{"path":"research/reasons-and-cases/literature-weight-of-evidence-and-exchangeability.md","content":"# Literature: do independent reasons add like cases moving a rule? (helper report, verbatim)\n\n*A research helper's report for [../reasons-and-cases.md](../reasons-and-cases.md), 2026-10-23, kept as it\narrived. Brief: the additivity of weight of evidence and when it holds; confirmation by instances against\nindependent evidence; how argumentation theory treats several examples; hierarchical models as the bridge;\nany source treating reasons and instances as one structure. Read-status tags are the helper's own; it was\nasked to stop and report before it had checked everything it planned (its last section lists what it did not\nread). \"My computation\" marks the helper's arithmetic; the note recomputes what it relies on.*\n\n---\n\n# Research report: do independent reasons add like cases moving a rule?\n\nRead-only task. No files written. Tags: [full text], [part], [abstract only], [secondary]. Items marked \"my computation\" are my own arithmetic, not literature.\n\n## Bottom line\n\n- Your theory (a)-(c) is mostly right. It needs three corrections.\n  1. Additivity needs both hypotheses to be simple. \"C versus not-C\" is composite whenever not-C has several rival explanations. So the rival-explanation test for reasons and the exact-rate problem for cases are the same structure: a latent variable integrated out inside a composite hypothesis.\n  2. The add-or-best choice for reasons is a crude stand-in for the same \"third option\" that the count gives for cases. The exact answer for dependent reasons is the conditional (chain-rule) term, which usually lies between \"best\" and \"sum\".\n  3. The direction of the error from naively summing per-case marginal factors is not fixed.\n- No source says outright that \"reasons for a claim\" and \"instances of a generalization\" are one structure at different levels. The closest are Keynes (1921), Peirce (1878) and the formal statements in Good and Jaynes.\n\n## 1. Additivity of weight of evidence\n\n**Good 1985, \"Weight of evidence: a brief survey\"**, Bayesian Statistics 2, pp. 249-270. https://www.waterboards.ca.gov/water_issues/programs/tmdl/docs/303d_policydocs/207.pdf [part: pp. 249-252, the Rubin reply]\n- Eq. (1), p. 251: W(H:E&E') = W(H:E) + W(H:E'|E). This is the always-valid chain rule, with a conditional second term.\n- p. 252 is the explicit statement of the composite-hypothesis point you asked for. Good says (my paraphrase) that the additive property simplifies when E and F are independent given H and given not-H. Then: \"This condition usually requires that both H and H̄ should be simple statistical hypotheses\". He credits Herman Rubin with the point.\n- p. 251: for general hypotheses the ratio P(E|H)/P(E|H̄) \"could be called a ratio of Bayesian likelihoods\".\n- In his reply, Good notes that P(E|H) is not well defined for composite H outside sharp Bayesian statistics.\n\n**Good 1979, \"Turing's statistical work in World War II\"**, Biometrika 66(2): 393-396. https://jhanley.biostat.mcgill.ca/bios601/CandH-ch0102/GoodOnTuring.pdf [full text]\n- §2: the Bayes factor equals P(E|H)/P(E|H̄). \"The result is especially 'Bayesian' if either H or H̄ is composite.\"\n- §3: Turing was an independent inventor of sequential analysis, using the log factor.\n- §5: the main 1941 use was with simple hypotheses, where the factor reduces to a likelihood ratio. Sometimes a \"weighted averages of factors\" theorem applied (Good 1950, pp. 68, 71). I did not read that theorem.\n- The decibans and banburismus passages describe summation of scores. No independence assumption is stated there.\n\n**Jaynes, *Probability Theory*, ch. 4** (chapter draft c4o, 9/19/2004; check equation numbers against the 2003 book). https://omega0.xyz/omega8008/ETJ-PDF/cc4o.pdf [part]\n- Eq. (4-10) expands the likelihood ratio by the product rule. Jaynes: \"we always have additivity, but not independent additivity unless the probabilities are independent.\"\n- Eq. (4-11) is the sum of per-datum terms. It holds when the data are independent given H and given not-H.