{"path":"research/pollock-against-accrual-and-the-overlap-check.md","content":"# Pollock against accrual, and the overlap check that was never switched on\n\n*2026-10-14. A session's reading, continuing the work a Codex session began on the founder's prompt\nand stopped on its usage limit. Three papers were read in full for it: Pollock, \"Defeasible\nReasoning and Degrees of Justification\" (the corrected version on his site, published in Argument\n& Computation 1, 2010); Prakken, \"A Study of Accrual of Arguments, with Applications to Evidential\nReasoning\" (ICAIL 2005); and Prakken, \"Modelling Accrual of Arguments in ASPIC+\" (ICAIL 2019). The\ncode was read where the note says what is built. Nothing here is ruled; § 8 lists what is put to\nhim. Companion: [claim-parts-and-inferential-reasons-2026-10-14.md](claim-parts-and-inferential-reasons-2026-10-14.md),\nthe outside audit of the same morning, and the seventeenth listening piece.*\n\n## 0. His three questions, and the short answers\n\nHis prompt, verbatim (2026-10-14, to Codex): *\"Give me a deep analysis of Pollock vs the accrual\nschool in the light of everything. Overlapping roots of evidence obviously shall not be added up.\nGiven that, what reasons does Pollock have to challenge us? And... Why has our check for overlapping\nevidence been turned off btw?\"*\n\n- **Pollock against the accrual school is not min against sum.** Read in full, both sides deny\n  that two reasons for one claim add up by their numbers. Pollock gives a claim with several\n  reasons the strength of its best reason, and explains the cases where two reasons seem to make\n  more as a *third* reason about the pair, which he writes out and which can be attacked. Prakken's\n  first principle is that an accrual's strength cannot in general be calculated from its elements,\n  and his 2019 model orders sets of reasons by supplied preference, never by arithmetic. The flat\n  sum this project runs on the reasons channel sides with neither; it comes from the numeric\n  gradual-semantics tradition (§ 3).\n- **Granting that shared origin counts once, Pollock still has five things against us** (§ 4).\n  The sharpest: the lift a second reason gives above the first is, for him, a claim somebody has to\n  make, and in the code it is arithmetic nobody can attack. The second: independence is not\n  decidable, so a sum can only be defended by detection, and detection is never complete; the best\n  reason alone is immune. That is this project's own payoff-removal law, which it applied to parts\n  and not to reasons.\n- **The overlap check was never on.** It was built on 2026-08-14 as a function with the docstring\n  sentence *nothing here changes shipped scoring*, measured once by the slicing instrument (it\n  changed nothing), made moot on the parts channel when the weakest-part default landed on\n  2026-08-31, and it never covered the reasons channel, which is the one his sentence is about.\n  Nothing discounts two reasons with one origin today (§ 5).\n\n## 1. What Pollock says, read in full\n\nThe paper's question is what changes in defeasible reasoning once conclusions carry degrees of\njustification rather than a yes or no. He catalogues three possible impacts (his § 4): diminishers,\nreasoning from several premises, and several arguments for one conclusion. Only the first survives\nhis examination.\n\n**1a. Several premises.** The argument's degree of justification is at most the minimum of its\npremises' degrees (his § 5.4), and may be lower: in a statistical syllogism it also depends on the\nprobability the syllogism rests on. He then names the one place the whole exceeds its parts.\nEnumerative induction's conclusion depends on the *number* of instances as well as their degrees,\nand may be *\"better justified than the conjunction of the premises\"*. He adds that this will hold of\nall defeasible inference schemes and that getting them formulated correctly will take considerable\nthought. So for Pollock a lift above the parts is a property of a particular scheme, never a rule\nof aggregation.\n\n**1b. Conjunction, and why min.