# ADR-0205 — modus_ponens + the Disagreement Rule (proof_chain's first inference rule) **Status:** Accepted (proof_chain phase 2.3 — the first inference rule + the wrong=0 mechanism) **Date:** 2026-06-02 **Relates to:** ADR-0204 (proof-graph builder — the proof-step shape this rule consumes), ADR-0201/0201.1/0202 (canonicalizer + contract), ADR-0203 (acyclicity guard), `generate.derivation.verify.select_self_verified` (the arithmetic twin of the disagreement rule). **Deferred to:** ADR-0206 (atom→carrier grounding). --- ## Context proof_chain has a DAG substrate (ADR-0204) and a canonical form (ADR-0201). Phase 2.3 adds the first **inference rule** (`modus_ponens`) and the **wrong=0 mechanism for proofs** (the disagreement/uniqueness rule). GPT-5.5 authored an independent adversarial corpus (`modus_ponens_cases.json`, 24 cases: 6 valid / 8 invalid / 10 disagreement) against the ADR-0202 grammar, with the proof-step shape left `[PENDING ADR-0204 SHAPE]` and the disagreement cases `[CONFIRM WITH OPUS]` — the contract-first handshake. ## Decision ### 1. The MP-rule contract (committed before the rule) - **Proof-step shape** = the already-landed ADR-0204 `Proof`/`ProofNode` (`proof_from_premises` desugars premises→premise-nodes, conclusion→mp-node). The `[PENDING ADR-0204 SHAPE]` placeholders resolve to this; the rule also accepts a bare `(premises, conclusion)` for corpus cross-check. - **Closed typed-reason set** (`generate/proof_chain/rules.py::MP_REASONS`): `unique_canonical_conclusion` (admit); `missing_implication`, `unestablished_antecedent`, `conclusion_mismatch`, `conclusion_disagreement` (refuse). ### 2. `evaluate_modus_ponens` — ROBDD-exact, bypassing unit-resolution In `generate/proof_chain/rules.py` (the proof-layer dispatch, ADR-0205 Option B): operates on proposition **formulas** via the canonicalizer; **never** calls `check_admissibility` / `_resolve_dep_units` (proofs have no units — the named 2.2 constraint, satisfied by construction). An implication `A→B` fires iff `key(A)` is an established premise key; the consequent `B` is recovered syntactically (`logic_canonical.parse_top_implication`) because the ROBDD form does not preserve which side is antecedent (`P→Q` and `¬P∨Q` share one diagram). ### 3. The disagreement / uniqueness rule (the wrong=0 mechanism) The literal twin of `select_self_verified`: **pool ALL admissible single-step MP derivations the premise set supports**, collect their canonical conclusion keys, and admit iff they collapse to exactly one key equal to the declared conclusion; ≥2 distinct keys → refuse (`conclusion_disagreement`); one key ≠ declared → `conclusion_mismatch`; no admissible derivation → `missing_implication` / `unestablished_antecedent`. **Pooling over the premise set — NOT filtering to the declared conclusion first — is the soundness mechanism.** Filter-first would admit-by-assertion when the same premises admit a different key (the `20/5 == 4` class one level up): e.g. a premise set that MP-derives both an unrelated tautology and the declared atom, or both the declared atom and a strictly stronger one. Collapse-vs-conflict is judged by **exact canonical key**, never surface — equivalent paths (`P∧Q`/`Q∧P`, `Q→R`/`¬Q∨R`, `Q`/`Q∧(R∨¬R)`) collapse and admit; subtly-different ones (`P∧Q`/`P∨Q`, `P→Q`/`Q→P`, `Q`/`Q∧R`) refuse. ### 4. Reason-set consolidation (the reconciliation finding) The mechanism makes one distinction per axis; the corpus's finer labels collapse onto the closed set — *the same redundancy confirmed for the disagreement labels applies to the invalid labels*: - 6 disagreement refuse-labels (conflicting / contradictory / tautology_vs_substantive / distinct_atom / near_miss / stronger) → **`conclusion_disagreement`** (the rule knows only: keys agree or they don't). - 4 antecedent-flavor labels (missing_antecedent / antecedent_mismatch / affirming_consequent / implication_direction_mismatch) → **`unestablished_antecedent`** (the rule knows only: the available implication's antecedent is or isn't an established premise; `affirming_consequent` and `implication_direction_mismatch` are the identical pattern `A→B, conclusion≡A, premise≡B` — no mechanical distinction). This consolidation is conveyed back to GPT-5.5 for the corpus's committed reasons. ## Honesty boundary (load-bearing — exact scope stated) Through phase 2.3, proof_chain is **sound over its declared atoms**, not grounded in recognized input (grounding is 2.4). And the disagreement rule's guarantee has a **precise scope**: it guarantees a unique conclusion among **single-step modus ponens derivations over the given premises** — **NOT** "uniquely entailed" by all proof strategies (a stronger, currently-false claim). This must never be read as "uniquely entailed." Same discipline as propositional-not-FOL. ## Evidence (cross-checked, not asserted) - **All 24 corpus cases agree on OUTCOME against the real rule** (6 admit / 8 refuse / 4 admit / 6 refuse) — no rule bug, no corpus outcome-misread. Reasons: 3 exact, 21 consolidate onto the closed set as above. - `tests/test_proof_chain_rules.py` — the 24 cases (transcribed, reproducible) + the pooling guards. **Mutation:** a filter-to-declared-conclusion-first variant makes MP-DISAGREE-007/010 wrongly admit → `test_pooling_*` fail. Pooling is load-bearing. - **Drive-by fix (cleanup-as-you-find):** the merged ADR-0204 `ProofNode.__post_init__` was dedented to module level, so **all `ProofNode` validation was silently dead** (smoke skips the dedicated test file, so the merge missed it — the "smoke ≠ full suite" hazard). Re-indented; validation restored; `test_self_dependency_refused_at_proof_model` passes. - Full binding-graph + admissibility surface green; smoke 67. Additive (no math path touched). ## Three deferrals — settled 1. **modus_ponens bypasses unit-resolution** — by construction (proof-layer dispatch). 2. **Conclusion typing** — the rule needs the declared conclusion's key + premise formulas, both reachable; `conclusion_symbol_id` suffices, **no `BoundUnknown`**. 3. **`semantic_role`** — proofs still not wired into serving; **stays `"unknown"`.** ## Deferred - **2.4** — atom→ADR-0144 `EpistemicNode` grounding (ADR-0206). Until then, "sound over declared atoms," never "reasons over input." - Multi-step proof composition and a proof *search* (the disagreement rule here is over a given premise set, single-step) — later, separately scoped.