{"path":"research/what-a-link-rests-on/outside-reading-argumentation-theory.md","content":"# Outside reading: what argumentation theory says about the unstated premise under a step\n\n*A research helper's report, kept as it came back on 2026-10-07 and not edited. The question put to it: how the missing premise is picked, how a real reason is told from an idle or question-begging one, how formal frameworks let the inference itself be disputed or weighed, and how chains are drawn.\nIt is a model's reading of the sources it lists, with its own statement of how far each was read;\nthe note that uses it is [what-a-link-rests-on.md](../what-a-link-rests-on.md). **Verify against\nthe sources before citing any of it outward.** Sections the helper marks as its own inference are\nits own, not this project's position.*\n\n---\n\n# Report: unstated premises, relevance tests, inference-level disputes and chains in argumentation theory\n\nResearch date 2026-10-07. Tools: web search and fetch; PDFs read with pdftotext; one scanned PDF (Walton and Reed 2005) read by in-memory OCR; no files created by me. Some PDFs were auto-saved by the fetch tool into its scratch folder; nothing was executed.\n\n## 0. Conventions and two caveats you need\n\n1. QUOTE LIMIT. My output rules allow only ONE short verbatim quotation per response, so I cannot supply the 1-2 verbatim quotes per source you asked for. Each entry gives a close paraphrase plus a locator (section, definition, principle, page) so you can check the wording in seconds. Paraphrases were checked against raw text (pdftotext, OCR, or raw Exa fetch). Where I relied on the WebFetch summariser instead of raw text I say so; treat those as lower confidence.\n2. READ DEPTH labels as requested. \"Substantial extract\" names the parts I read. Anything I could not open is marked, and what I say about it comes second-hand and is unverified.\n\n## 1. Bottom line by question\n\n**Q1 (picking the missing premise).**\n(a) Every source that addresses it agrees some premise always restores validity (at minimum the associated conditional, or the conclusion itself). So the mere existence of a bridge claim is no evidence the link is good. (Govier; van Eemeren and Garssen 2026; Gough and Tindale n.7; Ennis 2001.)\n(b) Informative versus trivial. The associated conditional (pragma-dialectics calls it the logical minimum) is rejected as the bridge by Govier, by van Eemeren and Grootendorst, and by Hitchcock 1985, for related reasons: it only restates the inferential commitment, and it is true whenever the conclusion is true, so it cannot expose a bad inference. The favoured replacement is a generalization: pragmatic optimum (van Eemeren and Grootendorst), covering generalization (Hitchcock), or an argumentation-scheme premise (Walton and Reed). Operational tests found: non-trivial truth (not true merely because the premises are false or the conclusion true), epistemic independence of the conclusion (Hitchcock 2011), and generalization at the level the arguer is plausibly committed to. Tension: the more general the bridge, the more informative, the more falsifiable, and the more interpretively risky (Govier; Ennis 2001).\n(c) Over-charity and straw man. Flagged by all. Converging mitigations: mark the bridge as analyst-supplied (Walton and Reed; Araucaria's missing flag), tie it to the arguer's commitments and context (van Eemeren Rule 5; Walton and Reed citing Gilbert), choose the reconstruction policy by purpose (Ennis 2001; Hitchcock 1985; Govier), extract rather than add (Gough and Tindale), and do not make every transparent hidden premise explicit (Rider and Thomason).\n\n**Q2 (real reason versus irrelevant or question-begging).** Johnson and Blair's relevance, acceptability, sufficiency triad is the standard frame, but Blair himself calls its terms placeholders. The sharpest operational test I found is Hitchcock (2011, p. 198): a premise is relevant iff some potentially accurate companion set (a) together with it suffices, (b) does not suffice without it, (c) is at least potentially accurate, and (d) can be verified without assuming the conclusion. Clause (b) is the idle-premise test; clause (d) is the question-begging test. Walton and Macagno distinguish topical from probative relevance. Freeman ties the relevance question to linked structure. I found NO dedicated named test for \"the supplied premise alone yields the conclusion, making the stated premise idle\"; it must be assembled from Hitchcock (b), the necessity amendment to Blair's account (reported by Hitchcock), and the Rationale No Danglers heuristic.\n\n**Q3 (disputing or weighting the inference).** Qualitative frameworks all treat the inference as separately attackable: Pollock undercutters; ASPIC+ rule names; AIF conflict-to-rule-application edges; Carneades exceptions; Rationale inference objections; recursive AFs (EAF, AFRA, RAF, RAFN, REBAF). Gradual semantics: see section 4 for the what-exists/what-does-not list. Short version: edge weights in QBAF families (DF-QuAD, quadratic energy, edge-weighted QBAF) are exogenous inputs; computed attack weights exist only for attack-only higher-order frameworks (Amgoud, Doder, Lagasquie-Schiex 2024/2026); a bipolar recursive continuous framework with edge values exists from one group (DECRAG, Feb 2026); capping by necessary co-premises exists at argument level (Pollock weakest link; Doder et al. Trimming; Selinger and Rago et al. products) but I found none attached to a support edge.\n\n**Q4 (chains).** Beardsley's diagram technique has intermediate conclusions; serial is the chain pattern. Freeman maps acceptability question to serial, relevance question to linked, ground-adequacy question to convergent. Wigmore charts chains (catenated inference, interim probanda) and deliberately does NOT state warrants, charting doubts and refutations instead. Araucaria and Rationale both let analysts add flagged missing nodes. The Rabbit Rule and Holding Hands are lexical heuristics for surfacing unstated co-premises (section 5). I found no source attributing them to van Gelder himself; the sources credit Neil Thomason.\n\n## 2. Q1 sources\n\n**Q1-1 Hitchcock (1985), Enthymematic arguments, Informal Logic 7(2-3): 83-97.** https://ojs.uwindsor.ca/index.php/informal_logic/article/download/2707/2148. FULL TEXT READ.\n- Thesis: an enthymeme implicitly assumes a universal generalization of its associated conditional over repeated content expressions, the maximal generalization consistent with plausibility (pp. 87-88; fully qualified statement pp. 94-95). Charity justifies the exceptions made on grounds of implausibility (p. 93).