** Kyburg's *conjunctivitis* (the conjunction is less justified than\neither conjunct, as a probability product would say) is rejected, on a computational ground rather\nthan an intuitive one (his § 5.1 to 5.3). To compute a product one must know whether the conjuncts\nare independent in the sense justification needs, and that cannot be computed: P and P are not\nindependent, (P&Q) and (Q&P) are not, two differently written logical equivalents are not, and\nlogical equivalence is undecidable by Church's theorem, with truth-functional equivalence not\nquickly decidable either. A resource-bounded reasoner needs a background computation that is linear\nin the number of conjuncts, and min is the only candidate anyone has proposed. He says this is not\nconclusive, only that nobody has found an alternative. Two consequences he draws: degrees of\njustification are computed by the reasoner and *the computation makes it true*, and they do not\nobey the probability calculus.\n\n**1c. Several arguments for one conclusion** (his § 6). He asks whether having two arguments for a\nconclusion should raise its degree of justification, calls that principle *the accrual of\narguments*, and answers no. Three moves carry the answer.\n\nFirst, the apparent accrual is a third argument. Carlos comes in and says it is raining; Susan comes\nin and says the same. Most people feel two reports beat one. Pollock agrees with the feeling and\ndenies the mechanism: we hold a third, contingent reason, that two people independently reporting\nthe same mundane thing are more likely right than one, and that reason is an instance of the\nstatistical syllogism appealing to a higher probability than either report alone. So there are\nthree arguments, the third stronger than the first two, and the conclusion takes the strongest\n(his Figure 5). The joint probability is what the third argument reads. Where the joint probability\nis *lower*, the feeling runs the other way, and he gives Bill and Stew, two jokesters each reliable\nalone and likely to be colluding when both report something surprising: knowing that, the wise\nresponse is to doubt them together.\n\nSecond, he concedes a case he cannot derive. Two unrelated tests for lactose intolerance, positive\non the first giving 0.7, on the second 0.75, joint probability unknown. Most people would say two\npositives give a better reason than one, and he cannot explain it through the joint probability\nbecause nothing is known about it: *\"we have no reason to expect having the two arguments to raise\nthe probability\"*. He leaves this standing against his own position.\n\nThird, the slope. Logical variants of one argument are not two arguments: the same argument\nduplicated, or run through (P&Q) rather than (Q&P), must not raise anything. Make the variants\nharder to detect and either the computation never raises a degree above what the best single\nargument gives, or it runs arbitrarily long, or, if the equivalences are first-order, it need not\nterminate. His conclusion: several arguments give only the degree the best one gives.\n\n**1d. What remains.** His § 7 and § 8: there is no way to make accrual work; the only impact of\ndegrees that needs more than the weakest link is the *diminisher*, a defeater too weak to defeat\nthat should still lower the conclusion (an argument at 0.9 against one at 0.89999999 should not\nleave the first unscathed). The whole paper's ground is the agent: a reasoner that must compute\ndegrees quickly in the background from the syntax of its own inference graph, with nobody's help.\n\n## 2. What the accrual school says, read in full\n\nThe corpus names Verheij and Prakken as the school. Prakken 2019 § 1 says what the school is built\non: Dung's abstract frameworks combined with *Pollock's* theory of defeasible reasons, with accrual\nadded as an inference the model must permit under constraints. So the school is a refinement of\nPollock's structure, not a rival arithmetic.\n\n**2a. Prakken 2005.** Two ways to model accrual: the knowledge-representation way, which writes the\naccrual into a rule's antecedent (rain and heat together → do not jog), and the inference way, which\nadds an accrual step applied after every individual reason has been built. He states three\nprinciples any treatment must satisfy (his § 3), each with its reason:\n\n1. *Accruals are sometimes weaker than their elements*, because accruing reasons may not be\n   independent. Heat and rain each a reason not to jog, together less unpleasant for one runner; or\n   witnesses *\"from a group of people who are more likely to confirm each other's statements when\n   these statements are false than when they are true\"*. From which: in general the strength of an\n   accrual cannot be calculated from the strengths of its elements.