact moves the group as if the group had learned it; it suits private,\n  non-overlapping evidence, and gives 0.988. Their own example: experts who all say 0.9 from independent\n  private evidence should be *\"amplified, above 0.9\"*.\n- **Averages of calibrated sources are too timid** (Ranjan and Gneiting 2010, Theorem 1), and forecasting\n  practice agrees: in the Good Judgment Project the best aggregate pushed the mean's odds to a power of 2.43\n  (self-rated experts) or 3.08 (non-experts) (Baron, Mellers, Tetlock, Stone and Ungar 2014). Those factors\n  also undo each forecaster's own caution, so they are not a count of independent sources.\n- **The overlap dial is one formula in two literatures.** Clemen and Winkler (1985): *k* experts with\n  pairwise error correlation ρ are worth *k* / (1 + (*k* − 1)ρ) independent ones, ceiling 1/ρ; at the\n  ρ = 0.8 they found in football and economic forecasts, all experts after the first are worth a quarter of\n  one. Satopää, Pemantle and Ungar (2016) derive the same expression for forecasters sharing information.\n- **Same evidence or different, in miniature** (Satopää, Pemantle and Ungar 2016, Example 3.1): a fair coin\n  and a two-headed coin, one drawn at random; each of two forecasters saw a flip come up heads and alone\n  says 2/3 that it is two-headed. If they saw the same flip the right answer is 2/3; if different flips,\n  4/5.\n- **Counting dependent evidence as independent overclaims, measured** (Domingos and Pazzani 1997; Zadrozny\n  and Elkan 2001; Niculescu-Mizil and Caruana 2005; Bennett 2000): naive Bayes probabilities are pushed\n  towards 0 and 1, its log-odds grow with every added word, and its ranking stays good, so the damage is\n  invisible to anyone who checks only which answer comes first.\n- **Corroboration and convergence** (Schum, as read in his 2000 and 2009 papers): several reports of one\n  event are redundant, and a second report matters only as far as the first is doubted; reports of\n  different events can be redundant or synergistic. Ignoring redundancy overstates; ignoring synergy\n  understates. L. Jonathan Cohen's Baconian conjunction takes the minimum, which matches Q38's first\n  sentence (read only through reviews).\n- **A shared failure route caps a sum** (Thompson et al. 2003): a DNA match with a one-in-a-billion random\n  match probability, at prior odds 1:1000, gives posterior odds of a million to one; allow a lab\n  false-positive rate of one in ten thousand and the posterior odds fall to ten to one.\n\n## 4. The arithmetic, made plain\n\nEach argument for a claim, taken alone, gives the claim a strength by Jeffrey's rule: *start + relevance\n× (the argument's strength − start)*. Call that the argument's *alone value*. How several arguments combine\ndepends on what they share:\n\n| The arguments | How they combine | Two arguments at alone value 0.9 |\n|---|---|---|\n| share their evidence entirely | the best one counts (equal to the average when they agree) | 0.900 |\n| share part of it, overlap ρ | log-odds add, scaled to *n* / (1 + (*n* − 1)ρ) arguments' worth | ρ 0.5: 0.949 · ρ 0.25: 0.971 |\n| share nothing doubtful (no one story outside the claim explains both) | log-odds add: the odds multiply | 0.988 |\n| shipped code (QEM over summed contributions) | none of the above | 0.695 |\n\nThe log-odds row is the multiplicative pool, naive Bayes and Potyka's logistic semantics written in one\nline: *logit(strength) = logit(start) + Σ [logit(alone value) − logit(start)]*. Attacks enter the same sum\nwith the opposite sign. Under it the order in which reasons arrive does not matter, a reason's contribution\ncan be shown as a factor on the odds (*this reason makes the claim nine times likelier than not*), and two\nequal independent reasons contribute equal halves of the evidence.