Part I - CONTRADICTION: the geometry of structural conflict

Part I – CONTRADICTION The geometry of structural conflict James D. Atkinson 2025 Abstract This paper introduces the contradiction trap: a dialectical and game-theoretic mechanism for detecting structural bias, motivated asymmetry, and narrative drift in institutional and algorithmic decision systems. Grounded in epistemic game theory, the trap recasts contradiction as a falsifiable evidential event: whenever a system’s stated rationale and its observable behaviour cannot be jointly sustained, the resulting inconsistency becomes a measurable signal of underlying deviation. The trap is formalised as a one-move, strictly competitive epistemic game in which every admissible response incurs coherence loss, generating informational payoffs that convert contradiction into diagnostic evidence. The framework provides a portable audit instrument for domains that claim impartiality but exhibit asymmetric behaviour – governance, organisational reasoning, and algorithmic architectures alike. By treating inconsistency not as a logical failure but as a data-bearing phenomenon, the contradiction trap establishes the epistemic foundations of the mathematics of integrity: a unified evidential paradigm in which legitimacy is demonstrated not through assertion, but through resistance to structured, adversarial challenge. Keywords: mathematics of integrity; epistemic game theory; dialectical inference; contradiction analysis; structural asymmetry; coherence loss; motivated deviation; epistemic diagnostics; adversarial reasoning; algorithmic accountability; institutional reasoning; reasoning integrity; philosophy of logic; evidential audit design. 1 Contents 1 Visual abstract 6 2 Notation and symbols 7 3 Introduction 9 4 Related works 11 4.1 Dialectical and logical foundations . . . . . . . . . . . . . . . . . . . . 11 4.2 Epistemic game theory and information dynamics . . . . . . . . . . . 12 4.3 Structural bias, auditing, and algorithmic accountability . . . . . . . . 12 4.4 Contemporary developments . . . . . . . . . . . . . . . . . . . . . . . 13 4.5 Synthesis................................... 13 5 Contribution and novelty 14 5.1 From proof to performance. . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 Quantification of contradiction. . . . . . . . . . . . . . . . . . . . . . . 14 5.3 Integration with algorithmic accountability. . . . . . . . . . . . . . . . 14 6 Definition and core structure 15 6.1 Formaldefinition .............................. 16 6.2 The generic R–N–B–f(P) framework . . . . . . . . . . . . . . . . . . . 17 6.3 Thecoreproperty.............................. 17 7 Origins and distinction 18 8 Coherence cost estimation methods: selection and validation 19 8.1 Comparative framework . . . . . . . . . . . . . . . . . . . . . . . . . . 20 8.2 Scaling and performance . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8.3 Null model and significance testing . . . . . . . . . . . . . . . . . . . . 22 8.4 Method selection guidance . . . . . . . . . . . . . . . . . . . . . . . . 22 8.5 Multi-method validation . . . . . . . . . . . . . . . . . . . . . . . . . . 23 9 Methodology for contradiction games 23 9.1 Purpose ................................... 23 9.2 Inputsandartefacts ............................ 23 9.3 Construction of the trap . . . . . . . . . . . . . . . . . . . . . . . . . . 24 9.4 Measurement and quantification . . . . . . . . . . . . . . . . . . . . . 24 9.4.1 Coherencecost........................... 24 9.4.2 Accumulation models . . . . . . . . . . . . . . . . . . . . . . . 25 9.5 Meta-moves and secondary signals . . . . . . . . . . . . . . . . . . . . 26 2 10 Structural contradiction analysis 26 10.1 Worked examples at full structural rigour . . . . . . . . . . . . . . . . 26 10.1.1 Example 1: organisational restructure . . . . . . . . . . . . . . 27 10.1.2 Example 2: algorithmic hiring audit . . . . . . . . . . . . . . . 