\n2. *An accrual makes its elements inapplicable*, since a comparison between accruals is meant to\n   consider all available information, while the elements take only part of it into account.\n3. *Flawed reasons do not accrue*: a reason shown false, or undercut, does not take part. This forces\n   a two-stage process, first testing each reason, then accruing those that pass.\n\nThree further things the paper holds. A refinement (his § 6.4): something can strengthen a reason\nwithout being a reason on its own (that the witness has been reliable before strengthens the\ntestimony and is no reason by itself), and such a strengthener blocks the plain rule rather than\naccruing with it. In the witness examples (his § 7.1), two witnesses who conferred count once, and\nconfirmation by an independent witness is modelled as a separate rule that lifts the conferred\nones back. And from Anderson and Twining he takes three kinds of combination (his § 7.2):\n*combining* (neither piece supports the conclusion alone), *corroborating* (two witnesses to one\nevent, doubtable only by attacking their credibility), and *converging* (witnesses to different\nevents, each supporting the conclusion, attackable on other grounds too). The paper's own weakness,\nnamed in 2019: it would not let strict reasons accrue (two witnesses placing a suspect in Amsterdam\nat different times did not combine into *Amsterdam that morning*).\n\n**2b. Prakken 2019.** The new model keeps the three principles and fixes the weakness. An accrual\nset is defined relative to a labelling of which arguments are In: only arguments with no\nimmediate subargument Out and no undercutter In enter it (his Definition 4.1). The preference\nrelation is on *sets* of arguments, assumed to satisfy only that any set containing a strict and\nfirm argument is on top (Definition 4.3). The strength of an accrual is supplied, by domain\nknowledge, never computed from its members. Lesser accruals are considered only when a larger one\nis subargument-defeated, which is how principle 2 and principle 3 are kept. Strict reasons now\naccrue (his Example 6.1). With no proper accruals the model reduces to ordinary ASPIC+. And his\n§ 6.5 relates accrual to the informal-logic vocabulary: following Wei and Prakken, two reasons are\n*linked* when neither alone suffices, *convergent* when either does, and an accrual is *cumulative*\nwhen the set is preferred to each member and *non-cumulative* otherwise. He adds that accruals, in\nthis classification, are a special kind of linked argument: all members are needed to build the\naccrual step, and the step may stand above any of them.\n\n## 3. The shorthand, corrected\n\nThe corpus's sentence (`weakest-link-arithmetic-and-the-merge-hunch.md` § 1b): the same paper\n*rejects accrual outright*, two arguments for one conclusion yield the maximum, *so our\ncorroborative channel sides against Pollock and with the accrual school (Verheij; Prakken 2005)*.\nThree things in it do not survive the full read.\n\n*Rejects accrual outright* overstates. He rejects accrual as a principle of aggregation and keeps\nthe phenomenon: two independent reports do justify more, through a third reason that says so, and\nthe lactose case he leaves unexplained against himself.\n\n*Sides with the accrual school* misplaces the school. Prakken does not sum. His first principle\ndenies that an accrual's strength follows from its elements', his 2005 model gives the accrual rule\nno arithmetic at all (rule priorities are set by hand), and his 2019 model orders sets by a\nsupplied preference. What the school adds to Pollock is permission: an accrual may exist as its\nown argument, stronger or weaker than its members, once somebody has said so.\n\n*The corroborative channel* is therefore a third position, held by neither. It comes from the\nnumeric gradual-semantics tradition (Potyka's QEM; Amgoud et al. 2017), where contributions add\nand are squashed, and where the padding problem is a known cost (Potyka & Booth; the D8 table in\n`strength-calculus-literature/computational-argumentation.md`). Where the three sit:\n\n| | Pollock 2010 | Prakken 2005, 2019 | Deliberus as built | Deliberus as ruled or proposed |\n|---|---|---|---|---|\n| Two reasons for one claim, nothing more said | the better one | each stands alone; an accrual may be formed and preferred above them | every support link adds, then QEM (`qbaf.py`) | open: ledger Q22, Q35 |\n| What lifts the claim above the best reason | a further reason about the pair (two independent reports are more reliable), itself defeasible | the accrual set, preferred by domain knowledge | the sum, silently | — |\n| The members once combined | folded into the stronger argument | inapplicable (principle 2) | each still counts | — |\n| Reasons that interact badly (jokesters; heat and rain) | the pair's joint reads lower | an accrual may be weaker than its elements (principle 1) | never below the strongest | — |\n| Strength computable from the members' strengths | no (independence undecidable) | no (principle 1) | yes, by construction | — |\n| A reason shown false | — | does not take part (principle 3) | drops out | RULED Q34 |\n| The parts of one claim | the weakest (min) | the weakest, inherited from Pollock | min (`necessary` default) | RULED 2026-08-31 |\n| A whole that exceeds its parts | inside a scheme: induction by count, a syllogism with a higher probability | a cumulative accrual | the sum, everywhere | Q22 |\n\nTwo rows deserve a sentence each. The parts row shows the project already *is* Pollock on one of\nits two channels: the weakest-part default of 31 August is his weakest link, and the corpus knew\nit. The first row shows the reasons channel is the odd one out, and that the code's rule there is\nnobody's.\n\nCorrections filed today: a dated line under § 1b of the weakest-link note; a dated line in\n`what-a-rule-rests-on.md` § 9b (the chain and the cable), which repeats *rejects adding outright*;\na dated line in `the-squirrel-and-the-strength-calculus.md` § 2, which names only adding as the\nrule for several reasons. The reader reports that quote the shorthand are records of what was read\nand are left as they are.\n\n## 4. Overlapping roots are not added. What does Pollock still have against us?\n\nTake his sentence as the premise: shared origin counts once, by whatever instrument (ledger Q4,\nsources as nodes; `what-counts-as-evidence.md` § 8c). Five challenges survive it, in order of weight.\n\n**C1. The lift above the best reason is a claim, and the code makes it arithmetic.** His lactose\ncase is exactly two reasons with independent origins. He concedes the feeling that two positives\nbeat one and refuses the number, because nothing known says what the pair is worth. The code\nanswers with a definite number: two links each contributing relevance times offset, summed and\nsquashed, above the better of them. Nobody asserted that number; nobody can attack it, since it is\nnot a claim; and it is the same number whether the two tests are known to be independent or\nmerely assumed so. His structure is that the lift is itself a reason, written down, with its own\nstanding and its own attackers (the jokesters attack it). In this project's own words that is\n*propose, never assert* and *contributions are added, never overwritten*, applied to the arithmetic:\na lift that moves a public badge is a contribution somebody should be able to point at. On this the\ncorpus agrees with Pollock against its own code.\n\n**C2. Independence cannot be computed, so a sum cannot be made padding-proof by detection.** His\nslope is the one the project knows from its own threat model: a detector of reformulations is\nnever complete, and against an adversary who models it, publishing it arms him. The project's law\nis *prefer payoff-removal to detection* (`fractal-scales-and-temporal-frame.md` § Three classes of\nadversary), and the weakest-link note's § 5c is the law working: under min, an entailed part can\nnever be the weakest, so padding on the parts channel is priced at zero with no detector. On the\nreasons channel the project chose detection instead: sameness before any relation (the keeper),\nmerge candidates, the proposed `SHARES_ORIGIN_WITH`. Those are good instruments and they are the\nkind Pollock says cannot close. The best reason alone is the payoff-removal form for reasons, as\nthe corpus's own table says (*the sybil-proof choice*, `weakest-link` § 0; `alternative_energy` in\n`support_semantics.py` already takes the max for parts tagged alternative). Since 2026-10-10 a\nmachine's sentence under a link is a claim (ledger AG12); many such sentences will be\nreformulations of each other, and on a channel that adds, each reformulation would add.\n\n**C3. The number is not a probability, and he says why.