\n\n## 5. Tests on real cases\n\nEvery number below is recomputed from the published counts or likelihood ratios (`cases.py`); the helper's\narithmetic agreed. Each piece of evidence enters as its alone value, every rule starts from the published\nprior, and the shipped code is given each observed fact as a reason at strength 1 whose relevance yields\nthe same alone value, so it receives the same information.\n\n**5a. Cases with a published combined probability.**\n\n| Case | What combines | Published outcome | Shipped code | Best | Log-odds | Share of the summed log-odds the data show |\n|---|---|---|---|---|---|---|\n| a coin, two different flips seen (exact) | 2/3 and 2/3 alone, start 0.5 | 0.800 | 0.550 | 0.667 | **0.800** | 1.00 |\n| biliary obstruction, both tests positive (93 children, Pereira and Pericchi 1990) | 0.784 and 0.879 alone, base rate 0.484 | 0.933 | 0.645 | 0.879 | **0.966** | 0.80 |\n| the same, both tests negative | 0.119 and 0.267 alone | 0.103 | 0.355 | **0.119** | 0.050 | 0.73 |\n| strep throat, all four signs present (Aalbers et al. 2011; Centor score 4) | likelihood ratios 1.46, 1.65, 1.65, 2.20; base rate 0.171 | 0.557 | 0.210 | 0.312 | **0.643** | 0.83 |\n| the same, all four absent (Centor score 0) | likelihood ratios 0.53, 0.71, 0.56, 0.58 | 0.025 | 0.112 | 0.099 | **0.025** | 0.99 |\n\nMean absolute error over the four clinical cells: shipped code 0.244, best argument 0.097, log-odds 0.043.\nThe shipped code is the furthest from the outcome in every row, and in every row it is also further than the\nstrongest piece of evidence taken alone. Log-odds is closest in four rows and overshoots slightly where the signs share a\nprocess. A damping of the summed log-odds to 0.8 fits the four clinical cells within 0.02, but it was read\noff those same cells, so it is a fit, not a test. The helper reports a lower share (0.62) for the strep\nrow against the validation study's pooled likelihood ratio of 3.85 at score 4; the Centor derivation's\n55.7% gives 0.83.\n\n**5b. The same law where nothing is a probability.**\n\n- **Hearing and sight locating one event** (Alais and Burr 2004): two independent cues should sharpen a\n  location judgment by √2 = 1.414; observers improved by 1.425.\n- **Election forecasts** (Graefe et al. 2014): four methods cut the error by 48%, against 50% for four fully\n  independent ones, so they were worth 3.7 independent methods. The pairs differ the way the rival test\n  predicts: *models + polls* were worth 1.45 methods (overlap about 0.38, because the models use the polls;\n  if the polls were wrong, one story explains both), *prediction markets + experts* 1.98 (overlap about\n  0.01).\n- **A clinical score and a blood assay** (Wells et al. 2001, pulmonary embolism, 930 patients): the\n  D-dimer's implied likelihood ratio stayed between 0.22 and 0.38 across a thirtyfold range of prior, which\n  is what multiplying needs; the two have different mechanisms.\n\n**5c. Where adding went wrong, and what the rival test says.** In each case a sum of apparently independent\nreasons overclaimed, and in each the question *suppose the claim is false: does one story explain several of\nthese at once?* has an answer that says count them once.\n\n| Case | What was added | What happened | The one story |\n|---|---|---|---|\n| intravenous magnesium after a heart attack | seven small trials, pooled odds ratio 0.45, p < 0.001 (Teo et al. 1991) | a 58,050-patient trial found 1.06 (ISIS-4 1995) | small trials with selective publication |\n| ego depletion | 198 tests, pooled d = 0.62 | a 23-lab preregistered replication found 0.04 (Hagger et al. 2016) | the same |\n| two cot deaths in one family | 1/8,543 squared to 1 in 73 million | four to five second deaths a year in the UK, not one a century (Hill 2004) | a shared genetic or environmental cause |\n| DNA partial matches, Arizona database | the product rule across loci | right within a factor of 1.8 at nine loci, far off in the tail, where the matches were relatives | kinship |\n| three laboratories dating one cloth (Damon et al. 1989) | quoted errors, 689 ± 16 years BP | the scatter was about three times what the errors allowed; the paper widened to 691 ± 31 | the critics name a gradient along the one sampled strip (Di Lazzaro et al. 2020, read only as a snippet) |\n\n**What the cases teach, in one line each.