\n- \"Non-Extensibility Beyond the Binary Case\", Exercise 4.1: with more than two exhaustive hypotheses, the independence conditions (4-26) and (4-27) cannot both hold except trivially.\n- Section \"Simple and Compound Hypotheses\" (eqs 4-63, 4-64): a composite hypothesis is handled by integrating out the nuisance parameter. With a uniform prior, P(a<f<b|data) is an incomplete Beta function. This is your pooled-count band probability.\n- The 50-of-50 bad widgets example shows the sum of per-datum db going absurd until a rival hypothesis is added.\n\n**Dawid 1984**, JRSS A 147(2): 278-292. https://www.cs.ubc.ca/~murphyk/MLRG/dawid84Prequential.pdf [part: pp. 278-285 read as page images]\n- §1: inference is for sequential forecasts, \"rather than to express information about parameters\".\n- §4: the Bayes forecast is ∫ P_θ(·|x^(n)) dΠ_n(θ). Example 1: for a uniform prior the forecast is (r+1)/(n+2), Laplace's rule.\n- §7: the prequential likelihood is the product of the one-step forecast densities. The ratio of two forecasters' prequential likelihoods is a product of successive ratios, and it is a martingale.\n\n**Kass & Raftery 1995**, JASA 90: 773-795, §3.2, p. 777. https://sites.stat.washington.edu/courses/stat527/s13/readings/KassRaftery_jasa1995.pdf [part]\n- The log Bayes factor is a difference of predictive scores. \"the Bayes factor can be viewed as measuring the relative success of H1 and H0 at predicting the data\".\n- They link this to Dawid's prequential analysis.\n\n**Grünwald 2004**, MDL tutorial, arXiv math/0406077. https://arxiv.org/pdf/math/0406077 [part]\n- Eqs (2.26)-(2.27): P(x^n) is the product of P(x_i|x^{i-1}).\n- Eq. (2.12): the Bayes marginal likelihood is a prior mixture.\n- Example 2.19: the Bayes predictive for a uniform prior is Laplace's rule.\n\n**Shafer, Shen, Vereshchagin & Vovk 2011**, Statistical Science 26(1): 84-101. https://arxiv.org/pdf/0912.4269 [part]\n- §8.1, simple against simple (θ=3/4 vs 1/2): the likelihood ratio is 3^k/2^t, a product of per-case ratios. \"The sequence of successive values of this likelihood ratio is a test martingale\".\n- §8.5, a band against its complement: θ≤1/2 against θ>1/2, each represented by a uniform measure. The test martingale is a ratio of integrals, equal to P{B_{t+1}≤k_t}/P{B_{t+1}≥k_t+1}. This is exactly your composite case.\n- §2 credits Wald and Barnard with sequential analysis. I did not read Wald directly.\n\n**Nikolakopoulos, Edmar & Ntzoufras**, \"Bayesian Evidence Synthesis for the common effect model\", arXiv 2103.13236 (v2 Jan 2026). https://arxiv.org/abs/2103.13236 [part: §3.2]\n- A product of study Bayes factors \"is not appropriate for the, usually assumed, common effect model\".\n- They write m(K) = f(y1) × f(y2|y1) × ..., a product of conditional marginal likelihoods, f(y_k|history) = ∫ f(y_k|θ) p(θ|history) dθ.\n- The product of Bayes factors can deliver evidence for the fixed-effects model, where each study has its own effect.\n- My reading, not their text: a product of per-case marginal factors is the exact Bayes factor of a different model, in which every case draws its own rate.\n\n**Rouder & Morey 2011**, Psychonomic Bulletin & Review 18(4): 682-689, DOI 10.3758/s13423-011-0088-7. Manuscript copy: https://legacy-www.math.harvard.edu/~knill/teaching/math19b_2011/exhibits/esp/rouder-morey.pdf [part: appendix]\n- \"Readers may have the intuition that one should simply multiply odds. Unfortunately, this approach is not valid.\"\n- They use a witnesses analogy: no single witness is compelling, but the totality shifts toward guilt. So the naive product can understate the evidence.\n\n**Chugg, Ramdas & Grünwald**, arXiv 2603.24421 (v2, dated Aug 2026). https://arxiv.org/abs/2603.24421 [part: §§5.1-5.2]\n- Likelihood ratios multiply for independent data.\n- The same holds for Bayes factors \"by updating each prior to the corresponding posterior after each study\".\n- \"Marginal likelihood-ratios or Bayes-factors from separate sources cannot in general be merged when their dependence structure is unknown.