** His ground against conjunctivitis is that\njustification-independence cannot be computed, so the degrees a reasoner can actually hold do not\nobey the probability calculus; a reasoner computes them and the computation makes them true. That\nis a direct answer to the founder's intuition of 30 September (ledger Q35: every claim's subgraph\nis eventually equivalent to a Bayesian estimate) and to closing question 1a (what the number on a\nclaim means). It is also the one challenge whose ground is weaker here than where he made it. He\ndesigned for OSCAR, an agent computing in real time from syntax with nobody to ask. Deliberus is a\nrecord with time, many hands and a judge for relevance; an independence judgement that an agent\ncannot compute can here be *contributed*, as a claim, and attacked. So his impossibility becomes a\nstatement about the default: with nothing contributed, what number is honest. His default is the\nbest reason. The founder's intuition is about the limit, once the independence structure has been\nstated. The two are not in conflict; they are two ends of one dial, and how much independence has\nbeen written down is the dial's position.\n\n**C4. Where a whole exceeds its parts, the excess lives inside a scheme.** Induction by count, the\nstatistical syllogism with a higher probability for two independent reports: the lift is a\nproperty of a named inference with its own critical question (*are the sources independent?*),\nnever of an aggregation rule applied to everything. The project has schemes on links and a\nproposed pass that types combinations (ledger Q22: corroborative, convergent, cumulative, after\nSchum), which is Prakken 2019 § 6.5 and Anderson and Twining's three kinds by other names. Pollock's\nversion is sharper on one point: the type is not a label on the combination, it is a *further\nclaim* with its own standing (two independent positive tests make the diagnosis more likely than\neither), and the typing pass should propose that claim, not a tag.\n\n**C5. The jokesters: a pair can be worth less than either member, and no sum can say so.** Prakken's\nfirst principle, Pollock's joint probability running below the constituents. The code's sum never\nfalls below the strongest link. Under any max-or-sum rule the pair's being worse than its members\nis representable only as attacks on each member's link (an undercutter: *given that the other also\nsaid it, this report is suspect*), which the graph can hold (a link's relevance claim, ledger Q33)\nbut nothing writes automatically. This is the one case neither default handles by arithmetic; it\nneeds the object Prakken calls the accrual, attackable as a whole.\n\nTwo things he agrees with the code about, worth saying so they are not counted as open. His\ndiminisher, a weak attack that lowers without defeating, is what QEM does with every attack link.\nAnd Prakken's third principle, a refuted reason does not take part, is ledger Q34 as ruled on 30\nSeptember (a reason that fails drops out; failure counts against a claim only through a\ncontribution that says why), and the corpus already cited him for it.\n\n## 5. Why the overlap check is \"off\": it was never on\n\nThe record, read in code and in git.\n\n- **2026-08-14, commit `3b31b81`**: `deliberus/evidence_routing.py` lands with routing, abstention,\n  the no-invention guard and `fair_share_discounts`, which gives a piece of evidence shared by *k*\n  siblings `1/k` of its weight at each. The docstring's last paragraph: *\"Nothing here changes\n  shipped scoring. The discount is offered as a function and measured against the live corpus;\n  adopting it alters how every strength in the graph is computed, which is the founder's decision\n  rather than an implementation detail.\"* So the function was never wired into the badge.\n- **Its only caller** is the slicing-stability instrument (`slicing_stability.py`), which recomputes\n  strengths under rival filings to see which claims are sensitive to how they were cut. The first\n  run (`slicing-stability-first-run.md`): *fair-share changed nothing anywhere measured*, and the\n  note explains why: the one known shared-evidence case, a fasting meta-analysis routed to several\n  children, had been dissolved earlier the same week by decomposing the evidence and superseding\n  the edges, which is the founder's own correction of 15 August at work (most apparent sharing is\n  coarse evidence).\n- **2026-08-31**: untagged parts aggregate by their weakest part (the padding ruling). Under min a\n  fact routed to three children cannot raise the parent, since the parent reads the weakest child\n  whatever the others share (`weakest-link` § 5c). The channel the discount was built for closed\n  behind it.