\n- The associated conditional alone cannot be the assumption: in the Depo-Provera example it comes out true merely because the conclusion is true, so the flaw cannot be exposed; refutation by parallel argument (Dalkon Shield) works only if a generalization is assumed (pp. 89-90).\n- He abandoned the criterion that a conditional making a stated premise redundant marks a non-sequitur, giving three reasons (p. 88). This matters for your idle-premise question.\n- Treats irrelevance as a slippery label and prefers to judge falsity of the implicit assumption (p. 87). Partial topical overlap separates enthymemes from non-sequiturs, with acknowledged counterexamples (Ennis's coat example, pp. 88-89).\n- Argues the implicit assumption is better seen as a rule of inference than a missing premise (pp. 94-95); Toulmin backing, Ennis back-ups and Scriven optimal assumptions support it (p. 96).\n- Different purposes call for different charity: identifying the arguer's assumption, strengthening, or discrediting (pp. 96-97). Says Govier's rule (justify additions from wording and context) lacks criteria (p. 83).\n\n**Q1-2 Hitchcock (1998), Does the traditional treatment of enthymemes rest on a mistake? Argumentation 12: 15-37.** https://link.springer.com/article/10.1023/A:1007738519694. ABSTRACT ONLY (paywalled). Abstract: the traditional unstated-premise treatment rests on the belief that only logical consequence can make a conclusion follow definitely; proposes a revised consequence notion extended to enthymematic consequence. Body content known only through his later papers (Q1-3 to Q1-5).\n\n**Q1-3 Hitchcock (2011), Inference claims, Informal Logic 31(3): 191-228.** https://informallogic.ca/index.php/informal_logic/article/download/3400/2709. SUBSTANTIAL EXTRACT (sections 1-2, Napoleon example pp. 196-198, section 8 on Gerlofs pp. 217-220).\n- Abstract: a conclusion follows iff an acceptable, counterfactual-supporting covering generalization non-trivially rules out true premises with untrue conclusion; a supposed unexpressed premise supplied to make an argument formally valid should be a covering generalization.\n- Napoleon argument (true premises, true conclusion, topical overlap, yet the conclusion does not follow): the associated material conditional is true and known, but cannot be known independently of the conclusion. This yields the relevance definition (a)-(d) quoted in section 1 (p. 198).\n- Calls the associated material conditional the logical minimum among validity-restoring additions that do not make a stated premise redundant (p. 218). Calls Gerlofs's proposal careful and likely equivalent to his own (pp. 218-219). Argues unstated premises are experienced as absent even in our own reasoning (section 2.2).\n\n**Q1-4 Hitchcock (2002), Sampling scholarly arguments: a test of a theory of good inference, in Argumentation and Its Applications (CD-ROM, OSSA, Windsor).** https://www.humanities.mcmaster.ca/~hitchckd/samplingastest.pdf. SUBSTANTIAL EXTRACT (method, results, discussion opening, conclusion).\n- 50 inferences sampled at random from library books. The covering-generalization test was directly applicable in 49; in 12 of those 49 the stated premises needed supplementing from nearby context or audience background knowledge (section 2). Formally valid inferences were rare; judging inferences usually needs field-specific substantive knowledge (conclusion). Defines non-trivial acceptability: not grounded merely in the unacceptability of the antecedent or acceptability of the consequent (p. 1). Useful base rate for your measurement.\n\n**Q1-5 Hitchcock, Non-logical consequence (author's PDF; journal details not verified).** https://www.humanities.mcmaster.ca/~hitchckd/non-logicalconsequence.pdf. SUBSTANTIAL EXTRACT (abstract; passage on covering generalization versus logical minimum). Requires truth-preservation to be non-trivial; says the supposed unstated premise is typically a covering generalization or something yielding one, not the logical minimum; cites the 49-of-50 result.\n\n**Q1-6 Hitchcock, Toulmin's warrants (author's PDF; the 2017 book contents list dates it 2003).** https://www.humanities.mcmaster.ca/~hitchckd/Toulminswarrants.pdf. SUBSTANTIAL EXTRACT (section 1 plus keyword checks). Reports Toulmin's challenge sequence: data answer what you have to go on; the warrant answers how you get there; backing supports the warrant. Toulmin (1958, p. 98) calls warrants general hypothetical statements acting as bridges authorising the step. Toulmin equivocates between warrant-as-statement and warrant-as-rule; Hitchcock treats it as a rule or licence and defends keeping it against criticisms by van Eemeren et al., Johnson and Freeman.\n\n**Q1-7 Gough and Tindale (1985), Hidden or missing premises, Informal Logic 7(2-3): 99-106.** https://informallogic.ca/index.php/informal_logic/article/view/2708/2149. FULL TEXT READ.\n- Prefers hidden to missing: the premise is extracted from the text, not added; the analyst is an archaeologist, not an architect (pp. 100-103).\n- Reports Govier's early scepticism that a missing premise is a product of the critic's mind, and that Govier rejects deductivism because premises could be added to make any argument valid (note 7, p. 105).\n- Charity should not mean building the best possible argument; extract as many premises as an adequate critique needs, but adding necessary fresh material risks a straw man (p. 103).\n\n**Q1-8 Govier (2018 [1987]), The problem of missing premises, ch. 5 of Problems in Argument Analysis and Evaluation (Windsor Studies in Argumentation 6).** https://ecampusontario.pressbooks.pub/wsia062018/chapter/5/ (WebFetch got 403; opened via Exa). FULL TEXT READ.\n- It is always possible to make an argument valid, with illuminating additions or with trivial ones; the associated conditional and the conclusion itself are always available but vacuous or question-begging (section 1).\n- For the Hardin sterilization example the conditional is redundant and useless; generalizations of it are non-redundant but go beyond what is said, so they are interpretively questionable (section 1).\n- Bans the associated conditional as a missing premise using Carroll's Tortoise regress: inference rules must be used, not stated, at some point; missing premises are only the gap-fillers (Ennis), a subset of unstated assumptions; pragmatic, meta-linguistic and methodological assumptions are excluded (section 2).\n- Charity can be taken too far (steelmanning); a reconstruction's clarity does not show it is accurate; balance charity against interpretive accuracy and purpose; no confident claim that some statement is the missing premise (section 3, closing list). Note 12 calls the interpretive cost of Hitchcock's generalizations a real objection.