\"\n\n**Lavine & Schervish 1999**, American Statistician 53(2) [secondary, via search summaries only]. Bayes factors for composite hypotheses are not coherent as absolute support, and they depend on the within-hypothesis prior. Posterior odds are coherent. Not read.\n\n## 2. Confirmation on instances versus independent evidence\n\n**Good 1967**, \"The White Shoe is a Red Herring\", BJPS 17(4): 322. https://mail.joelvelasco.net/teaching/5330/good67-white_shoe.pdf [full text]\n- Two worlds: 100 black crows and no others, against 1,000 black crows plus one white. A black crow gives a factor of about 10 against \"all crows are black\". Good calls it a \"Bayes-Jeffreys-Turing factor\".\n- \"it is simply not true that a 'case of a hypothesis' necessarily supports the hypothesis\".\n- He also gives W(H:Black Crow) = W(H:Crow) + W(H:Black|Crow). This is a chain rule on a single instance, where the two terms can be of opposite sign.\n\n**Fitelson & Hawthorne 2010**, \"How Bayesian confirmation theory handles the paradox of the ravens\", in Eells & Fetzer (eds.), *The Place of Probability in Science*. https://mail.joelvelasco.net/teaching/5311/FitelsonHawthorneRavens.pdf [part: pp. 28-36 and the proof appendix]\n- They say that \"under plausible conditions\" a sequence of n instances gives a ratio of likelihood ratios \"on the order of (1/p)^n\". For p=.95 and n=100 they give ≈169.\n- Footnote 28 notes that p = P[Ba|Ra·~H·K] averages over a range of alternative frequency hypotheses H_f. A single raven is weak because many instances are needed to separate H from alternatives with f near 1.\n- The (1/p)^n reading treats instance factors as multiplicative. My Jensen argument (§6) shows it is an upper bound when not-H is a mixture. I did not find the conditional-independence assumption stated there.\n\n**Hawthorne, SEP \"Inductive Logic\"**. https://plato.stanford.edu/entries/logic-inductive/ [part: §1.3, §2.4-2.5]\n- §1.3: outcomes of distinct experiments are \"usually probabilistically independent\" given h_k with auxiliaries, so likelihood ratios multiply.\n- §2.5, Rule BE-C: the posterior probability of a band h[v,u] is ≥ 1/(1+K/BD(u,v,m,n)), with BD the normalised Beta function and K a bound on prior-density ratios.\n- The same entry therefore uses the product form for point hypotheses and the Beta integral for bands. It does not remark on the difference.\n\n**Fitelson 2001**, \"A Bayesian Account of Independent Evidence with Applications\", Phil. Sci. 68(S3): S123-S140. Slides only: http://fitelson.org/psa_2000.pdf [part; paper behind a paywall, not read]\n- Peirce additivity desideratum (A): mutually confirmationally independent evidence has c(H,E1&E2) = c(H,E1) + c(H,E2).\n- Screening-off desideratum (S): if H screens off E1 from E2, they are independent evidence for H. Only the log-likelihood-ratio measure satisfies S.\n- The slides credit Sober 1989 for the screening-off idea. I did not read Sober.\n\n**Keynes 1921**, *Treatise on Probability*, Ch. XX §§2-6. https://archive.org/stream/treatiseonprobab007528mbp/treatiseonprobab007528mbp_djvu.txt [part: OCR text]\n- Pure induction is \"the process of strengthening the probability of any proposition by adducing numerous instances of known truths which follow from it\".\n- It rests on \"the number of independent consequences which the conclusion and the premisses have in common\".\n- Each instance raises the probability \"so long as the new instance could not have been predicted with certainty from a knowledge of the former instances\". This is the conditional (chain-rule) condition.\n- Convergence to certainty needs a finite prior and a finite uncertainty about each next instance given the earlier ones, assuming the generalisation is false.\n- He contrasts mere repetition with varying the negative analogy.\n\n**Wrinch-Jeffreys prior.** Wagenmakers blog preview of a draft book chapter: \"the Bayes factor in favor of the general law equals n+1\" against a uniform alternative. https://bayesianspectacles.org/a-free-course-book-on-bayesian-inference-9-wrinch-and-jeffreys-to-the-rescue [part; secondary]. I did not read the Wrinch & Jeffreys 1921 primary.