{"path":"research/what-a-case-counts-for.md","content":"# What a case counts for: induction, evidence as a subgraph, and what an explanation does\n\n**Date**: 2026-10-23 · **Type**: the founder's challenge to two cells of a table, worked through the corpus,\noutside sources and five cases · **Status**: Claude's reading and proposals (ledger Arc AY). Nothing here is\nruled. **Arithmetic**: [cases.py](what-a-case-counts-for/cases.py), its output\n[what-cases-printed.txt](what-a-case-counts-for/what-cases-printed.txt), the trial counts\n[bcg_trials.csv](what-a-case-counts-for/bcg_trials.csv). **Sources checked**:\n[literature-as-read.md](what-a-case-counts-for/literature-as-read.md).\n\n**His words**, 2026-10-23, to a table in\n[normic-rules-normative-reasons-and-trying-rules-out.md](normic-rules-normative-reasons-and-trying-rules-out.md)\n§ 7 that said a one-case claim's explanation *\"takes nothing back\"* and that a counterexample to a\nstatistical rule *\"counts only through frequency\"*: *\"I love where we're going with this, but... Think deeply\nbefore we proceed about \"takes nothing back\" for the \"one case\" and about the \"counts only through frequency\"\nfor a statistical rule. This is actually primed to become core to how Deliberus handles induction arguments\nand how evidence-as-subgraph is realized. Think this through. Ultrathink\"*\n\n**What is ruled and what is open.** Ruled: a rule claim carries one number, how sure we are that the rule as\nwritten holds (AT1); every writer of a rule asks whether a failure calls for an explanation (AT2); for a\nnormic rule a failed case is an objection until an explanation naming a kind attacks its relevance (AT3, ruled\nthe same day, with a statistical clause this note corrects); how often a statistical rule's outcome follows\nsets the link's relevance in a case (AT8); independent reasons add in log-odds and the best counts alone\nuntil independence is read (Q38, Q46). Open, and the question here: **what a single case counts for, and\nhow a rule's number is read from its cases.**\n\n## 0. The short version\n\n1. **Two cells of the table were wrong.** An explanation of a one-case claim takes nothing back from the case,\n   and it does two things the cell hid: it says which link of the reasoning that predicted the case failed,\n   and it moves the case to the rule it counts for (the gate shut on a holiday stops counting against *the\n   gate is open every Tuesday* and starts counting for *the gate is closed on public holidays*). And a\n   statistical rule is not moved only by a run of cases: **every case moves it, and which way depends on where\n   its stated proportion sits.** From a flat start, one wet red-sky morning lowers *about a third of red-sky\n   mornings bring rain* from 0.17 to 0.11, and one dry morning raises it to 0.23.\n2. **One computation serves all three kinds of rule.** A rule's wording states a band for how often its\n   outcome follows: about a third (statistical), nearly always once its named exceptions are set aside\n   (normic), always (strict). Its one number is how sure its cases make us that the true rate lies in that\n   band. The cases are pooled as a count, *k* of *n*, with no model.\n3. **What an explanation does depends on the rule.** It always names a kind. For a normic rule it takes the\n   case out of the rule's count, an exception. For a statistical rule it proposes a narrower rule with its own\n   proportion, kept if it carries spread the broad rule left unexplained. For a strict rule nothing saves it,\n   and the repair is a narrower wording. For a one-case claim it places the blame.\n4. **This is evidence as a subgraph, made exact.** A study is a bundle of cases filed under a rule; a\n   meta-analysis pools bundles; heterogeneity is unexplained spread between bundles, which calls for a\n   narrower rule; a moderator is an explanation tested by how much of the spread it carries. In 13 real BCG\n   vaccine trials one effect leaves 92 % of the variation between trials unexplained; latitude carries about\n   three quarters of it (75.6 % by the usual estimator), and the narrower rule gives a child in Madras a risk ratio of 0.88 where the broad rule says 0.49.\n   `heterogeneity` and `study_count`, extracted and inert today, get their jobs.\n5. **The reasons channel cannot take a rule's cases.** With their independence unread, thirty cases count as\n   their best one (0.23 where the count gives 0.69), and the independence reading would likely read two cases\n   of one rule as one story, since they share the rival explanation. Cases need a channel of their own: pooled\n   there by count, the count then enters the rule's reasons as one argument, its shift the log of the count's\n   Bayes factor, beside any mechanism argued for the rule.