\n\n**Q1-9 van Eemeren and Grootendorst (2004), A Systematic Theory of Argumentation (CUP), ch. 1 note 8.** https://api.pageplace.de/preview/DT0400.9780511162459_A23689806/preview-9780511162459_A23689806.pdf. SUBSTANTIAL EXTRACT (publisher preview: front matter and ch. 1 only). Taken literally an argument with an unexpressed premise is invalid; the premise logically required usually has too little informative content, so analysis should aim at the pragmatic optimum rather than stop at the logical minimum. Note 9: convergent = multiple, linked = coordinative, serial = subordinative.\n\n**Q1-10 van Eemeren and Grootendorst (1992), Argumentation, Communication, and Fallacies, pp. 60-72 (logical minimum p. 64; discussion of contexts forcing the logical minimum to count as optimum p. 66).** COULD NOT OPEN. Second-hand only via Marraud (Q1-12), Gerlofs (Q1-14) and Blair (Q1-13): the logical minimum adds nothing new because proposing the argument already commits the arguer to it, so it is superfluous; the pragmatic optimum is the most informative validating candidate consistent with the arguer's commitments. UNVERIFIED at primary level.\n\n**Q1-11 van Eemeren and Garssen (2026), The pragma-dialectical approach to argumentation, in Aikin, Casey, Stevens (eds.), Routledge Handbook of Argumentation Theory, pp. 54-62.** https://pure.uva.nl/ws/files/305459267/The_Pragma-Dialectical_Approach_to_Argumentation_26_03_05_18_04_33.pdf. FULL TEXT READ.\n- Rule 5 (Unexpressed Premise Rule): no false attribution of unexpressed premises to the other party and no disowning one's own; an antagonist distorts when the reconstruction goes beyond the protagonist's commitments (section 6.3).\n- Rule 7 gloss: checking validity of non-explicit reasoning would be futile because it can always be made valid by adding the logical minimum, if reason then standpoint. Rule 8: standpoints defended by non-explicit reasoning count as conclusively defended only via an appropriate, correctly applied argument scheme.\n- Rule 4 (Pertinence) bars argumentation irrelevant to the standpoint; Rule 6 covers petitio principii.\n\n**Q1-12 Marraud (2024), The logical perspective in pragma-dialectics, Topoi (open access), doi 10.1007/s11245-024-10065-4.** https://link.springer.com/article/10.1007/s11245-024-10065-4. SUBSTANTIAL EXTRACT (section 4 on the pragma-dialectical periods). Reproduces van Eemeren and Grootendorst 1992 p. 64 (Bart-is-Dutch example): logical minimum states the inferential commitment only. Describes the pragmatic optimum as the logical minimum generalized as far as possible without implausible commitments, and argues it cannot count as a hidden premise. Useful taxonomy: objecting attacks a premise; rebutting attacks the inference (an argument incompatible with the logical minimum); refuting attacks the conclusion.\n\n**Q1-13 Blair (2010), Logic in the pragma-dialectical theory, ISSA Proceedings 2010.** https://rozenbergquarterly.com/issa-proceedings-2010-logic-in-the-pragma-dialectical-theory/. FULL TEXT READ. Reports pragma-dialectics wants the unexpressed premise to make the argument valid, be a speaker commitment and be the most informative candidate. For defeasible arguments (analogy, induction, best explanation, conductive) a validity-ensuring premise such as no further evidence or all else equal misrepresents the force and attributes an unreasonable commitment. Proposes treating argumentation schemes as inference licences and revising the procedure so it does not rely on reconstructive deductivism.\n\n**Q1-14 Gerlofs (2011 [ISSA 2010]), Hidden premises, hidden treasures? ISSA Proceedings 2010, ch. 52, pp. 577-586.** https://rozenbergquarterly.com/issa-proceedings-2010-hidden-premises-hidden-treasures/. FULL TEXT READ. Argues the logical minimum IS an adequate unexpressed premise and should stay in the structure as a premise, with the pragmatic optimum (the general rule) as its separate ground. Benefit: a critic can accept the ground yet reject the conditional (torn-wings bat example), and critical questions of each scheme sort into attacks on a stated premise, the ground, or the ground-to-conditional connection. Admits two difficulties: how to phrase the ground-to-conditional link, and the risk of making invalid reasoning look valid. This is the closest published match to your design.\n\n**Q1-15 Ennis (1982), Identifying implicit assumptions, Synthese 51: 61-86.** https://link.springer.com/article/10.1007/BF00413849. COULD NOT OPEN (preview shows only the reference list). Content known second-hand: needed assumptions (needed to support the conclusion or make the argument good) versus used assumptions (unstated reasons the arguer actually relied on), per Walton and Reed (Q1-19, p. 341 quoting Ennis pp. 63-64); gap-fillers versus back-ups (Grennan, Q1-18); three criteria for needed assumptions (fidelity, contribution to gap-filling, plausibility; pp. 70-71 per Grennan). UNVERIFIED at primary level.\n\n**Q1-16 Ennis (2001), Argument appraisal strategy: a comprehensive approach, Informal Logic 21(2): 97-140.** https://exa.ai/library/publication/hyczg9mxwwr (search-engine copy of excerpts). SEARCH SNIPPET ONLY (long excerpts). Five assumption-ascription policies (no-added, simple deductive, used, needed, goal-oriented). The simple deductive gap-filler is often a false universal. The used-assumption policy avoids improving on the arguer (too much charity would be wrong when appraising the arguer). The needed-assumption policy builds the strongest unbiased argument. The deductive policy is useful for probing and for guessing implicit parts that might deserve challenge.\n\n**Q1-17 Ennis, Problems in testing informal logic critical thinking reasoning ability, Informal Logic (c. 1984; volume/year not captured).** https://informallogic.ca/index.php/informal_logic/article/view/2717/2158. First page opened; cited passages SEARCH SNIPPET ONLY. Used assumptions are an empirical claim judged by inference to the best explanation; for needed assumptions maximal charity applies; significant assumptions are never logically necessary (there is always a way around); the Josie example shows deductive gap-fillers can be uncharitable.\n\n**Q1-18 Grennan (1994), Are gap-fillers missing premisses? Informal Logic.** https://ojs.uwindsor.ca/index.php/informal_logic/article/view/2456. ABSTRACT ONLY (plus search excerpt). Argues an unstated major premise in a syllogism is not a gap-filler but a back-up for the implicit inference claim.\n\n**Q1-19 Walton and Reed (2005), Argumentation schemes and enthymemes, Synthese 145: 339-370.