28 10.1.3 Example 3: legal consistency test . . . . . . . . . . . . . . . . 29 10.2 Meta-move classification . . . . . . . . . . . . . . . . . . . . . . . . . . 29 10.3 Field deployment (mini-study) . . . . . . . . . . . . . . . . . . . . . . 30 10.4 Evasion composite index (ECI) . . . . . . . . . . . . . . . . . . . . . . 30 10.5 Robustness under perturbation . . . . . . . . . . . . . . . . . . . . . . 31 10.6 Synthesis: contradiction as diagnostic evidence . . . . . . . . . . . . 31 11 Applications across domains 31 11.1 Legal and regulatory analysis . . . . . . . . . . . . . . . . . . . . . . . 31 11.2 AI fairness and algorithmic auditing . . . . . . . . . . . . . . . . . . . 32 11.3 Organisational governance and decision systems . . . . . . . . . . . . 32 11.4 Philosophical and epistemic inquiry . . . . . . . . . . . . . . . . . . . 32 11.5 Media systems (neutral case study) . . . . . . . . . . . . . . . . . . . 33 12 Analytical function 33 12.1 Overview................................... 33 12.2 Controlledframing ............................. 34 12.3 Response inevitability . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 12.4 Diagnostic inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 12.5 Documentaryvalue............................. 35 12.6 From logic to measurement . . . . . . . . . . . . . . . . . . . . . . . . 36 13 The core principle: asymmetry without necessity shifts the burden toward intent 36 13.1 Evidential interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 37 13.2 Boundaries and caveats . . . . . . . . . . . . . . . . . . . . . . . . . . 37 13.3 Multi-agent and recursive cases . . . . . . . . . . . . . . . . . . . . . . 38 14 Game-theoretic formalisation 38 14.1 Formaldefinition .............................. 39 14.2 Epistemic constant-sum . . . . . . . . . . . . . . . . . . . . . . . . . . 39 14.3 Payoffs and information . . . . . . . . . . . . . . . . . . . . . . . . . . 40 14.4 Equilibrium analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 14.5 Information-theoretic interpretation . . . . . . . . . . . . . . . . . . . 41 14.6 Comparative game-theoretic structure . . . . . . . . . . . . . . . . . . 41 14.7 Strategicdynamics ............................. 41 14.8 Boundedcoherence ............................ 42 14.9 Interpretive consequence . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 14.10 Ethicalguardrails .............................. 42 14.11 Prohibiteduses ............................... 43 15 Future research programme 44 15.1 Empirical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 15.2 Theoretical extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 15.3 Methodological development . . . . . . . . . . . . . . . . . . . . . . . 45 15.4 Applied implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 45 16 Summary of contributions 46 17 Conclusion 46 A Quick reference card 48 B Methodology for contradiction games 49 B.1 Purpose ................................... 49 B.2 Pre-registration (recommended) . . . . . . . . . . . . . . . . . . . . . 49 B.3 Inputsandartefacts ............................ 49 B.4 Construction (designing the trap) . . . . . . . . . . . . . . . . . . . . . 49 B.5 Deployment ................................. 50 B.6 Measurement and quantification . . . . . . . . . . . . . . . . . . . . . 50 B.6.1 Coherencecost........................... 50 B.6.2 Information gain . . . . . . . . . . . . . . . . . . . . . . . . . . 50 B.6.3 Meta-moves............................. 51 B.7 Analysis and outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 B.8 Reporting template (one page) . . . . . . . . . . . . . . . . . . . . . . 51 B.9 Ethics and safeguards . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 C Quick-start checklist (practitioner version) 52 Appendix C: Quick-Start Checklist (Practitioner Version) 52 D Glossary of specialist terms 52 Appendix D: Glossary of Specialist Terms 52 E Worked example (generic) 53 E.1 Contextandsetup.............................. 