\n\n## 1. One example, as it would sit in the graph\n\n- Claim, one case: *the gate is open now*.\n- Reason, one case, linked as support: *today is Tuesday*.\n- The rule under that link: *this gate is open every Tuesday* (normic).\n- Cases filed under the rule: *the gate was open on Tuesday 7 October*, *the gate was shut on Tuesday 14\n  October*, *the gate was open on Tuesday 21 October*.\n- An exception claim: *the gate is closed on public holidays*, with *14 October was a public holiday*.\n\nWhat the exception does: *shut on 14 October* stops counting against the Tuesday rule and starts counting for\nthe holiday rule. The case itself is untouched; it was true and stays true. If *the gate is open now* had been\npredicted on a holiday Tuesday and the gate was shut, the one-case claim is false, and the exception says why:\nthe rule's application failed today, not the rule and not *today is Tuesday*.\n\n## 2. What the corpus already holds\n\n| Angle | Where | What it says |\n|---|---|---|\n| Four shapes of a step, *example* is from cases to a rule | [what-a-rule-rests-on.md](what-a-rule-rests-on.md) § 6 | 84 live links carry *example*; its standing questions ask whether the example is representative |\n| A case's two homes, counted twice | same, § 7 | *\"as further links into the conclusion, or as support for the rule or for the likeness. Counted in both, they are counted twice.\"* Left open there; § 7 below answers it |\n| Mill's question, why one case sometimes suffices | same, § 5 | a wider rule one level up, learned from cases; *overhypothesis* named, no arithmetic |\n| His own words on induction | [scheme-bounded-decomposition-and-evidence-as-subgraph.md](scheme-bounded-decomposition-and-evidence-as-subgraph.md) | 2026-03-29: *\"But of course there's also inductive arguments, right? And the whole thing about what constitutes a good bayesian inductive argument and so on.\"* And: a meta-analysis *\"should be decomposed as far as possible\"* |\n| Weight of evidence adds under conditional independence | [what-counts-as-evidence/philosophy-of-evidence.md](what-counts-as-evidence/philosophy-of-evidence.md) § 2.4 | Good's additivity, and that summing is licit only under conditional independence |\n| Confirmation by degree, the ravens | same, § 2.2 | a count that treats every confirming instance as one unit is led astray |\n| Blame after a failed prediction is computable | same, § 3.1 | Duhem and Quine, with Dorling and Strevens: once the auxiliaries are explicit |\n| Variety of evidence is no theorem | same, § 3.2; [what-a-rule-rests-on.md](what-a-rule-rests-on.md) § 7 | a repeat can beat a varied case; a shared source becomes a node |\n| A universal law has probability zero | philosophy-of-evidence § 2.7 | Popper's starting point |\n| The start comes from a reference class | [starting-values-and-the-problem-of-the-prior.md](starting-values-and-the-problem-of-the-prior.md) | argued in the open; Hájek, the reference class problem is everyone's |\n| Normic laws imply high frequency | [normic-rules-normative-reasons-and-trying-rules-out.md](normic-rules-normative-reasons-and-trying-rules-out.md) § 1 | Schurz: a normic rule still has a frequency of exceptions |\n| Heterogeneity weighs nothing | [long-form-sources-and-meta-analysis-weighing.md](long-form-sources-and-meta-analysis-weighing.md) | a twelve-study meta-analysis at I² 0.73 scores as a forty-study one at 0.05 |\n\nNot in the corpus before today: Laplace's rule of succession, the Beta-binomial reading, de Finetti's\nexchangeability, Carnap's continuum and its zero for universal laws, the point-mass repair, Reichenbach's\nnarrowest reference class (four passing mentions), random-effects meta-analysis and meta-regression. § 3 adds\nthem, from the checked sources.\n\n## 3. What the literature adds\n\nEach item as checked in [literature-as-read.md](what-a-case-counts-for/literature-as-read.md), with how much\nof the source was read.\n\n- **Laplace's rule of succession** (1814, *Essai philosophique*, read in English translation): after *k*\n  successes in *n* trials the next succeeds with probability (*k* + 1)/(*n* + 2), from a flat start, the\n  Beta(1, 1) of today's terms. His own example is the sunrise, and he adds that knowing the mechanism makes the\n  probability *\"incomparably greater\"*: the count is a floor that other knowledge raises, which is AY2's count\n  beside a mechanism argued for the rule. De Morgan called its results *\"merely minima\"*.