** http://arg-tech.org/people/chris/publications/2005/synthese2005.pdf (scanned; read by OCR). SUBSTANTIAL EXTRACT (pp. 339-345 and 361-370; case-study pages 346-360 not read).\n- Attribution of premises is hard to verify and risks straw man; charity as strengthening alone is too crude because it pushes toward the best possible argument and may distort a weak one (p. 342).\n- Reconstruct with argumentation schemes and critical questions; most enthymemes rest on defeasible generalizations (pp. 339-340, 363).\n- Inserted premises must be clearly marked as the critic's hypothesis, and should be plausible to the audience and fit the arguer's apparent position, following Gilbert 1991 (p. 365). Araucaria marks analyst-added premises with a missing attribute (note 15). The reconstructed argument is closely related to but not identical to the original (Verheij, pp. 361-366).\n\n**Q1-20 Toulmin (1958), The Uses of Argument.** COULD NOT OPEN. Only via Hitchcock (Q1-6).\n\n**Q1-21 Hitchcock, reply to Bermejo-Luque on inference claims (file named 'so'.pdf at his McMaster site; title not captured).** SEARCH SNIPPET ONLY. Says the regress from justifying a covering generalization is not vicious. UNVERIFIED.\n\n(Rider and Thomason 2008, Q4-10, adds the Forest Formula caution about not making every hidden premise explicit.)\n\n## 3. Q2 sources\n\n**Q2-1 Johnson and Blair (2000), Informal logic: an overview, Informal Logic 20(2): 93-107.** https://informallogic.ca/index.php/informal_logic/article/download/2262/1706. SEARCH SNIPPET ONLY. Confirms the relevant-sufficient-acceptable triad originates with Johnson and Blair (1977), adopted by Govier (adequate for sufficient), Freeman and others. Primary Logical Self-Defense (1977/1994/2006): COULD NOT OPEN; its missing-premise definition is known from Gough and Tindale (Q1-7).\n\n**Q2-2 Hitchcock (2014), review of Blair, Groundwork in the Theory of Argumentation, Informal Logic (volume/pages not captured).** https://exa.ai/library/publication/ltdv2jj34yf. SUBSTANTIAL EXTRACT (about two-thirds; ends mid-sentence). Blair's premissary relevance: a premise is relevant iff it belongs to a set authoritatively warranting the inference. Hitchcock's critique: this makes irrelevant premises relevant (the Stoic grain-market example), so add that the premise must be necessary for the warrant to hold; it also excludes relevant-but-insufficient premises. Blair (2007): relevance is local, a placeholder, and mainly bites when deciding whether a passage is an argument or whether a premise has any probative bearing. The review also gives Hitchcock's revised relevance definition (a)-(d).\n\n**Q2-3 Hitchcock 2011 (Q1-3).** Clause (b) and the necessity amendment supply the idle-premise test; clause (d) supplies the question-begging test.\n\n**Q2-4 Freeman (1991), Dialectics and the Macrostructure of Arguments (Foris).** https://api.pageplace.de/preview/DT0400.9783110875843_A19822678/preview-9783110875843_A19822678.pdf. SUBSTANTIAL EXTRACT (front matter, contents, ch. 1 start, ch. 1 discussion of relevance versus modality). The basic dialectical questions are acceptability, relevance (why is that relevant?) and ground adequacy; contents show acceptability to serial, relevance to linked, first ground-adequacy question to convergent (ch. 4). Ch. 1: warrants function to explain relevance, while modalities concern argument strength; Toulmin's virtue is to separate relevance from modality. Ch. 3 is titled Data, Warrant, Backing or Just Plain Premises?\n- Fisher (1992), critical study of Freeman, Informal Logic 14(2-3), https://informallogic.ca/index.php/informal_logic/article/download/2542/1983, search snippet plus first page: confirms the question-to-structure mapping, that Freeman rejects Toulmin's warrant, backing and data as separate roles, and gives Freeman's definition of linked premises as taken together to show a relevant reason.\n- Reed (1999), Building monologue, http://arg.tech/people/chris/publications/1999/ossa99.html, search snippet only: confirms Freeman 1991 pp. 38-39 questions and treats backing as just another premise answering the relevance question.\n\n**Q2-5 Walton and Macagno (2016), Profiles of dialogue for relevance, Informal Logic 36(4): 523-562.** https://philarchive.org/archive/WALPOD-2. SUBSTANTIAL EXTRACT (sections 1-3). Topical relevance is subject-matter overlap; P is probatively relevant to Q if some sequence of argumentation pro or con Q starts at or contains P. Texts can be coherent and topically related yet probatively irrelevant (Windsor red-herring example, section 1.3). Method: compare a descriptive graph with a normative graph and use Carneades' Find Arguments assistant (knowledge base as a source of implicit premises) to test whether a path to the ultimate conclusion exists. Walton (2004), Relevance in Argumentation: COULD NOT OPEN (snippet only).\n\n**Q2-6 Selinger (2015), A formal model of conductive reasoning, ISSA Proceedings 2014 (Sic Sat), c. pp. 1331-1339.** https://rozenbergquarterly.com/issa-proceedings-2014-a-formal-model-of-conductive-reasoning/?print=pdf. FULL TEXT READ. Proposes a relevance test for an undercutter: it must raise the inference's internal strength once its negation is added to the premises, and the hybrid argument (premises plus the negated rebuttal) must not itself be acceptable. States the sufficiency of this test is left open.\n\n**Q2-7 van Eemeren and Garssen 2026 (Q1-11).** Pertinence Rule (4) and petitio principii under Rule 6. **Govier (Q1-8):** adding the conclusion as the missing premise is vacuous and question-begging. **Rationale No Danglers (Q4-11).**\n\nIdle-premise summary: no named test. Pieces: Hitchcock 2011 clause (b); Hitchcock's necessity amendment to Blair; Hitchcock 1985 explicitly rejecting redundancy as a criterion (his reasons: redundant premises can leave arguments valid; redundancy depends on the contested truth-functional reading; the criterion misses some formal fallacies).\n\n## 4. Q3 sources\n\n**Classic and structured frameworks**\n\n**Q3-1 Pollock (1987), Defeasible reasoning, Cognitive Science 11: 481-518.** http://www.horty.umiacs.io/courses/readings/pollock-1987-defreasoning.pdf. SUBSTANTIAL EXTRACT (sections 1-4.2). Definitions 2.2-2.5: rebutting defeaters give reason to deny the conclusion; undercutting defeaters attack the connection between reason and conclusion, being reasons to deny that P would not be true unless Q were. Defeat condition 4.1 targets the conditional from the premises to the conclusion, so undercutters can themselves be defeated (levels of arguments).