{"path":"research/support-semantics-and-the-conjunction-problem.md","content":"# Support Semantics and the Conjunction Problem\n\n**Date**: 2026-08-17\n**Status**: Literature findings. **The distinction Deliberus needs is already named, formalised and dual-analysed in the argumentation literature** — the corpus does not have to invent it. Nothing decided.\n**Prompted by**: the founder, on the subtree-energy finding — *\"The subtree energy and evidence computation is solvable, I'm sure. We just need theoretical clarity.\"*\n\n---\n\n## 1. The problem, restated\n\n`subclaim_energy` treats every decomposition child identically: additive, weighted 0.3, no correlation term. But two different relations are wearing that one name.\n\n**Dr Smith's four parts are jointly required.** Publication record, institutional appointment, publication-implies-expertise, domain match. If the domain does not match, credibility *in that domain* collapses — the other three do not rescue it.\n\n**Three studies finding one effect are corroborative.** Losing one leaves the case standing.\n\nAdditive energy scores both the same, so a mother can read strong while a *necessary* part is weak. And it compounds with the splitting incentive: **decomposing a conjunctive claim into more parts raises its strength while making it strictly more fragile.**\n\n---\n\n## 2. The literature named this, in four ways, and then unified them\n\nThere is no single interpretation of support in bipolar argumentation. Four specialisations were developed independently and later restated in a common setting:\n\n**Deductive support** (Boella et al.). *a supports b* means accepting a implies accepting b. Characterised by the **closure principle**.\n\n**Necessary support** (Nouioua & Risch). *a supports b* means accepting b implies accepting a — **a is necessary for b**. Characterised by the **inverse closure principle**.\n\n**Evidential support** (Oren & Norman). An argument **cannot stand at all** unless supported by a chain reaching actual evidence. Support is not a bonus; it is a precondition.\n\n**Backing support** (Cayrol & Lagasquie-Schiex).\n\nThe unifying result worth carrying: **deductive and necessary support are duals**, and each is modelled by inducing a different *complex attack* — so the two interpretations differ not in how much weight they carry but in **what attacking a part does to the whole**.\n\n---\n\n## 3. The mapping, which is close to exact\n\n**Dr Smith's parts are necessary support.** Accepting \"Smith is a credible expert in labour economics\" requires accepting \"labour economics is the relevant domain\". That is the inverse closure principle verbatim. And the literature's mechanism gives the behaviour the corpus wants without inventing a formula: under a necessary reading, **an attack on any one part induces an attack on the whole**. Conjunction falls out of the induced-attack structure rather than being bolted on as a minimum.\n\n**Corroborating studies are closer to deductive or evidential support.** Each raises the case; none is individually necessary; attacking one does not induce an attack on the whole.\n\n**So the fix is ontological before it is arithmetic.** `DECOMPOSES_INTO` must record *which interpretation* holds, because the two generate different induced attacks. Choosing `min` instead of `sum` would approximate the symptom; recording the interpretation addresses the thing.\n\n**And evidential support names something the corpus has separately been circling.** *An argument cannot stand unless supported by a chain reaching evidence* is the sorry-marker intuition in formal dress, and it is a stricter reading than Deliberus currently implements — where an unsupported claim sits at an agnostic 0.5 rather than being unable to stand.\n\n---\n\n## 3b. Three shapes, read out of the live graph rather than reasoned about\n\n*Added 2026-08-17. The founder asked what decompositions actually look like. Reading real trees found a\nshape the implementation does not have, which is why this section quotes the corpus rather than examples.*\n\n**All of these must hold — necessary.** *\"Justice outweighs equality morally\"* decomposes into\n*institutional legitimacy as covering value*, *equality as derivative legal value*, *shift in value\npriority conditions*, and others. Van den Haag's argument needs every one: if equality is **not** a\nderivative legal value the weighing collapses, and three strong siblings do not rescue it. **The weakest\npart decides.** Implemented — minimum plus a ceiling.\n\n**Each of these adds a bit — corroborative.** *\"Evaluate net benefits for families\"* draws on\n*Card-Krueger evidence*, *monopsony in low-wage markets*, and *minimal job loss from increases*. Knock out\nCard-Krueger and monopsony theory still stands. **Contributions accumulate.** Implemented, and this is\nwhere the splitting incentive lives.\n\n**At least one of these must work — alternative. NOT implemented.** *\"Financing determines net benefit\"*\nhas children that are **competing financing routes**: AI taxation, diffuse taxation, supernormal profits.\nYou do not need all of them and they do not each add a little. You need one that survives. **The strongest\npart decides**, which is a *maximum* — neither the minimum nor the sum.