** Reasons with different mechanisms add in log-odds almost\nexactly. Signs of one process keep about four fifths of their sum. Reasons one story outside the claim\nexplains together are worth the best of them, and adding them is how every overclaim above happened.\n\n## 6. Recommendations (RULED 2026-10-22, with Q46)\n\nIn order of dependency.\n\n1. **Independent arguments add in log-odds, and the squash goes** (ledger Q46). The open half of Q38: a\n   claim's strength is *logit(start) + Σ shifts* over the arguments read as independent, the best argument\n   standing for any group read as one story, each shift computed from the ruled two factors as now. One\n   fully relevant 0.9 reason then gives 0.9 and two independent ones 0.988. Evidence: § 5a, where the\n   shipped code was the worst rule in every case. It changes what every badge shows, so it ships with 4.\n2. **Build the keeper's independence reading (Q40) and test it on these cases first.** The rival question\n   should answer *one story* for the magnesium trials, the cot deaths, the related DNA profiles and\n   *models + polls*; *different stories* for the two coin flips, the score and the blood assay, *markets +\n   experts*. They are fixtures copied from reality, the way the ruled wording was tested on its six pairs\n   (`retrofitting-the-reasons-channel.md` § 9k). It is the lever with the largest payoff: in § 5a the gap\n   between the best argument and log-odds is up to 0.33.\n3. **Give the reading a middle answer, *partly one story*.** Signs of one process kept about 0.8 of their\n   summed log-odds; Clemen and Winkler's *n* / (1 + (*n* − 1)ρ) is the form. Proposed starting overlap\n   for *partly*: ρ = 0.25, inside the range these cells imply (0 to 0.37), to be measured as claims\n   resolve.\n4. **Honest starts stop being optional.** Under the squash the start barely mattered; under log-odds it\n   multiplies. Strep's four signs from a 0.5 start would read 0.90, where the outcome at the real base rate\n   is 0.56. So wherever a claim belongs to a kind with a known base rate, recommendation 1 needs the start\n   (ledger Q35; `starting-values-and-the-problem-of-the-prior.md` § 9 question 1); elsewhere 0.5 stays, as\n   now.\n5. **Replace the squash's accidental humility with a stated one.** The squash has been hiding uncalibrated\n   inputs. In its place: shrink each machine-proposed relevance towards its presumed value by the pass's\n   measured reliability (`bayesian-and-informal-logic.md` § 2.9), score calibration on claims that resolve\n   (Q41's row 6), and cap a sum by its shared failure routes (one misread source behind several reasons,\n   which sources as nodes, Q4, make readable).\n6. **Show contributions as evidence, not percentage points.** Each reason as a factor on the odds (*nine\n   times likelier than not*), equal independent reasons as equal shares, with the effective number (Q43)\n   and the record's weight (Q41, ruled tonight) beside the claim's strength.\n\n## 7. What he decided, and what follows\n\n**Ruled**: all of § 6 with Q46, the starting overlap ρ = 0.25 for *partly one story*, and the pick below (log-odds ships where a start exists, 0.5 elsewhere). **Owed**: the build (`TODO.md`); a walkthrough that teaches him the reasons, the mathematics and the philosophy under it, at his request; and, for his blessing, a clause on Q38's resident line: *independent arguments add in log-odds: two independent reasons at 0.9 give 0.988, the same evidence twice counts once, and one story explaining several reasons makes them count as their best.*\n\nThe questions as they were put:\n\n- **Q46**: do independent arguments add in log-odds, with QEM's squash retired on the reasons channel?