\n\n**Q3-2 Pollock (2001), Defeasible reasoning with variable degrees of justification, Artificial Intelligence 133: 233-282.** http://www.horty.umiacs.io/courses/readings/pollock-2001-var-deg-just.pdf. SUBSTANTIAL EXTRACT (abstract, introduction, diminishers). Weakest link principle: an argument's degree of justification for its conclusion is the minimum of its premises' degrees and the strengths of the reasons used. Rejects accrual of independent reasons. Defeaters too weak to defeat can still diminish; limited collaborative defeat. This is the standard capping principle.\n\n**Q3-3 Modgil and Prakken (2014), The ASPIC+ framework for structured argumentation: a tutorial, Argument and Computation 5(1): 31-62.** https://webspace.science.uu.nl/~prakk101/pubs/ASPICtutorial.pdf. SUBSTANTIAL EXTRACT (Defs 3.2, 3.10; section 3.3; weakest- and last-link orderings). A naming function n maps each defeasible rule to a formula meaning the rule is applicable; an argument concluding the negation of n(r) undercuts any argument whose top rule is r (Def 3.10). Strict inferences cannot be attacked. Weakest- and last-link orderings (Defs 3.21+) are ordinal preferences, not numeric.\n\n**Q3-4 Chesnevar et al., Towards an argument interchange format, Knowledge Engineering Review (c. 2006-07; preprint, final pagination not verified).** http://arg.tech/people/chris/publications/2006/aif_final.pdf. SUBSTANTIAL EXTRACT (section 3.2, examples, Toulmin extension). Nodes are information (I) nodes and scheme-application (S) nodes; inferences are RA-nodes, conflicts CA-nodes, preferences PA-nodes. Edges between two I-nodes are forbidden because every connection needs an explaining S-node; S-to-S edges are called warrant edges. An attack on an RA-node is the undercut. In the Toulmin extension the warrant is an RA-node and backing an I-node. **Q3-5 Bex, Modgil, Prakken, Reed (2013), On logical specifications of the AIF, J. Logic and Computation (volume/pages not verified).** https://webspace.science.uu.nl/~prakk101/pubs/aifsem12.pdf. SUBSTANTIAL EXTRACT. Formal AIF graph definition (RA-node needs at least one premise edge and exactly one conclusion edge) and mapping to ASPIC+.\n\n**Q3-6 Gordon, Prakken, Walton (2007), The Carneades model of argument and burden of proof, Artificial Intelligence 171: 875-896.** https://webspace.science.uu.nl/~prakk101/pubs/GordonPrakkenWalton2007a.pdf. SUBSTANTIAL EXTRACT. Premise types: ordinary, assumption, exception. The proponent must establish ordinary premises and, once challenged, assumptions; exceptions are the respondent's burden. Critical questions of a scheme become assumptions or exceptions according to who should bear the burden of production. Undercutters are modelled directly as exceptions; linked arguments require all premises; convergent ones are multiple arguments.\n\n**Q3-7 Verheij (2014), Argumentation and artificial intelligence, Handbook of Argumentation Theory ch. 11.** https://www.ai.rug.nl/~verheij/publications/pdf/handbook2014ch11.pdf. SEARCH SNIPPET ONLY. In his assumption-based model the rules are conditionals inside the language, so an undercutter is an attack on a conditional (Pollock's red-light example).\n\n**Q3-8 van Gelder (2007), The rationale for Rationale, Law, Probability and Risk 6: 23-42, doi 10.1093/lpr/mgm032.** https://exa.ai/library/publication/v42l6rh95ls (aggregator copy; OUP page not opened). SUBSTANTIAL EXTRACT. Linked premises are shown enveloped in one colour. An inference objection counters not a stated premise but the relationship between premise and claim; it attaches to the support relation itself. A Show Ancestors command displays the chain from any claim to the main contention. Semi-formality means unstated claims must be identified and made explicit. No Rabbit Rule text in this paper.\n\n**Recursive and higher-order abstract frameworks (extension-based, unweighted)**\n- **Q3-9 Modgil (2009), Reasoning about preferences in argumentation frameworks, AIJ 173: 901-934** (EAF; second-order attacks on attacks to carry preferences). https://exa.ai/library/publication/b71f810jkwy. ABSTRACT ONLY. Modgil and Prakken (2010) structured EAFs, https://webspace.science.uu.nl/~prakk101/pubs/comma10-eaf.pdf, SEARCH SNIPPET ONLY.\n- **Q3-10 Baroni, Cerutti, Giacomin, Guida (2011), AFRA, IJAR 52: 19-37.** https://arxiv.org/abs/1810.04886. ABSTRACT ONLY (unlimited recursive attacks; compatibility with Dung).\n- **Q3-11 Cayrol, Fandinno, Farinas del Cerro, Lagasquie-Schiex (2020), RAF** (Annals of Mathematics and AI): COULD NOT OPEN; described via Q3-14 and Q3-15.\n- **Q3-12 Cayrol, Fandinno, Farinas del Cerro, Lagasquie-Schiex (2018), Structure-based semantics of AFs with higher-order attacks and supports, COMMA 2018, pp. 29-36 (RAFN).** https://hal.science/hal-02181911/document. FULL TEXT READ. Necessary supports and attacks may target attacks or supports; extensions become triples (arguments, valid attacks, valid supports). Legal example: a lawyer's argument attacks the support from a witness statement to intention to kill. Extension-based, unweighted.\n- **Q3-13 Cayrol, Cohen, Lagasquie-Schiex (2021), Higher-order interactions (bipolar or not) in abstract argumentation: a state of the art, Handbook of Formal Argumentation vol. 2.** https://ri.conicet.gov.ar/bitstream/handle/11336/157653/CONICET_Digital_Nro.3fa7c988-c6fd-4c36-be63-dc84094a0df3_A.pdf. SEARCH SNIPPET ONLY (abstract and outline: EAF, HLAF, AFRA, RAF; ASAF, RAFN, REBAF for supports).\n\n**Gradual and quantitative**\n\n**Q3-14 Amgoud, Doder, Lagasquie-Schiex (2024), Higher-order argumentation frameworks: principles and gradual semantics, IJCAI 2024.** https://www.ijcai.org/proceedings/2024/0357.pdf. FULL TEXT READ. First gradual semantics for frameworks where attacks carry weights and can be attacked by arguments. Equations (1)-(2): for any element x (argument or attack), strength = base weight divided by (1 plus the sum, over attacks on x, of an aggregate of the attack's strength and its source's strength). So an attack's strength IS computed from the arguments attacking that attack. Nine principles (Equality, Precedence, Local Equality...). Attack-only: sources are arguments; no supports. Hbs/Mbs families with aggregation operators; one hybrid satisfies all principles.\n**Q3-15 same authors (2026), Gradual semantics for weighted higher-order argumentation frameworks, JAIR 86, Article 31.