\n- **De Finetti's representation** (1931, 1937; cited): an exchangeable sequence behaves as independent draws\n  *given* the unknown rate, and the counts are all that matter. Cases are conditionally independent given the\n  rate and dependent without it, which is § 8's reason the reasons channel cannot take them. For a finite run\n  the theorem holds only approximately (Diaconis and Freedman 1980, cited).\n- **Carnap's continuum** (1952, read in part; 1945, read): for a yes-or-no property his methods give\n  (*s* + λ/2)/(*n* + λ); λ = 2 is Laplace's rule. In an infinite domain his c* gives a universal law\n  probability 0 *\"for either law\"*, whatever the cases (1945, p. 92). The straight rule (λ = 0) does not, so\n  the zero belongs to the smoothed methods.\n- **The point-mass repair**: Broad (1918) showed that a flat start makes *all* members alike improbable in\n  any large population; Jeffreys called the result *\"preposterous\"*. Wrinch and Jeffreys (1919, 1921) gave the\n  general law a finite start of its own, and Haldane (1932) and Jeffreys (1933) wrote the mixture. With half\n  the start on the law and a flat start on the rest, *n* conforming cases give the law (*n* + 1)/(*n* + 2)\n  (Wagenmakers, Zabell and Gronau 2024, read): case C's figures exactly. Hintikka (1966) and Zabell (1996)\n  built continua that confirm universal laws (neither read).\n- **Overhypotheses** (Kemp, Perfors and Tenenbaum 2007, read): a model that learns from earlier bags how\n  uniform bags are reads a single black marble from a new bag as near certain (0.99), where a model with the\n  uniformity fixed in advance learns nothing about the new bag from twenty earlier ones. Their tribe example:\n  one member is almost as informative as twenty about skin colour, not about weight. This is AY3's start from\n  sibling rules, and Mill's question (*\"Why is a single instance, in some cases, sufficient for a complete\n  induction\"*, *System of Logic* III.iii.3, read) answered. The paper does not cite Mill; the link is ours.\n- **Reichenbach and Kyburg** (secondary): Reichenbach's rule is the narrowest class *\"for which suitable\n  statistics can be compiled\"*, not *reliable*, as this note first said. Kyburg's refinement (1974, 1983) uses\n  intervals: a narrower class overrides a broader one only when its interval disagrees with the broader one's.\n  AY6 now carries that condition.\n- **Schurz on normic laws** (2001, abstract; SEP's account): normic laws imply that most A are B with *high*\n  probability, because self-regulating systems are in their normal states most of the time. No threshold is\n  given. The band *at least 0.95* in case B is this note's reading, not his.\n- **Random effects and moderators**: DerSimonian and Laird (1986, abstract) propose the random-effects model\n  and note that covariates *\"would reduce the heterogeneity\"* and give *\"more specific therapeutic\n  recommendations\"*. Thompson and Higgins (2002, abstract): meta-regression is observational and *\"data\n  dredging is the main pitfall\"*, so moderators should be named in advance. AY4 and AY5 now carry that.\n- **The BCG trials**: the 13 trials are in the metadat R package (`dat.bcg`); Colditz et al. (1994) report\n  14 trials and that latitude and study validity together *\"explained 66% of the heterogeneity\"*. The share\n  latitude carries alone depends on the estimator: 79 % by the moment estimator this note first used, 75.6 %\n  by restricted maximum likelihood (REML, the usual one, recomputed here and matching the metafor package),\n  41 % by the empirical Bayes estimator of Berkey et al. (1995). Why the dataset holds 13 trials where Colditz\n  counts 14 was not found.\n\n## 4. The corrected table\n\n| Reading | Example | A failing case | An explanation of it |\n|---|---|---|---|\n| one case | *the gate is open now* | is its denial | places the blame in the reasoning that predicted it, and moves the case to the rule it counts for |\n| strict rule | *no four fifth powers sum to a fifth power* | ends it | saves nothing; the repair is a narrower wording |\n| normic rule | *the gate is open every Tuesday* | counts against it | an exception: out of this rule's count, into the exception rule's |\n| statistical rule | *about a third of red-sky mornings bring rain* | moves it a little, as every case does, either way | proposes a narrower rule with its own proportion |\n\n## 5. One computation for three kinds of rule\n\nA rule's wording states a band for how often its outcome follows when its condition holds. Its one number\n(AT1) is how sure its cases make us that the true rate lies in that band: with a Beta start and *k* of *n*\ncases conforming, the mass of the Beta(start + *k*, start + *n* − *k*) between the band's ends. Code\ncomputes it; no model is asked.