\n- Whether the independence reading gets a middle answer, and its starting overlap.\n- Whether recommendation 1 waits for honest starts everywhere, or ships where a start exists and keeps 0.5\n  elsewhere (Claude's pick: the second, since 0.5 is no worse under log-odds than under the squash for a\n  claim with no known base rate).\n\nNot found by either helper: a dataset with separate premise strengths and an observed outcome for their\nconjunction, which would test Q38's first sentence or its proposed revision (§ 9j there); none was found\neither way.\n\n## 8. How the outcome-frequency rule (ledger AT8) meets this one\n\n*2026-10-22, late, on his question whether today's insight contradicts the row another session proposed\nthe same night (AT8, `what-relevance-is.md` § 3f): in a case, a link's relevance is the rule's strength\ntimes how often its outcome follows in cases like this one. His comment on it: \"Outcome-tracking\ndefinitely makes sense to me, seems bayesian enough...\"*\n\n**They agree in aim and meet in one spot of arithmetic.** Under the ruled two factors a link's relevance\nis Δ = P(claim | the reason holds) − P(claim | it does not) (`strength-calculus-literature/bayesian-and-informal-logic.md`\n§ 2.1). AT8's statistical half sets relevance to the rule's strength times the stated proportion, which is\nthe first term alone. That equals Δ only when the outcome never comes without the condition, that is when\nthe claim starts at 0. Red sky, *about a third of red-sky mornings bring rain* at strength 0.9, the\nreason observed (computed 2026-10-22; the rule's failure falls back to the start):\n\n| The claim's start (rain on any morning) | AT8 as written gives | The proportion read as the outcome gives | Relevance that implies | Odds factor |\n|---|---|---|---|---|\n| 0 | 0.30 | 0.30 | +0.30 | — |\n| 0.20 | 0.44 | 0.32 | +0.15 | 1.88 |\n| ⅓ | 0.53 | 0.33 | 0 | 1.00 |\n| 0.40 | 0.58 | 0.34 | −0.10 | 0.77 |\n| 0.50 (no base rate known) | 0.65 | 0.35 | −0.30 | 0.54 |\n\nSo a rule that says one red-sky morning in three brings rain would, as written, lift *rain today* to 0.65\nfrom an agnostic start: firmer than the rule itself allows. Under QEM's squash a relevance of 0.3 moved\nalmost nothing, so the gap was invisible; under log-odds with honest starts (§ 6 recommendations 1 and 4)\nit shows. AT8's normic half agrees exactly (a normic rule at strength 0.95 whose exception has a one in ten\nchance: 0.927 both ways), because there the outcome follows whenever the rule applies.\n\n**The repair, Claude's reading, unruled:** for a statistical rule the stated proportion sets the case's\nalone value (the rule's strength times the proportion, plus the start where the rule fails), and the\nrelevance is whatever moves the start there; a proportion below the start makes the link an attack. In\nCarnap's terms the proportion is *firmness* (how probable the outcome is given the condition) and a\nreason's relevance is *increase in firmness* (how much the condition raises it), the distinction in\n`bayesian-and-informal-logic.md` § 3.3. It needs no second number on the rule, so AT1 stands: the\ncomparison number is the claim's start, which recommendation 4 already makes the base rate where a kind\nhas one. Outcome-tracking is Bayesian once it tracks how much more often the outcome follows than it\nwould anyway.\n\n## 9. Does it contradict modern argumentation theory?\n\n*2026-10-23, his question: \"oh I meant to ask, does this new insight contradict modern argumentation theory\nand what the ppl who built QEM/QBAF or other modern argumentation theory software etc think?\" The helper's\nreport is kept verbatim in `how-independent-reasons-add/argumentation-builders-on-probability.md`; this\nsection is Claude's reading of it together with `strength-calculus-literature/`.*\n\n**No. It picks a formula the field already has, and answers the field's main worry its own way.