\n- **It never covered the reasons channel.** The function discounts one fact shared by several\n  *parts* of one claim. The case in his sentence is several *reasons* for one claim with one origin\n  (three studies on one dataset; two witnesses who conferred). `computational-argumentation.md`\n  § D8 says this plainly: the discount *covers one fact routed to several parts, not several\n  supporters with one source*, and *shared origin between supporters is not represented on the\n  argument channel*. The squirrel note's table (§ 2) says the same in one line: links add, nothing\n  discounts reasons that share an origin.\n- **What was proposed for that channel** and sits open: sources as nodes with reliability as\n  attackable claims, inputs sharing an origin forming one channel, a `SHARES_ORIGIN_WITH` proposal\n  with copy detection keyed on shared errors (ledger Q4, `what-counts-as-evidence.md` § 8c); and\n  typed combinations (Q22). The similarity-adjusted semantics the weakest-link note found in the\n  literature (§ 9b) is not wired either.\n- **The adoption gate was never reached** because nothing was there to measure: the next-steps list\n  in `evidence-division-and-the-foundation.md` says to put the discount to the founder *with\n  measured before/after strengths*, and the one measurement found no difference.\n\nSo: not switched off. Built for a channel that min then closed, never built for the channel he is\nasking about. The Codex session's interim reading of the same history, before it stopped, was the\nsame: *built but never connected to scoring* is the accurate description, with no later switch-off\nfound.\n\nOne public sentence overstates it. `docs/open-problems.md` § 3 says evidence shared between parts\n*is discounted so it isn't counted twice*. Built and unadopted is the state; the sentence is on a\nserved page and is put to him in § 8, not edited here.\n\nHis own sentence in the prompt, *\"Overlapping roots of evidence obviously shall not be added up\"*,\nis the first stated position on the reasons channel. It is recorded in the ledger's Q4 row as his\nwords, not as a ruling on the mechanism.\n\n*The next morning (2026-10-15), on his pointer to the keeper of the seven keys: the detector Pollock said no reasoner has exists in two of its three parts. The keeper's `same` catches reformulation and its `contains` and `overlaps` catch dependence in content, both live and neither run between the sibling reasons of one claim; shared origin is provenance, ledger Q4, unbuilt; the fair-share discount above is none of these. [retrofitting-the-reasons-channel.md](retrofitting-the-reasons-channel.md) § 4, § 5; ledger Q40.*\n\n## 6. What this means for us (a proposal, unruled)\n\nThree ways forward were weighed. The dominated one first, in a sentence: keep the sum and bolt the\ndiscount on through Q4's origin channels. It is detection-shaped, so Pollock's slope stands\nagainst it, and it leaves C1 untouched: the lift is still a number nobody asserted.\n\nThe second, a numeric middle: bound the sum by its parents (Potyka & Booth 2024) or discount by\nsimilarity (Amgoud et al. 2018). Immune to unbounded padding, still lifts without a stated reason,\nstill not a claim.\n\nThe third, which is the pick, composed from what is already built or ruled:\n\n1. **On the reasons channel the default with nothing further said is the best reason**, as on the\n   parts channel it is the weakest part. Both are Pollock's; both remove the payoff of adding\n   links; the max primitive exists (`alternative_energy`). A claim with ten modest independent\n   reasons then reads as strong as its best one until somebody says what the ten are worth\n   together, which is honest thinness (*conservative relevance beats semantic reach*,\n   `ux-principles.md`).\n2. **A lift above the best reason is a claim with the lifted reasons as its required parts.**\n   Pollock's Figure 5 in this ontology: *two independent positive tests make the diagnosis more\n   likely than either* is a claim; its parts are the two test results, tagged necessary, so its own\n   standing is the weaker of them (min, built); its link to the diagnosis carries a relevance above\n   either test's (Q33, built: effect is standing times relevance); the conclusion takes the max over\n   links, so the joint link governs. The lift lives in the relevance of the joint claim's link,\n   which is exactly where Pollock puts it (a syllogism with a higher probability), and it is\n   addressable: attack the joint claim (*these tests are not independent*, the jokesters' attack on\n   the pair) and the lift falls to the best member.