** https://www.jair.org/index.php/jair/article/download/21469/27335. SUBSTANTIAL EXTRACT (abstract, introduction, section 2, section 8, conclusion). Extends Q3-14 with 16 principles and experiments (convergence in under 20 iterations). Related work: the only earlier attempt at gradual semantics for frameworks in which arguments support or attack relations, with weighted relations, is Barringer, Gabbay and Woods (2005), which gave a formula but no well-defined semantics (the equation system can have many solutions). The authors state they treat only attacks from arguments; attacks from attacks or sets are future work.\n**Q3-16 Barringer, Gabbay, Woods (2005).** COULD NOT OPEN; characterised only via Q3-15.\n\n**Q3-17 Mossakowski and Neuhaus (2018), Modular semantics and characteristics for bipolar weighted argumentation graphs, arXiv 1807.06685.** https://arxiv.org/abs/1807.06685. ABSTRACT ONLY plus search excerpts (aggregation function plus influence function; convergence results).\n**Q3-18 Rago, Toni, Aurisicchio, Baroni (2016), Discontinuity-free decision support with quantitative argumentation debates, KR 2016, pp. 63-73 (DF-QuAD).** https://cdn.aaai.org/ocs/12874/12874-57532-1-PB.pdf. SEARCH SNIPPET ONLY (long excerpts).\n**Q3-19 Potyka (2018), Continuous dynamical systems for weighted bipolar argumentation, KR 2018, pp. 148-157** (quadratic energy model). Tutorial https://arxiv.org/html/1811.12787v1: ABSTRACT ONLY. The QE definition below comes from Q3-20 (second-hand).\n**Q3-20 Yin, Potyka, Rago, Kampik, Toni (2025 arXiv 2507.11323 as Contestability in quantitative argumentation; KR 2026 as Contestability in edge-weighted quantitative bipolar argumentation frameworks, pp. 676-687).** https://arxiv.org/pdf/2507.11323 and https://proceedings.kr.org/2026/64/. SUBSTANTIAL EXTRACT (abstract, introduction, section 3). In an edge-weighted QBAF each attack or support has a given weight in [0,1]; a supporter or attacker contributes its strength times the edge weight; aggregation is by sum (QE, REB) or product (DF-QuAD); influence functions differ. Weights are inputs (for example user preferences or network connection weights); the paper tunes them by gradient-based explanations to meet a target strength. No arguments about edges.\n**Q3-21 Doder, Vesic, Croitoru (2020), Ranking semantics for argumentation systems with necessities, IJCAI 2020, pp. 1912-1918.** https://www.ijcai.org/proceedings/2020/0265.pdf. FULL TEXT READ. Necessity relation between arguments: an argument cannot be accepted if a necessary argument is not. Principle 16 Trimming: an argument's degree is at most that of each argument necessary for it. Principles N-neutrality, N-weakening, N-reinforcement. Semantics Npe: base weight times the product of necessary arguments' degrees, divided by an exponential of the probabilistic sum of attackers' degrees; degree never exceeds base weight; acyclic graphs, attacks only or necessities only. Necessity behaves neither as support nor as attack. Necessities target arguments, not edges.\n**Q3-22 Lagasquie-Schiex (2026), Is the necessity relation a real support? ICAART 2026, pp. 1306-1317.** https://www.scitepress.org/Papers/2026/142707/142707.pdf. SUBSTANTIAL EXTRACT (introduction, section 4, section 6). Says gradual semantics for weighted bipolar frameworks with necessities are scarce (Doder et al. 2020 is the only prior one), proposes translating them to attack-only weighted frameworks, and concludes necessity is not a support (does not reinforce its target).\n**Q3-23 Yamaguchi and Matsuda (2026), DECRAG, Argument and Computation (online Feb 2026), doi 10.1177/19462174251376186.** https://sage.cnpereading.com/doi/10.1177/19462174251376186. SUBSTANTIAL EXTRACT (abstract, section 1, Definitions 1-4). Graph where an edge's destination can be another edge (recursive attacks, supports, negative supports); a differential-equation system over a cost model yields values for both nodes and edges, and an edge's effect on its target is scaled by the edge's own value variable; says necessity and base scores can be simulated. Single research group; no independent replication found.\n**Q3-24 Tang (2025), Encoding higher-order argumentation frameworks with supports to propositional logic systems, arXiv 2512.23507.** https://arxiv.org/abs/2512.23507. ABSTRACT ONLY plus background definitions. Defines frameworks where attacks and supports can be both sources and targets with [0,1]-valued equational semantics; single-author preprint with no citations at time of access; formulas not verified.\n**Q3-25 Rago, Vasileiou, Toni, Son, Yeoh (2025), A methodology for incompleteness-tolerant and modular gradual semantics for argumentative statement graphs, KR 2025.** https://arxiv.org/pdf/2410.22209v7.pdf. SUBSTANTIAL EXTRACT (abstract, Defs 1-2, 10-12). Statements with conjunctive premises; the dialectical-conjunction semantics multiplies the strengths of the premise literals, each evaluated dialectically from its supporters and attackers, and so caps a statement by its weakest-evidenced conjunct. Terminology trap: here undercutting means attack on a premise, which ASPIC+ and Pollock call undermining.\n**Q3-26 Selinger (2014), Towards formal representation and evaluation of arguments, Argumentation 28(3): 379-393.** https://exa.ai/library/publication/rwf7w4tj56y. SEARCH SNIPPET ONLY. Conclusion strength = product of linked premises' acceptability times a predefined conditional acceptability of the inference; independent arguments cumulate by a probabilistic-sum-like operator. Selinger (2015, Q2-6) states attack on inference is not yet defined and sketches an extension.\n**Q3-27 Freedman, Dejl, Gorur, Yin, Rago, Toni (2025), Argumentative large language models for explainable and contestable claim verification, AAAI 2025.** https://ojs.aaai.org/index.php/AAAI/article/download/33637/35792. SEARCH SNIPPET ONLY. LLM generates supporters and attackers and assigns each a base score; DF-QuAD aggregates. There are no edge weights: the child's base score has to stand for how strongly it supports or attacks its parent.\n\n**What exists and what does not (as of my search; absence claims are absence of evidence)**\nEXISTS:\n1. Computed attack weights from arguments attacking the attack: attack-only (Q3-14, Q3-15).\n2. A bipolar recursive continuous framework valuing nodes and edges together: DECRAG (one group, 2026); a related preprint (Q3-24).\n3. Qualitative recursive support/necessity frameworks where supports can be targeted: RAFN, REBAF, ASAF (no weights).\n4. Capping at argument level: Pollock weakest link (min); Doder et al. Trimming and Npe (product); Selinger (product); Rago et al. (product over premises).\n5. Fixed edge weights: edge-weighted QBAFs (Q3-20).\nDOES NOT (found none):\n- A mainstream QBAF semantics (DF-QuAD, QE, REB, edge-weighted) in which a support edge's weight is derived from arguments about that edge.