\n\n**The gap matters in the flattering direction.** Under the current code, three weak alternatives *sum* into\na strong-looking case, so a claim with several unconvincing escape routes reads better than one with a\nsingle solid route. That is the same failure shape as the conjunction problem, mirrored.\n\n**And it is a sequencing decision, not just a feature.** Adding a third value after tagging has begun means\nre-reading everything already tagged, because a decomposition marked `corroborative` today might have been\n`alternative` all along. **Decide the vocabulary before the first tag is written**, not after.\n\n**What the graph looks like for all of this: nothing new.** No node type, no extra edge. One word on the\nexisting `DECOMPOSES_INTO` edge. All 131 currently carry nothing, which is why nothing has moved.\n\n## 4. Two further findings, one useful and one a warning\n\n**The aggregation layer has a 2026 paper of its own, and it has now been read.** *Aggregative Semantics for Quantitative Bipolar Argumentation* (Munro, Bloch & Lesot, arXiv:2603.06067) addresses exactly the layer Deliberus implemented without a specification. **Its central move: attackers and supporters should be aggregated separately, in a three-stage computation with three independently constrainable aggregation functions, precisely because the two do not play symmetric roles.** So the answer to the remaining strength questions is more parametrisation rather than one better formula — and the three shapes above are an instance of exactly that, different aggregation functions for groups playing different roles. Reading it also exposed two divergences from canonical QE that were accidental and are now deliberate: **we feed the energy term with edge strengths where the paper feeds supporters' own acceptability degrees recursively**, and **we sum offsets from 0.5 where the paper sums raw degrees** (defensible here, since 0.5 means \"unanswered\" rather than \"half-strength\", with a one-line path back at `NEUTRAL_POINT`). And it supplied a ready-made integrity rule: **no argument may both support and attack the same target** — measured live at zero violations, and the first concrete constraint for the approved ontology layer.\n\n**And aggregating support across people hits an impossibility.** *Aggregation of Support-Relations of Bipolar Argumentation Frameworks* (AAMAS 2020) shows that when a group disagrees about *what supports what*, any aggregation rule satisfying basic axioms and preserving certain properties **must be a dictatorship**. This is the judgment-aggregation family the shipped discursive-dilemma flag already belongs to, arriving one layer down: the support relation inherits it. Deliberus does not aggregate competing views of the support structure today. **If it ever does — and a multi-user graph eventually must — this result says a neutral rule is not available**, so the design question is which non-neutrality to adopt openly rather than how to avoid one.\n\n---\n\n## 5. What this changes, and what it does not\n\n**Changes.** The conjunction problem is a solved *modelling* question with a named answer, a dual analysis, and a mechanism (induced attacks). The remaining work is choosing an interpretation per decomposition and recording it, which is an ontology addition.\n\n**Does not change.** Whether Deliberus should adopt necessary support as the default for decomposition is a founder decision with real consequences — under a necessary reading, one weak part drags the whole down, which is stricter than today and will lower many strengths. And none of this touches the splitting incentive, which is about *count*, not interpretation.\n\n**Sequencing note, updated.** The aggregative-semantics paper has been read, and it moves the sequencing: **settle the vocabulary before the first tag is written.** A third shape (`alternative`) is already in the live graph and unimplemented, and adding it after tagging begins means re-reading everything already tagged.\n\n---\n\n## Sources\n\nCayrol & Lagasquie-Schiex on bipolar frameworks and gradual semantics · Boella et al. (deductive support) · Nouioua & Risch (necessary support) · Oren & Norman (evidential support) · *An axiomatic approach to support in argumentation* (IRIT) · *A Principle-Based Analysis of Bipolar Argumentation Semantics* (28 semantics against 10 principles) · *A survey of different approaches to support in argumentation systems* (Knowledge Engineering Review) · *Aggregative Semantics for Quantitative Bipolar Argumentation*, Munro, Bloch & Lesot, arXiv:2603.06067 (**read 2026-08-17**; see §4) · *Aggregation of Support-Relations of Bipolar Argumentation Frameworks*, AAMAS 2020 · Baroni et al. 2019 for the QBAF introduction.\n\n## Cross-references\n\n[strength-layer-audit.md](strength-layer-audit.md) (the defects this answers) · [self-similar-decomposition-and-claim-ontology.md](self-similar-decomposition-and-claim-ontology.md) (the mother-claim pattern) · [decomposition-axes.md](decomposition-axes.md) (evidence routing and fair-share discounting, which address sharing rather than logical relation) · [qbaf-gradual-semantics-research.md](qbaf-gradual-semantics-research.md) (the original QEM selection)\n"}