\n3. **The typing pass of Q22 proposes that claim, not a tag.** A machine that reads two reasons as\n   convergent with independent origins (Q4 supplies the origins) writes the joint claim as a\n   proposal, provenance machine, under the rule that machine work proposes on contested ground.\n4. **Prakken's second principle comes free.** Under max the members cannot double count once the\n   joint link stands above them, as under min an entailed part cannot be the weakest: the same\n   closure (`weakest-link` § 5c) on the other channel, with no supersession needed. His first\n   principle (a pair worth less than its members) stays a contribution: an undercutter on each\n   member's link, or an attack on the joint claim, written by a person or proposed by the pass.\n5. **The founder's limit survives.** Where every independence fact has been written down as joint\n   claims, the graph reads as the Bayesian picture reads; where none has, it reads as Pollock's.\n   Closing question 1a decides the default between them, and the eight test cases of the Codex\n   note § 13 are the spec: the sixth (a redundant middle step must not change the score) and the\n   two-independent-reasons case are the ones this proposal changes.\n\nThe cost is a change to a shipped calculus (the energy sum over support links, live since the\nbadge repair of 17 August and ruled in its two-factor form on 28 September), and a stretch during\nwhich many claims with several weak reasons read lower than today. The gain is that every number\non a public badge is either the reading of one argument or the reading of a claim somebody made.\n\n## 7. What does not fit\n\n- **His ground was tractability in an agent.** Much of his argument is about what a background\n  computation can do in linear time from syntax. This project does not reason in real time and has\n  contributors; his impossibility weakens to a default argument (§ 4 C3). What survives in full is\n  the slope (C2), which is about adversaries, not speed.\n- **He concedes two cases against himself**: the lactose tests, and induction by count. Any reading\n  of him as *min and max, nothing else* is too clean, and the corpus's was.\n- **Prakken 2019 cuts across the corpus's own table.** The weakest-link note maps linked premises to\n  min and convergent premises to the sum. Prakken calls an accrual a *linked* argument (all members\n  needed to build the step) that may be *cumulative* (above every member). So linked does not\n  entail min; the audit note's point that the false-without-it test measures need and not content\n  is the same cut from another side.\n- **Prakken's strengths are ordinal and ours are numbers.** His preference on sets is assumed, not\n  computed; nothing in either paper says how a number should be assigned to an accrual. The joint\n  claim of § 6 is this project's answer and neither author's.\n- **QEM is nobody's school.** Potyka's semantics was chosen in March for other properties (the\n  strength-layer audit and `qbaf-gradual-semantics-research.md`); its sum was never a position on\n  accrual, which is why the D8 decision has stayed open for a month without a home.\n- **The jokesters** fall outside both defaults (C5): the only rule that lowers a pair below its\n  members is a contribution saying so, and the pass that would write it does not exist.\n\n## 8. Filed\n\n- Corrections, dated lines in place: `weakest-link-arithmetic-and-the-merge-hunch.md` § 1b;\n  `what-a-rule-rests-on.md` § 9b; `the-squirrel-and-the-strength-calculus.md` § 2.\n- Ledger: his sentence on Q4, as his words; a new row Q37 for the reasons-channel default and the\n  joint claim, open, with § 6 as the recommendation.\n- `TODO.md` § RELEVANCE: the overlap-check finding and what defends the reasons channel; the public\n  sentence to put to him.\n- Corpus graph: connections from this note to the weakest-link note, the D8 decision, the evidence\n  dossier's § 8c, the evidence-division note, the squirrel note, the relevance note and the audit\n  note.\n- **To him, one at a time**: the `open-problems.md` § 3 sentence (*is discounted*), which says in\n  force what is built and unadopted; then Q37.\n"}