\n\n**Statistical** (case A, by construction; band 0.25 to 0.42 for *about a third*, flat start):\n\n| Cases | How sure the rule holds |\n|---|---|\n| none | 0.170 |\n| one wet morning | 0.114 |\n| one dry morning | 0.226 |\n| 10 wet of 30 | 0.691 |\n| 20 wet of 30 | 0.003 |\n| 100 wet of 300 | 0.998 |\n\nA wet morning, the outcome the rule names, lowers it, because a flat start puts most of its weight on rates\nabove a third. *Confirming instance* has no meaning for a statistical rule: a case is data about the rate, and\nits direction depends on where the stated rate sits. This is the cell the table had wrong.\n\n**When the count lands outside the band, it also says where the rate is.** Twenty wet mornings of thirty\nleave *about a third* at 0.003 and hold *about two thirds* (0.58 to 0.75) at 0.691; the next morning's chance\nof rain read from the count is 0.656. AT8, ruled today, reads a case through a rule as the rule's strength\ntimes its stated proportion plus the start where the rule fails, which here gives 0.499: the start, while\nthirty cases say two in three. AT8 is right as ruled once the count's rival wording exists as a claim beside\nthe first (AT2's *a dispute about the kind is a rival wording*; AT12's repair), because the case is then read\nthrough the wording that holds. So the count does two jobs: it gives the written rule its number, and where\nthat number falls it proposes the wording the cases support, computed with no model.\n\n**Normic** (case B; *the gate is open every Tuesday, except on public holidays*, read as unexplained failures\non at most 5 % of Tuesdays, flat start): 10 open Tuesdays give 0.431; one unexplained shut Tuesday drops it to\n0.118; when the shut Tuesday turns out to be a holiday it leaves the count and the rule is back at 0.431; 50\nopen Tuesdays with two explained holidays give 0.927, with two unexplained shut Tuesdays 0.498. A normic rule\nis a statistical rule whose band sits near always (Schurz), which is also why it escapes Carnap's zero: its\nband has width.\n\n**Strict** (case C, real): Euler conjectured in 1769 that no four fifth powers sum to a fifth power.\n27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵ = 61,917,364,224 (recomputed). With a flat start on the failure rate,\n*exactly no failures* has probability zero however many cases conform. With half the start placed on it, 10\nconforming cases give 0.917, a thousand 0.999; one counterexample gives 0 exactly. A strict rule therefore\nneeds a point mass in its start, and that start is a claim of its own (*this is the kind of rule that holds\nwithout exception*), which can be argued.\n\n## 6. What an explanation does\n\nAlways the same move: it names a kind that applies to other cases and can fail on its own (AT3's guard). What\nthe kind does depends on the rule.\n\n- **Normic**: the case leaves the rule's count, and enters the count of the exception rule.\n- **Statistical**: the case stays in the broad rule's count, and the explanation proposes a narrower rule\n  (the reference class split by the named kind) with its own count. The narrower rule is kept if it carries\n  spread the broad rule left unexplained.\n- **Strict**: nothing saves it; the repair is a rival, narrower wording, a new claim.\n- **One case**: it places the blame among the links of the reasoning that predicted the case, which is how\n  Duhem and Quine's holism becomes computable once the links are explicit.\n\n**Case D, real: one move, two outcomes.** In the 1840s Uranus strayed from the orbit Newton's law gave it over\nthe known planets. Adams and Le Verrier explained the stray by an unseen planet, an exception claim that named\na kind (a massive body beyond Uranus) and could fail on its own (it said where to look). Galle found Neptune in\n1846 close to Le Verrier's position: the explanation survived its own test, and Newton's law kept its count.\nIn 1859 Le Verrier made the same move for the drift of Mercury's perihelion and proposed a planet inside its\norbit. No such planet was found, so the drift stayed unexplained and counted against the law, until general\nrelativity accounted for it in 1915. (History as commonly told; dates from memory, not checked this session.)\nRead in the graph: the observed orbit of Uranus is a one-case claim that was never taken back; the explanation\nmoved the blame from the law to an auxiliary (the list of planets), and Neptune's sighting became a case of\nthe explanation itself.\n\n## 7. Evidence as a subgraph: a study is a bundle of cases\n\n**Case E, real.