\n\n**Peirce 1878**, \"The Probability of Induction\", Popular Science Monthly 12: 705-718, §II. https://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_12/April_1878/Illustrations_of_the_Logic_of_Science_IV [full text of §II and the sampling passage]\n- Independent arguments that concur ought \"to produce a belief equal to the sum of the intensities of belief\". Peirce then argues that belief intensity is the log of the chance.\n- In the same essay, a judgment from sampled beans needs \"not one number but two\" (the probability, and the amount of knowledge behind it).\n- Good 1985 says Peirce \"failed to arrive at the correct definition of weight of evidence owing to a mistake\". Levi agrees but names a different mistake. So cite Peirce for the additivity remark, not for a correct definition.\n\n## 3. Argumentation theory\n\n- **Walton, Reed & Macagno 2008, p. 314: argument from example** [secondary, via Aberdein, arXiv 1904.02593, https://arxiv.org/pdf/1904.02593].\n  - Scheme: individual a has F and G, therefore generally F implies G.\n  - Five critical questions: is the premise true; does the example support the generalization; \"Is the example typical of the kinds of cases the generalization covers?\"; how strong is the generalization; do special circumstances impair generalizability.\n  - Aberdein says enumerative induction \"can be characterized in terms of this scheme\".\n  - The scheme is a single presumptive example. No question asks about the number of examples or how they accrue. I did not read Walton's book.\n- **Aggregation functions in gradual semantics**: Potyka, arXiv 1809.07133, Table 1. https://arxiv.org/pdf/1809.07133 [part].\n  - The aggregators are sum (Euler-based and quadratic-energy models), product (DF-QuAD), and top (max).\n  - This maps to your \"add\" and \"best\". No dependence modelling appears there.\n- **Pollock versus Selinger**, via Hitchcock, ISSA 2014. https://rozenbergquarterly.com/issa-proceedings-2014-the-linked-convergent-distinction/ [part; secondary, via summarizer].\n  - Pollock assumes \"the maximum of the strengths\" of separate arguments. Selinger accrues them with x⊕y = 2x+2y−2xy−1.\n  - Hitchcock concludes the dispute \"comes down to a conflict of intuitions\".\n- **Yanal 1991** (Informal Logic 13(3): 137-144), via Goddu 2003, Informal Logic 23(3): 215-236. https://distantreader.org/stacks/journals/il/il-2172.pdf [part; Yanal secondary].\n  - Independent reasons combine as the \"ordinary sum\" pr1 + pr2 − pr1·pr2, a noisy-OR rule. This is not a log-odds sum.\n  - Goddu argues the test fails to separate linked from convergent arguments.\n- **Pilditch, Hahn & Lagnado 2025**, \"The problem of dependency\", Synthese 205:143, DOI 10.1007/s11229-025-04969-w. https://ora.ox.ac.uk/objects/uuid:88c2a9ec-fbe8-49dd-82a5-0d40baf33f0b [part: §§1-2].\n  - This is the closest hit to your repetition question.\n  - Repeated measurements with the same device might be construed as independent events, \"in the same way as repeated tosses of a coin or throws of a die\". That ignores shared design.\n  - Repeated positive tests from one device raise confidence in its reliability, so they help the hypothesis. This is the shared-latent structure again.\n  - The paper also argues the Bayesian-network toolkit does not scale to social networks of sources.\n- **Pilditch, Hahn, Fenton & Lagnado 2020**, \"Dependencies in evidential reports: the case for informational advantages\", Cognition. https://discovery.ucl.ac.uk/id/eprint/10108145/ [part: abstract and introduction].\n  - Dependence is usually read as redundancy.\n  - They show cases where a structural dependency gives more support under partial or contradicting observations.\n  - They list three forms of dependence: shared background or reliability, direct information flow, and shared evidence.\n- Belkin, Madsen & de-Wit 2026 (CogSci), https://escholarship.org/uc/item/6f6597d3 [part]: a Bayesian model of source-dependency inference. Agreement is stronger evidence if the sources are independent.\n- Harris & Hahn 2009 (JEPLMC 35(5): 1366-1373), Hahn, Harris & Corner 2009 and 2016 (Topics in Cognitive Science 8(1): 180-195), Hahn & Hornikx 2016 (Synthese 193: 1833-1873), Madsen, Hahn & Pilditch 2020 (JEPLMC 46(9): 1795-1805): all [abstract only or secondary]. None addresses repeating an argument from one source.