53 E.2 Branch outcomes and coherence costs . . . . . . . . . . . . . . . . . 53 E.3 Sensitivity and robustness . . . . . . . . . . . . . . . . . . . . . . . . . 54 E.4 Summary................................... 54 4 F Estimator pseudocode (RB–C, GI–C, SD–C) 54 F.1 RB–C: Rule-based coherence . . . . . . . . . . . . . . . . . . . . . . . 54 F.2 GI–C: Graph-informed coherence . . . . . . . . . . . . . . . . . . . . . 54 F.3 SD–C: Semantic-distance coherence . . . . . . . . . . . . . . . . . . . 55 F.4 Aggregation and normalisation . . . . . . . . . . . . . . . . . . . . . . 55 F.5 Sanitychecks ................................ 56 5 1 Visual abstract Commitments R,N,B Framed Proposition 𝑓(𝑃) Responses 𝐺/𝐷 Coherence Cost 𝐶(𝑟) Information Gain Δ𝐼 Figure 1: Visual abstract for The Contradiction Trap: symmetry-based framing converts contradiction into measurable coherence cost and information gain. 6 2 Notation and symbols Symbol Meaning Core Reasoning Structure (R–N–B–f(P)) R Rationale set: formal, stated commitments of the system. N Narrative set: contextual or situational justifications used to defend behaviour. B Behaviour set: observed behaviours (empirical events with evidential weight). 𝑓(𝑃) Framed proposition applying symmetric pressure across R, N & B. {𝐺,𝐷} Canonical response set: grant or deny the framed proposition. ΓMinimal unsatisfiable closure (diagnostic contradiction subset). Γ ⊆ 𝑅∪𝑁 The smallest commitment set producing inconsistency. C Clarity configuration mapping pathways to outcomes. 𝑃1,𝑃2Symmetric pathways whose outcomes jointly exhaust admissible responses. Reasoning Graphs and Inference 𝐺0Base reasoning graph: (𝑉,𝐸0)with 𝑉 = 𝑅∪𝑁. 𝐺+Augmented graph after adding framed-proposition constraints. 𝐸0Original justificatory edges. 𝐸𝑓Constraint edges induced by 𝑓(𝑃). ⊢IInference operator under reasoning system I. ⊥Logical inconsistency (used for closure detection). Contradiction Dynamics 𝐶(𝑟) Coherence cost incurred by response R ∈ {𝐺,𝐷}. Δ𝐼 Information gain: reduction of uncertainty induced by contradiction. ECI Evasion Composite Index: aggregate measure of meta-moves. MM Meta–move: reframing, delay, avoidance, or misdirection tactic. ECI(𝑡) Time-indexed evasion score during an interaction sequence. Epistemic Game-Theoretic Structure 𝐶-game Contradiction game: ⟨𝐴,𝐵,𝑆𝑅,𝑈𝐴,𝑈𝑆⟩. 𝑆𝑅Response set for the system under test (usually {𝐺,𝐷}). 𝑈𝐴Auditor utility: positive coherence cost. 𝑈𝑆System utility: negative coherence cost. 7 Symbol Meaning BR𝐵Best-response set; empty in contradiction traps. 𝑈𝑆(𝑟) < 0 No admissible response preserves coherence. Δ(𝑟) Epistemic update caused by response R. Coherence-Cost Estimators RB–C Rule-Based Coherence estimator. GI–C Graph-Informed Coherence estimator (topology-based). SD–C Semantic-Distance Coherence estimator (embedding/semantic). 𝐶RB,𝐶GI,𝐶SD Output coherence-cost values from the three estimators. 𝐶agg Aggregated coherence cost across estimators. N Normalisation operator for multi-model cost scaling. Information Dynamics Δ𝐻 Change in reasoning entropy due to contradiction exposure. Δ𝑅 Reduction of admissible explanations after contradiction. 𝐼post Posterior informativeness after observing a response. 𝐻norm Normalised entropy of the reasoning distribution. Structural Metrics and Diagnostics ECI Meta-evasion score (repeated for clarity). 𝑑sem Semantic distance between narrative elements. 𝑑top Topological inconsistency distance in 𝐺+. 𝜌Structural sensitivity parameter for multi-method validation. 𝜔Weight assigned to estimator output in 𝐶agg. Quantification and Measurement 𝔼[𝐶] Expected coherence cost over responses. Var(𝐶) Variance of coherence-cost estimator outputs. 𝑍Normalised contradiction score across branches. Θaudit Audit temperature: measure of adversarial pressure. Simulation and Experimental Parameters 𝑇Number of rounds or interaction steps. 𝑚Number of estimator models used. N Dimensionality of semantic/graph representation. 