** Thirteen controlled trials of the BCG vaccine against tuberculosis (Colditz et al. 1994, as\ndistributed in the metadat R package; counts in [bcg_trials.csv](what-a-case-counts-for/bcg_trials.csv),\nrecomputed in [cases.py](what-a-case-counts-for/cases.py)). Read as one rule with one effect, *BCG lowers\ntuberculosis risk*:\n\n| | |\n|---|---|\n| pooled risk ratio (random effects) | 0.49 |\n| Cochran's Q | 152.2 on 12 degrees of freedom |\n| I² (share of the variation not due to chance) | 92 % |\n| spread between trials, τ² (REML; moment estimator 0.309) | 0.313 |\n| with latitude named: residual τ² (REML; moment 0.063) | 0.076 (75.6 % carried by latitude; 79 % by the moment estimator, 41 % by empirical Bayes) |\n| narrower rule's risk ratio at 13° (Madras), REML (moment: 0.89) | 0.88 |\n| at 55° | 0.26 |\n\nMapped onto the graph: each trial is a bundle of cases (*k* cases of tuberculosis among *n* vaccinated, and\namong *n'* controls) filed under the rule; the meta-analysis is the pooled count; the 92 % is unexplained\nspread between bundles, a call for an explanation; *the vaccine protects more at higher latitude* is an\nexplanation that names a kind (distance from the equator, a proxy for exposure to other mycobacteria) and can\nfail on its own; it carries most of the spread, so it is kept as a narrower rule. A moderator found by\nsearching the same trials it is tested on is a fit, not a test (Thompson and Higgins's data dredging): it is\nkept only if it was named in advance or holds on bundles not used to find it. Whether latitude was named before the\ntrials were pooled was not checked here; the explanation usually given is exposure to other mycobacteria\nnear the equator. A rival that fits one trial\nonly (*the Madras trial was unlucky*) would carry none of it and explain nothing, which is AT3's guard measured.\n\nAnd the one-case claim changes: *BCG will lower the risk of this child in Madras* is read with a risk ratio\nof 0.49 under the broad rule and 0.88 under the narrower one, nearly no protection. Which rule a single case is read through is Reichenbach's question,\nthe narrowest class for which suitable statistics can be compiled, refined by Kyburg so that a narrower\nclass overrides only where its interval disagrees with the broader one's, and in the graph it is a visible link someone can argue (the\nreference class of [starting-values-and-the-problem-of-the-prior.md](starting-values-and-the-problem-of-the-prior.md)).\n\nSo the extracted fields get their jobs: `study_count` is the number of bundles, `sample_size` and the effect\nare a bundle's counts, `heterogeneity` is the unexplained spread that calls for a narrower rule. None of it\nneeds a model to compute once the cases are filed.\n\n**Heterogeneity weighs, and through the computation, not an attack.** Read with the spread (moment estimator), the broad\nrule says less about one new setting than about the average: its mean effect protects with probability above\n0.999, while one new setting's risk ratio has a 95 % range of 0.16 to 1.54 and protects with probability\n0.889. That is the corpus's August wish (*heterogeneity should move strength*,\n[long-form-sources-and-meta-analysis-weighing.md](long-form-sources-and-meta-analysis-weighing.md) § 2,\n[scheme-bounded-decomposition-and-evidence-as-subgraph.md](scheme-bounded-decomposition-and-evidence-as-subgraph.md))\nmet without the step the stress suite showed was missing: a minted heterogeneity claim attacking the\nmeta-analysis moved nothing until someone scored its link, while a spread computed from filed bundles moves\nthe reading of every case at once. Where only a reported I² is known and no bundle counts are filed, the\nAugust route stands: the spread is a claim someone states, and it is argued like any other.\n\n## 8. Why the reasons channel cannot take a rule's cases\n\nThe reasons channel adds separate arguments for a claim (Q46). A rule's cases are not separate arguments:\nunder a rule whose rate is unknown they are exchangeable, so one case tells us about the rate and so about the\nnext. Three ways to aggregate thirty red-sky mornings, ten of them wet (case A):\n\n| Aggregation | How sure the rule holds |\n|---|---|\n| pooled as a count (Beta-binomial) | 0.691 |\n| summed as thirty independent reasons | 0.702 |\n| the best single case alone (Q46 with independence unread) | 0.226 |\n\nSumming came close here, because a narrow band behaves almost like a single rate; it drifts further as bands\nwiden. The real failure is the third row, which is today's default: until independence is read, a rule with\nthirty cases reads as strong as its best one. And the independence reading would likely not rescue it. Its\nquestion, *suppose the claim is false: is each reason's most likely explanation the same?*, gets the same\nanswer for every case of one rule (the rate is not a third), so it would read them as one story and count\nthem once. That is right for separate arguments and wrong for cases. **Cases are pooled by count in a channel\nof their own, and the reasons channel never sees them.