\n- Computational argumentation that treats several examples differently from several reasons: not found.\n\n## 4. Hierarchical models as the unifier\n\n- **Gelman et al., BDA3 §5.2**, pp. 104-105. https://sites.stat.columbia.edu/gelman/book/BDA3.pdf [part]. Exchangeable units are not independent: \"yj are exchangeable but they are not independent\". Exchangeability is a mixture of iid models.\n- **Higgins, Thompson & Spiegelhalter 2009**, JRSS A 172(1): 137-159. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2667312/ [part].\n  - \"A 'fixed effect' model assumes that a single parameter value is common to all studies.\"\n  - Under exchangeability the study effects are \"independent and identically distributed conditional on\" the random-effects distribution f(Φ).\n- **Henderson, Goodman, Tenenbaum & Woodward 2010**, Phil. Sci. 77(2): 172-200. https://cocolab.stanford.edu/papers/HendersonEtAl2010-PhilosSci.pdf [part]. \"Higher-level theories may also specify how homogeneous data obtained from different trials or experimental settings are expected to be.\"\n- **De Pretis, Landes & Osimani 2019**, \"E-Synthesis\", Frontiers in Pharmacology 10: 1317. https://www.frontiersin.org/articles/10.3389/fphar.2019.01317/full [part, first half, via summarizer; secondary].\n  - A shared sponsorship-bias variable modulates a group of studies.\n  - They say this \"leads to what Bayesian statisticians call a hierarchical model\".\n  - Sequential updating uses repeated application of the chain rule.\n- Landes, Osimani & Poellinger 2018 (EJPS 8, DOI 10.1007/s13194-017-0169-1) [abstract only]. Stegenga 2013 (Synthese 190: 2391-2411) [abstract only]. Lehtinen 2013 [part]: Stegenga's \"independence\" is Arrow-style IIA, not statistical independence. Claveau & Grenier 2019 (Synthese 196: 3001-3028) [abstract and introduction]: the variety-of-evidence thesis is an exception-prone rule of thumb.\n\n## 5. Head-on comparisons\n\nNo source found that explicitly compares combining independent pieces of evidence with accumulating instances as one operation. The nearest:\n- Keynes Ch. XX defines pure induction as strengthening a proposition by independent consequences, with the sequential \"not predictable from earlier instances\" condition.\n- Peirce 1878 treats both in one essay but separately: independent arguments add on a log scale, while a sample needs two numbers.\n- Good 1985 and Jaynes ch. 4 give the single formal rule covering both: chain rule, then conditional independence given H and not-H.\n- Pilditch et al. 2025 shows the repeated-measurement-as-coin-toss question in practice.\n\n## 6. My computations (not literature)\n\nMethod: flat Beta(1,1) start, numerical integration, Python.\n\n- **Reproducing your numbers.**\n  - Band [0.25, 0.42], 10 wet of 30: pooled posterior 0.691, matching your 0.691.\n  - Best single case: 0.226, matching your 0.226. So your strengths are probabilities, and the band is probably [0.25, 0.42]. That is my inference, not given.\n  - Naive sum of per-case marginal log factors: 0.702.\n  - Pooled log BF 2.393, and the sum of the chain-rule conditional increments 2.393. The two are equal.\n- **Direction of the naive-sum error varies by band** (pooled vs naive log BF): [0.2, 0.5] gives 3.464 vs 3.223 (under); [0.3, 0.37] gives 1.847 vs 1.956 (over).\n- **The error grows with n** at the same one-third frequency, band [0.25, 0.42] (pooled vs naive):\n\n| n | Pooled | Naive |\n|---|---|---|\n| 30 | 2.39 | 2.44 |\n| 300 | 7.99 | 24.44 |\n| 3000 | 52.14 | 244.44 |\n\n- **Point against point**, 1/3 vs 1/2, 10 wet and 20 dry: the log likelihood ratio is 10·ln(2/3) + 20·ln(4/3) = +1.699 nats, exactly the per-case sum.\n- **Conditional increments are order-dependent.** In one order of the 30 cases they run −0.470, +0.813, +0.379, −0.248, ... down to about ±0.1. They sum to the order-invariant total.\n- **Universal law against a uniform alternative.** The Bayes factor is n+1, so the log factor is ln(n+1): 2.398 at n=10, 4.615 at n=100, 6.909 at n=1000. The 1000th instance adds about 0.001 nats. A naive n·ln2 gives 693 at n=1000.