𝜎Noise applied during semantic-distance estimation. 𝑘Meta-move sampling depth. Sets, Operators, and Miscellanea 8 Symbol Meaning 𝑉Vertex set of reasoning graph. 𝐸Edge set (dependencies). ∘Function composition (e.g., narrative chaining). 𝜕Γ Boundary of a minimal unsatisfiable closure. 𝜒Closure characteristic: 1if inconsistent, 0if consistent. 3 Introduction Contradiction has long served as the philosopher’s stress test of truth. From the Socratic elenchus to Aristotle’s law of non-contradiction and the modern reductio ad absurdum (Socratic method: (Benson, 2021); reductio: (Groarke, 2023)), contradiction has functioned as philosophy’s diagnostic heartbeat: when a position collapses in on itself, it radiates its own refutation. Yet, while classical logic isolates contradiction within propositions, organisational and institutional reasoning often conceals contradiction within systems. This paper formalises a method for locating those contradictions, not in abstract syntax, but in an applied reasoning framework – a procedural trap that converts rhetoric into data (R. J. Aumann & Brandenburger, 1995; Floridi, 2011; Skyrms, 2010). The contradiction trap extends logic from proof to performance. By structuring a question where every available answer contradicts a distinct part of the respondent’s declared logic, it transforms a qualitative dispute into an epistemic experiment. In doing so, it reframes contradiction as evidence: a measurable, reproducible signal of bias or concealed motive. This situates logical analysis within practical domains such as governance, ethics, and algorithmic accountability (Ananny & Crawford, 2018; Barocas et al., 2019; Binns, 2018), providing a portable tool for interrogating systems that claim neutrality yet behave with asymmetry. Glossary note. Key technical terms used throughout this paper include: coherence cost (the measurable strain within a reasoning system when commitments conflict), epistemic game (a structured interaction in which agents’ beliefs depend on one another’s reasoning consistency), and meta-evasion score (a behavioural index capturing secondary avoidance tactics such as reframing or delay). A complete glossary is provided in Appendix D. While the contradiction trap introduces a novel formal and epistemic framework, it emerges from a broader lineage of research in dialectical reasoning, epistemic game 9 Granting the proposition preserves one set of commitments but breaks another; denying it does the inverse. The contradiction therefore exposes the system’s operative motive. 6.1 Formal definition We model a contradiction trap as the structured tuple 𝑇 = ⟨𝑅,𝑁,𝐵,𝑓(𝑃)⟩ (6.1) (cf. (Benthem, 2001; Woods & Walton, 1982)), where: •𝑅is the Rationale Set: formal principles, policies, or commitments constituting the system’s declared logic. •𝑁is the Narrative Set: contextual explanations or situational claims used to defend asymmetries in prior decisions. •𝐵is the Behaviour Set: empirically observed actions that carry evidential weight independent of narratives. •𝑓(𝑃)is a Framed Proposition: a question engineered to apply symmetric pressure across 𝑅,𝑁, and 𝐵, revealing latent asymmetry via forced inconsistency. In the canonical case, 𝑓(𝑃) elicits responses {𝐺,𝐷} (grant, deny). A continuous relaxation, 𝑓(𝑃) ∶ ℝ → [0,1], (6.2) permits graded or probabilistic answers, allowing coherence to be measured on a continuous scale rather than mapped to discrete branches. The defining property of a contradiction trap is simultaneous unsatisfiability: 𝑅∪𝐵∪{𝑓(𝑃)} ⊢I⊥(6.3) and 𝑁 ∪𝐵∪{¬𝑓(𝑃)} ⊢I⊥, (6.4) so that no consistent closure exists under either branch. Contradiction thereby functions as an experiment: two symmetric claims, one underlying logic, and a measurable signal of bias when the system cannot satisfy both. 16 Interpretation. Contradiction is not accidental but structurally inevitable. The proposition 𝑓(𝑃)imposes constraints on the combined reasoning graph of 𝑅,𝑁, and 𝐵, yielding a minimal unsatisfiable closure: the smallest jointly incompatible subset of commitments under an inference system I (cf. (Liffiton & Sakallah, 2008; Reiter, 1987)). This closure is the system’s diagnostic fingerprint—the point at which its declared rationale, protective narrative, and observable behaviour can no longer be jointly maintained. 