\n- A gradual semantics in which a necessity or co-premise attaches to an edge and caps that edge's contribution (necessities target arguments).\n- Gradual semantics for higher-order frameworks where attacks may originate from attacks or sets (named as future work in Q3-15).\n- A defined attack-on-inference in Selinger's numerical model (open per Selinger 2015).\n\n## 5. Q4 sources\n\n**Q4-1 Beardsley (1950), Practical Logic (later Thinking Straight).** COULD NOT OPEN. Second-hand: Freeman (ch. 1) says Beardsley originated the diagram technique (basic reasons on top, arrows to intermediate conclusions, then final conclusion); Hitchcock (Q4-2) says serial, divergent and convergent come from Beardsley 1950 p. 19; Fisher (Q2-4) says Beardsley and Thomas introduced the four-way distinctions.\n**Q4-2 Hitchcock (2014), The linked-convergent distinction, ISSA Proceedings 2014.** https://rozenbergquarterly.com/issa-proceedings-2014-the-linked-convergent-distinction/. FULL TEXT READ. Thomas (1977) was first to name linked; convergent, divergent and serial come from Beardsley 1950 p. 19. Argues linkage is a way support works, not a type of argument, so there may be no fact of the matter; criteria based on suspending a premise are futile. Also reports Pollock's max-not-accrual versus Selinger's accrual formula.\n**Q4-3 Freeman 1991 and Fisher 1992 (Q2-4).** Serial = answer to why should I believe that premise (support for a premise). Linked = answer to the relevance question (supplying a connecting premise). In Freeman's scheme a missing connecting claim is linked; an omitted intermediate conclusion is serial (my reading of his mapping).\n**Q4-4 Goodwin (2000), Wigmore's chart method, Informal Logic 20(3): 223-243.** https://informallogic.ca/index.php/informal_logic/article/download/2278/1722. SEARCH SNIPPET ONLY (long excerpts). Wigmore reasons that evidentiary inference is always inductive and that stating the warrant only moves the court's doubt to the major premise; he charts no ordinary warrants (one exception: expert-backed generalizations, with a special D symbol). Strength of a step is shown by charting alternative explanations (left) and their refutations (right); each step marks a determinate doubt, and no doubt means no step to represent.\n**Q4-5 Tillers and Schum (1988), Charting new territory in judicial proof: beyond Wigmore, Cardozo Law Review 9 (pagination not verified).** https://larc.cardozo.yu.edu/cgi/viewcontent.cgi?article=1309&context=clr. SEARCH SNIPPET ONLY. Evidence varies in remoteness from the ultimate probandum; chains pass through interim probanda; each intermediate stage exposes a possible source of doubt; ancillary evidence strengthens or weakens a link in a chain set up by other evidence; Wigmore called such inference catenated. This is precedent for evidence about a link.\n**Q4-6 Reed and Rowe (2004), Araucaria: software for argument analysis, diagramming and representation, Int. J. Artificial Intelligence Tools (volume/pages not verified).** http://arg.tech/people/chris/publications/2004/ijait.pdf. SUBSTANTIAL EXTRACT. Supports convergent, linked and chained structures, missing premises (editable analyst-added text) and refutations; reconstruction is a critical phase because arguments are heavily abridged. A teaching-use PDF (https://computationalrhetoricworkshop.uwaterloo.ca/wp-content/uploads/2016/07/Araucaria-as-a-Tool-for-Diagramming-Arguments-in-Teaching-and-Studying-Philosophy.pdf, authors not captured, SEARCH SNIPPET ONLY) adds that analyst-added premises appear as grey nodes, schemes can be selected per link, and strength can be marked on inference arrows.\n**Q4-7 Davies (2012), Computer-aided argument mapping and the teaching of critical thinking, Part 1, Inquiry 27(2): 15-30** (PhilArchive metadata lists 27(3):16-28, which appears to be wrong). https://philarchive.org/archive/DAVCMA. SUBSTANTIAL EXTRACT. Missing rungs: prose omits connecting information, and mapping exposes it (section 5.1f); co-premises are helping premises; implicit assumptions should be made explicit even when trivial. Credits Holding Hands and the Rabbit Rule to Neil Thomason; Rabbit Rule is Holding Hands applied between premises and contention.\n**Q4-8 Rider and Thomason (2008), Cognitive and pedagogical benefits of argument mapping: LAMP guides the way to better thinking, in Knowledge Cartography (Springer), from p. 113.** https://www.reasoninglab.com/wp-content/uploads/2013/10/riderthomason_cognitive_pedagical_benefits_of_am_2008.pdf. SUBSTANTIAL EXTRACT (sections 1-3 and references; roughly the first 40%). Source of the exact statements below plus the Forest Formula caution (Q4-10).\n**Q4-9 Reasoning Lab, Argument mapping: the rules (Critical Thinking Skills BV sheet).** https://www.reasoninglab.com/wp-content/uploads/2013/10/Argument-Maps-the-Rules.pdf. FULL TEXT READ (2 pages, read as images).\n**Q4-10 Rationale Online, Tutorial 2 (print page and pages 2.6 Rabbit Rule, 2.8 Holding Hands, 2.9 No Danglers).** https://rationaleonline.com/explore/en/tutorials/tutorials/Tutorial_2/print.htm (and the 6_Rabbit_Rule, 8_Holding_Hands, 9_No_Danglers pages). SUBSTANTIAL EXTRACT, WEBFETCH SUMMARY ONLY (wording normalised; lower confidence).\n**Q4-11 Laronge, Replacing inference-rule heuristics with defeasible class-inclusion transitivity (n.d.).** https://inferenceincourt.com/wp-content/uploads/2009/12/replacing-rough-inference-rule-heuristics-with-dcit-laronge.pdf. SEARCH SNIPPET ONLY. Calls the Rabbit Rule and Holding Hands rough heuristics resting on an incomplete model of structural correctness, with vague term-matching.\n\n### The Rabbit Rule and Holding Hands, stated precisely (from Q4-8, Q4-9, Q4-10)\n- RABBIT RULE (vertical, between a claim and its reason). Every important term of the claim must also occur in at least one premise of each reason offered for that claim. Wordings differ slightly: the Reasoning Lab sheet speaks of a claim and each of its reasons; Rationale's tutorial says the contention's terms must appear in one of the premises of the simple argument; Rider and Thomason (2008) say: \"You can't conclude something about rabbits if you haven't been talking about rabbits.\"\n- HOLDING HANDS (horizontal, within a reason). Every important term occurring in a premise but not already accounted for by the claim must also occur in another premise of the same reason (Rider and Thomason: in another premise or the conclusion). No term may stand alone in one box.\n- NO DANGLERS (Rationale) = the two combined: every important term in any claim of a simple argument appears in another claim of it.