\n- **Jensen bound.** For H \"all F are G\" against a mixture not-H with mean frequency p, E[f^n] ≥ p^n, so the true Bayes factor is ≤ p^(-n). The product of single-instance factors overstates.\n- **Shared rival explanation** (assumed numbers: P(E|C)=0.95, P(dew|¬C)=0.3, P(E|dew)=0.9, P(E|other ¬C)=0.05).\n  - Best-one gives ln 1.136; the true joint is ln 1.305; the naive sum is ln 2.272.\n  - The second reason's conditional shift is 0.169.\n\n## Not checked (planned, not done)\n\n- Wald 1945 and 1947 primary text (method of weights): only secondary.\n- Good 1950 weighted-average theorem; Good 1981b and Levi 1982 on Peirce.\n- Fitelson 2001 full paper, Sober 1989, Fitelson's other papers, Hosiasson-Lindenbaum 1940, Maher 1999 body, Myrvold 2003 body.\n- Walton, Reed & Macagno 2008 book text; Pollock's statistical induction; Prakken 2005 (covered by the project already).\n- Lavine & Schervish 1999 full text; Morey & Rouder 2011 (interval-null Bayes factor, secondary only).\n- Stegenga 2013, Landes et al. 2018 and Hahn-group abstracts: full text not read. De Pretis et al. second half (about 96k characters) not read.\n- Jaynes ch. 5; the 2003 book equation numbering.\n- Naive-Bayes literature (Hand & Yu 2001): not verified.\n\n## Verdict on the working theory\n\n**(a) Holds, with a qualification.** With two point hypotheses, each case contributes a fixed log likelihood ratio and the total is the sum.\n- Decisive sources: Good 1985 p. 252, Shafer et al. §8.1, Jaynes (4-11) and Good 1979 §5.\n- It is a statement about a pair of simple hypotheses, not about \"H versus not-H\".\n- Jaynes' non-extensibility exercise says per-hypothesis additivity fails once the alternative is a disjunction of two or more rivals.\n- \"Rate = 1/3 vs 1/2\" gives 1.699 nats, but that is not the weight of evidence for \"about a third\" against its complement.\n\n**(b) Holds.** The right wording: log factors always add under the chain rule, but each term is conditional. The conditional term equals the unconditional one only if the case is independent of the earlier ones given H and given not-H.\n- Decisive sources: Good 1985 eq. (1) with p. 252; Jaynes (4-10) with the compound-hypotheses section; Dawid 1984 §§4 and 7; Kass & Raftery §3.2; Nikolakopoulos et al. §3.2; Rouder & Morey appendix.\n- Refinements: each conditional term is a ratio of posterior-predictive probabilities, with the within-hypothesis weights re-updated by the earlier cases. Individual terms depend on order; only the total is invariant.\n- The error of the naive sum has no fixed sign (Rouder & Morey say it can understate; the Jensen bound says it overstates for universal-against-mixture; my band results go both ways). It also grows with n.\n- The band does not screen cases off from each other; the exact rate does (Fitelson's S property, Sober). A prior inside each side is a modelling choice. SEP Rule BE-C bounds it with K, and Lavine & Schervish warn about it [secondary].\n\n**(c) Mostly right, with two corrections.**\n1. The same latent-variable structure sits behind reasons. Reasons sharing a rival explanation are conditionally independent only given that explanation, which is integrated out inside not-C. This is the dew case, and Jaynes' non-extensibility result applies. So \"add or best\" is a binary approximation to a third option for reasons too.\n   - \"Best\" is exact only when the second reason is fully predictable from the first under not-C. \"Add\" is exact only when the reasons are independent given C and given not-C.\n   - The rival-explanation question is a test of whether not-C is simple relative to the pair of reasons.\n   - Supporting sources: Pilditch et al. 2025 (shared design, reliability learning), De Pretis et al. 2019 (shared modulator), Bovens & Hartmann (already covered).\n2. No literature source states that reasons and instances are one structure. Keynes and Peirce come closest, Good and Jaynes give the formal rule, and Higgins et al. and BDA3 give the hierarchical mathematics for the case side. The unification is your synthesis and mine, not an established claim.\n"}