6.2 The generic R–N–B–f(P) framework We represent the reasoning architecture as a directed acyclic graph 𝐺0= (𝑉,𝐸0), (6.5) with vertex set 𝑉 = 𝑅∪𝑁 ∪𝐵, (6.6) where edges in 𝐸0encode justificatory or causal dependencies. The framed proposition induces a constraint set 𝐸𝑓, producing the augmented graph 𝐺+= (𝑉,𝐸0∪𝐸𝑓). (6.7) A contradiction arises when 𝐺+contains a minimal unsatisfiable closure: ∃Γ ⊆ 𝑉 ∶ Γ ⊢I⊥(6.8) and ∀ Γ′⊊ Γ, Γ′⊬I⊥. (6.9) Graph convention. Contradiction cycles refer to minimal unsatisfiable closures in 𝐺+, not to structural cycles in the baseline DAG 𝐺0. The trap introduces inconsistency; it does not require any pre-existing flaw. 6.3 The core property The fundamental property of a contradiction trap is that all admissible responses incur positive coherence cost: 𝐶(𝑟) > 0 (6.10) 17 for every 𝑟 ∈ {𝐺,𝐷}. (6.11) This yields a strictly competitive epistemic game: 𝐺𝐸= ⟨𝐴,𝐵,𝑆𝑅,𝑈𝐴,𝑈𝑆⟩, (6.12) with utilities 𝑈𝑆(𝑟) = −𝐶(𝑟), 𝑈𝐴(𝑟) = 𝐶(𝑟), (6.13) and 𝑆𝑅= {𝐺,𝐷}. Let 𝑐𝑟denote the coherence configuration induced by response 𝑟. Since 𝐶(𝑟) > 0for all responses, max 𝑟∈𝑆𝑅𝑈𝑆(𝑟) < 0, (6.14) BR𝐵= ∅. (6.15) No response is coherence-preserving; the system has no Nash equilibrium (cf. (Dufwenberg & Lindén, 1996)). Contradiction becomes the unavoidable terminal state of play—a diagnostic signature of structural asymmetry rather than a logical failure. 7 Origins and distinction The contradiction trap inherits its conceptual lineage from classical dialectic and modern epistemic audit models. Woods and Walton’s cumulative dialectical games (Woods & Walton, 1982), Veraksa’s structural–dialectical psychology (N. Veraksa et al., 2013), and Benthem’s dynamic epistemic logic (Benthem, 2001) each highlight contradiction as a generative epistemic act. Contemporary audit frameworks extend this lineage. Mökander’s ethics-based audits (Mökander, 2023) and Buhmann et al.’s institutional accountability mechanisms (Buhmann et al., 2020) treat contradiction as a condition of transparency. Yang et al. (K. Yang & Kudenko, 2023; Y. T. Yang et al., 2025) use Stackelberg-style epistemic games to formalise trade-offs between privacy and accountability – precisely the structure operationalised by coherence cost. Distinction from classical forms. Reductio ad absurdum reveals the falsity of a proposition by deriving contradiction from assuming it true. The contradiction trap differs by operating at the systemic level: it cross-tests an entire network of commitments by deploying a proposition whose affirmation and denial contradict different elements of the 18 system’s rationale. Socratic elenchus cross-examines beliefs to induce aporia. The trap adopts the structure but not the moral purpose: its aim is evidential exposure, not intellectual humility. It transforms dialectical pressure into a measurable diagnostic of structural bias. Table 3: Distinction between classical logical methods and the Contradiction Trap. Aspect Reductio ad Absurdum Socratic Elenchus Contradiction Trap Primary Goal Prove proposition false Induce self-knowledge Expose bias or motive. Scope Single proposition Speaker’s beliefs System of commitments. Method Derive contradiction Questioning dialogue Forced binary choice exposing inconsistency. Adversariality Non-adversarial Non-adversarial Explicitly adversarial. Evidence Type Logical proof Qualitative insight Measurable contradiction. Measurability Binary (valid/invalid) Qualitative Quantitative (𝐶,Δ𝐼). Applications Mathematics, logic Philosophy, education Law, AI auditing, governance. 