\n- PURPOSE. A lexical, structural screen to surface unstated co-premises, and a teaching device for reading and writing maps; the Rabbit Rule is described as the most powerful use of Holding Hands for noticing hidden premises that do much of the argumentative work. NOT a goodness test: the tutorial says satisfying them does not make an argument good. Cautions from Rider and Thomason: universal quantifiers can legitimately dangle; experienced mappers need not and should not make every hidden premise explicit; they cite van Heuveln's (2004) Forest Formula (make explicit only claims with which the inference is not sufficiently transparent) and note transparency is audience-relative. I did not open van Heuveln (2004).\n- ATTRIBUTION. Davies credits both rules to Thomason; Rider and Thomason co-author the main exposition; Rationale was developed by van Gelder's company. I found no text in which van Gelder states them, and one search summariser's attribution to van Gelder is unsupported by any source I opened.\n\n## 6. Where sources disagree\n\n1. Should the bridge be a premise at all? Premise: Gerlofs (with a ground), Freeman (just plain premises), Rationale co-premises, Walton and Reed (scheme premises, marked as analyst hypotheses). Not a premise but a rule or inference claim: Hitchcock 1985/1998/2011 (though in 2011 he says Gerlofs's approach is probably equivalent), Govier (associated conditional banned; rules used not stated; Carroll regress), Toulmin (warrant separate from data), Wigmore (do not state warrants; chart doubts). Neither: ASPIC+ makes rule applicability a nameable claim whose negation undercuts, but strict rules cannot be attacked.\n2. Is the associated conditional trivial? van Eemeren and Grootendorst, Govier, and Hitchcock 1985 say superfluous or inadequate; Gerlofs says adequate and useful as a premise; Hitchcock 2011 says it is the logical minimum and is true iff a covering generalization exists. Marraud reads pragma-dialectical rebuttal as targeting it.\n3. Level of generality. Hitchcock: maximal plausible generalization; van Eemeren and Grootendorst: generalize as far as commitments allow; Govier and Ennis 2001: generalizations are often false or go beyond what the arguer committed to; Ennis and Blair 2010: for defeasible inference, validity-restoring universals distort.\n4. Deductivism. Reconstructive deductivism underlies the pragma-dialectical procedure (and Groarke); Blair, Govier, Godden and Walton and Reed reject it for non-deductive schemes.\n5. Charity. Ennis: maximal charity for needed assumptions but not when appraising the arguer; Gough and Tindale: extract, do not build the best argument; Govier: strong charity distorts; Walton and Reed: charity too crude; Rider and Thomason: leave most hidden premises implicit.\n6. Hidden versus missing (Gough and Tindale versus Govier, who says hiding is itself disputable).\n7. Origin of serial and divergent: Freeman credits Thomas with naming the patterns; Hitchcock 2014 says Beardsley 1950 p. 19 (not verified in the primary).\n8. Linked versus convergent: Freeman gives a question-based criterion; Hitchcock 2014 says suspension-based criteria are futile and the distinction concerns support not arguments.\n9. Accrual and capping: Pollock rejects accrual and uses minimum; Selinger multiplies and accrues; Doder et al. multiply over necessities; DF-QuAD aggregates supporters and attackers.\n10. Necessity versus support: Doder et al. say necessity is neither; Lagasquie-Schiex says necessity is not a support and criticises some of Doder et al.'s principles.\n11. Relevance: Hitchcock 1985 distrusted irrelevance as a logical charge, then gave the 2011 operational definition; Blair's premissary relevance is criticised by Hitchcock as over-inclusive.\n12. Rabbit Rule wording and scope, and attribution (section 5).\n\n## 7. Could not verify or not found\n\n- Could not open: Ennis 1982 (paywalled); Hitchcock 1998 body; Toulmin 1958; van Eemeren and Grootendorst 1992; Beardsley 1950; Johnson and Blair 1977/1994/2006; Blair 2012 chapter; Walton 2004; Hitchcock 1992 Relevance; Freeman 2011; Pollock 1995; Barringer, Gabbay, Woods 2005; Cayrol et al. 2020 (RAF); van Heuveln 2004; Gilbert 1991 Enthymeme Buster; Gerlofs 2009 dissertation; Wigmore's own text; Potyka 2018 KR paper itself; Mossakowski and Neuhaus beyond abstract.\n- Search snippet only (excerpts verbatim from search tool, page not read in full): Ennis 2001; Johnson and Blair 2000; Goodwin 2000; Tillers and Schum 1988; DF-QuAD paper; Selinger 2014; Freedman et al. 2025; Verheij 2014; Cayrol, Cohen, Lagasquie-Schiex 2021; Laronge; Walton 2004 snippet.\n- Walton and Reed pages 346-360 (case studies) not read.\n- Citation fields I could not confirm: Chesnevar et al. KER volume and pages; Bex et al. 2013 journal volume and pages; Reed and Rowe 2004 volume; Tillers and Schum pagination; Hitchcock 2014 review volume; Hitchcock Toulmin's warrants and Non-logical consequence venues; Ennis c. 1984 volume.\n- Not found: any named, published test specifically for a supplied premise that alone entails the conclusion and so renders the stated premise idle; any peer-reviewed gradual semantics with a support-edge weight computed from arguments about it (DECRAG and the HAFS preprint are the only candidates, both unvetted); any gradual semantics attaching a necessary co-premise to an edge.\n- DECRAG's formal properties and HAFS's formulas were not verified beyond the definitions I read.\n\n## 8. Design implications (my inference, not from the sources)\n\n- Storing the bridge as its own claim is defensible (Gerlofs, Freeman, Rationale) but contested; a safer hybrid is a bridge claim plus a separate undercutter or exception slot (Carneades, ASPIC+, Rationale inference objection), so critics can attack the claim by evidence or attack the step by naming a defeating circumstance.\n- Do not auto-generate the associated conditional; require a general or scheme-level bridge and a check that it is non-trivially true and checkable without assuming the conclusion (Hitchcock 2011 (b), (d)).\n- Flag machine-supplied bridges as hypotheses (Walton and Reed; Araucaria missing flag) and record which reconstruction policy was used (used versus needed, Ennis).\n- Chains: an omitted connecting claim is a linked premise; an omitted intermediate conclusion is a serial step (Freeman); Wigmore suggests expanding a step only when there is a determinate doubt.\n- If you want numeric strength: min or product capping is well precedented at argument level (Pollock, Doder, Selinger, Rago); for an edge whose weight comes from arguments about it, the closest templates are the higher-order attack semantics (IJCAI 2024) and DECRAG; neither is off the shelf for supports.\n"}