** The rule reaches a one-case conclusion through one\napplication link, whose relevance AT8 sets from the rule's stated rate. A case is counted once, under the rule,\nwhich answers the double count [what-a-rule-rests-on.md](what-a-rule-rests-on.md) § 7 left open.\n\n**How the count meets the rule's other arguments.** A rule can also be argued for without cases: a mechanism\n(*red sky at dawn means clearing to the west and weather arriving from the west*), a cited authority. Those\nare separate arguments for the rule claim and go in its reasons channel (Q46). The count joins them there as\n**one** argument, whose shift in log-odds is the log of the count's Bayes factor: for ten wet mornings of\nthirty, the band's start of 0.170 (log-odds −1.59) and the count's result of 0.691 (log-odds 0.80) make a\nshift of 2.39, a factor of about 11. Q46's arithmetic then adds the mechanism beside it, unless the two share\nan origin (the mechanism was itself read off the same mornings). So the whole path is: cases, pooled into a\ncount; the count, one argument for the rule; the rule, through one application link, to the one-case claim.\n\n## 9. Proposals (ledger Arc AY; AY1 ruled 2026-10-23, the rest unruled), in order of dependence\n\n1. **AY1, a correction to the ruled exception row (AT3)**: its statistical clause says *only a run of cases\n   departing from its stated proportion counts against it*, and its canon check says *one failure leaves a\n   statistical rule unmoved*. Case A shows both are wrong: every case moves a statistical rule a little,\n   either way; what no single case does is call for an explanation. **RULED 2026-10-23**: he chose the\n   reply *\"Rule AY1 (every case moves a statistical rule) as proposed: amend the exception row's statistical\n   clause and its canon check.\"* (a button whose wording the session wrote); AT3 is amended in the ledger.\n2. **AY2, a case counts once, under the rule whose condition it meets**: cases are filed under rules (the\n   *example* links already in the graph) as conforming, failing, or explained, and reach a one-case conclusion\n   only through the rule's application.\n3. **AY3, a rule's number is read from its count**: the band from its wording, the cases pooled, the\n   computation in code; the start from sibling rules where they exist (Mill's question answered as a learned\n   start), else flat; a strict rule's start carries a point mass, argued. A count outside the band proposes\n   the rival wording it supports, so a case is never read through a failed wording back to the start (§ 5).\n4. **AY4, what an explanation does depends on the rule** (§ 6), each tested by what it carries.\n5. **AY5, a study is a bundle of cases** (§ 7): meta-analysis as a pooled count, heterogeneity as a call for a\n   narrower rule, a moderator kept by the spread it carries.\n6. **AY6, a one-case claim is read through the narrowest rule for which a suitable count exists**, a\n   narrower rule overriding only where the counts disagree (Kyburg), chosen in a visible, arguable link.\n\nWhat only he decides: whether cases get their own channel (AY2, AY3), which is an ontology decision; how a\nband is read from a wording such as *about* (AY3). Whether AT3's clause is amended (AY1) he ruled the same\nday: it is.\n\n## 10. Read and not checked\n\nThe history in case D is as commonly told, from memory, and was not checked. Schurz's account was read only\nin its abstract and an encyclopedia summary; Reichenbach and Kyburg only at second hand. The BCG counts are the published dataset; the\nfigures in § 7 are recomputed with DerSimonian and Laird's moment estimator, so they differ slightly from\nfigures fitted by restricted maximum likelihood. The literature in § 3 is as checked in\n[literature-as-read.md](what-a-case-counts-for/literature-as-read.md), with each source's read status there.\n\n## Cross-references\n\n[normic-rules-normative-reasons-and-trying-rules-out.md](normic-rules-normative-reasons-and-trying-rules-out.md)\n§ 7 · [what-a-rule-rests-on.md](what-a-rule-rests-on.md) § 5–§ 7 ·\n[how-independent-reasons-add.md](how-independent-reasons-add.md) ·\n[scheme-bounded-decomposition-and-evidence-as-subgraph.md](scheme-bounded-decomposition-and-evidence-as-subgraph.md) ·\n[long-form-sources-and-meta-analysis-weighing.md](long-form-sources-and-meta-analysis-weighing.md) ·\n[starting-values-and-the-problem-of-the-prior.md](starting-values-and-the-problem-of-the-prior.md) ·\n[what-counts-as-evidence/philosophy-of-evidence.md](what-counts-as-evidence/philosophy-of-evidence.md) ·\n[what-relevance-is.md](what-relevance-is.md) § 3f.\n"}