**\n\n- **The formula is already in the family.** Potyka's MLP-based semantics (AAAI 2021) sets a claim to\n  logistic(logit(base) + Σ weight × strength), the logit-additive shape of Q46; he chose it for a simple\n  reading on 0 to 1 and found it satisfies the field's principles almost perfectly without being designed to;\n  he never names probability, log-odds or naive Bayes in it (searched). Mossakowski and Neuhaus's sigmoid\n  aggregation is a log-odds sum on another scale, and Munro, Bloch and Lesot (2026) combine two values by\n  x₁x₂ / (1 − x₁ − x₂ + 2x₁x₂), which is multiplying odds around 0.5, without naming Bayes. So Q46 moves\n  from one of Potyka's formulas to another of his, used on offsets.\n- **Why QEM was quadratic.** Potyka built it (KR 2018) to fix two faults: DF-QuAD's saturation (one\n  argument at strength 1 makes the others irrelevant) and Euler-based semantics' asymmetry between attack and\n  support. Why the square, and what it does to small contributions, is not stated (searched).\n- **The 0.57 is half ours.** Potyka's QEM on raw strengths gives one 0.9 supporter 0.72 and two 0.88.\n  Deliberus feeds it the offset from 0.5 (0.4), and the square of a small offset is smaller still. § 0 item 1\n  was corrected the same night; the figures 0.57 and 0.70 stand for the shipped code.\n- **What the number means is the real difference.** Most of the field treats strength as a measure of the\n  debate and leaves probability out of the output; Amgoud, Doder and Vesic (AIJ 2022) say probability, where\n  it enters, belongs in the base weight, and Potyka and Booth (KR 2024) call the comparison with probabilistic\n  approaches open. Reading strength as probability given the record is Deliberus's choice (Q41, R1 open),\n  and Prakken's three kinds of strength (inferential, dialectical, rhetorical) place it: Q46 is about the\n  inferential kind, which Prakken himself gave a probabilistic reading in 2018.\n- **The objection a builder would lead with: padding** (Potyka and Booth 2024). An unbounded sum lets many\n  near-worthless arguments swamp a claim, and their own table shows the logistic semantics fails both their\n  conservativeness bounds. Q46's answer has two halves: reasons that one story explains count once (Q40, as\n  Amgoud et al.'s similarity semantics count redundant arguments once, KR 2018), and truly independent weak\n  reasons should accrue (Godden and Zenker's four reasons, `bayesian-and-informal-logic.md` § 2.6). No\n  published semantics pairs a logit-additive sum with an independence discount (searched): Deliberus's\n  pairing is new.\n- **A choice the field says must be made.** Amgoud, Doder and Vesic show a semantics picks one of cardinality\n  precedence, quality precedence or compensation (their example: does a Fields medalist outweigh three\n  students?). Q38 with Q46 picks quality precedence until independence is read (the best argument), then\n  compensation (log-odds) among independent arguments.\n- **Two differences stay, owed to the build.** Directionality: in probability, news about a conclusion flows\n  back to its reasons (explaining away); the reasons channel flows forward only, which is what Potyka's\n  tutorial names the key difference. And the edges: a fully relevant reason at strength 0 or 1 gives an alone\n  value whose log-odds are infinite (the saturation Potyka criticised in DF-QuAD), so the build clamps alone\n  values short of 0 and 1 (`TODO.md`, *How independent arguments add: build Q46*).\n- **Software.** Carneades weighs an argument by its weakest premise and compares the best pro with the best\n  con, on an ordinal scale, having rejected a 0–1 probability scale as more than users can estimate; Arg&Dec,\n  ArgLLM and ArgRAG use DF-QuAD or QEM; only Bayesian-network tools (Agena, BARD, Fenton and Neil's idioms)\n  combine evidence by probability, jointly over the whole network. No argument-mapping tool found adds\n  log-odds across reasons.\n"}