8 Coherence cost estimation methods: selection and validation This section outlines practical methods for estimating the coherence cost 𝐶, allowing practitioners to choose between rule-based, graph-informed, and semantic estimators depending on data structure, interpretability constraints, and computational resources. Each method quantifies the internal strain a reasoning system exhibits when its commitments conflict. 19 8.1 Comparative framework Three principal estimation methods are presented below, each offering distinct tradeoffs between interpretability, scalability, and computational complexity. Their comparative properties are summarised in Table 4. Table 4: Comparative properties of coherence-cost estimators (transposed view). Property RB-C (Rule-Based) GI-C (Graph-Informed) SD-C (Semantic Distance) Inputs Explicit rule sets or commitments Dependency or causal graph Embedding vectors or text corpora Complexity 𝑂(𝑛) 𝑂(𝑛log 𝑛) 𝑂(𝑛2) Interpretability High Medium Low Transparency High Medium Low Implementation Effort Low Moderate High Weighting Support Manual Automatic Implicit Typical Domain Legal / Policy Governance / Decision Systems NLP / Model Auditing Note: SD-C methods incur high upfront training cost but low inference cost once embeddings are established. (1) Rule-based coherence (RB-C) RB-C operates on explicit commitments expressed in a rule language R, identifying contradictions as minimal violation sets (D. M. Gabbay & Guenthner, 2003; Gärdenfors, 1988; Meyer & Wieringa, 1993; Prentzas & Hatzilygeroudis, 2012). The logical substrate may be propositional, deontic, or modal depending on the domain. 𝐶RB(𝑐𝑟) = min􏿺|𝑆| ∶ 𝑆 ⊆ (𝑅∪𝑁), (𝑅∪𝑁)∖𝑆is consistent under I􏿽.(8.1) Complexity: 𝑂(𝑛). Highly interpretable and reproducible; ideal for regulatory, contractual, or policy corpora. 20 (2) Graph-informed coherence (GI-C) GI-C models commitments as a directed graph 𝐺 = (𝑉,𝐸) (8.2) 𝑉 = 𝑅∪𝑁 (8.3) and quantifies contradiction as the size of the minimal hitting set required to restore consistency (Dung, 1995; Hunter, 2008; Pearl, 2009b; Reiter, 1987): 𝐶GI(𝑐𝑟) = |HitSetmin(𝑟)| (8.4) Complexity: 𝑂(𝑛log 𝑛). GI-C aligns with conflict-set detection and model-based diagnosis in classical AI. (3) Semantic distance coherence (SD-C) SD-C estimates latent contradiction in unstructured or natural-language corpora. Each proposition 𝑞 ∈ 𝑄𝑟is embedded as a vector  𝑣and compared to its coherence-preserving projection  𝑣′(J. Li et al., 2016; MacKay, 2003; Reimers & Gurevych, 2019): 𝐶SD(𝑐𝑟) = 􏾜 𝑞∈𝑄𝑟􏿴1−−cos( 𝑣, 𝑣′)􏿷(8.5) Complexity: 𝑂(𝑛2). KL divergence or Wasserstein distance may be used where embeddings encode stance or implication. Interpretation. SD-C is well-suited to NLP-based audits, latent contradiction detection, and model interpretability contexts. 8.2 Scaling and performance Approximate computational scaling: RB-C: 𝑂(𝑛), GI-C: 𝑂(𝑛log 𝑛), SD-C: 𝑂(𝑛2)(8.6) RB-C and GI-C support live or iterative audits; SD-C is better suited to retrospective, high-fidelity analysis. Practical scaling note. GI-C uses graph operations that become costly on dense, high-degree structures; use sparsification or community-aware subgraphs for > 104nodes. SD-C scales with the em21 bedding index; approximate nearest-neighbour search or minibatch scoring keeps large corpora tractable. RB-C complexity is driven by rule expansion-cache clause normal forms and prune dominated rules. 8.3 Null model and significance testing To assess significance, randomise labels within symmetric inputs 𝑋(permutation test) to obtain bootstrap samples (P. I. Good, 2005): C0= {𝐶(𝑏) 0}𝐵 𝑏=1 (8.7) Normalise using: 𝑧 = 𝐶(𝑟)−−𝜇0 𝜎0, 𝐴 = |𝐶𝐺−−𝐶𝐷|(8.8) Flag incoherence when A ≥ 𝜏𝐴and 𝑧 ≥ 𝜏𝑧. 8.4 Method selection guidance Estimator choice should follow data structure: •RB-C: explicit rules or commitments. •GI-C: interdependent or hierarchical reasoning. •SD-C: semantic drift, narrative contradiction, unstructured domains. Agreement between methods indicates coherence: |𝐶1−−𝐶2| < 𝜖 ⇒ coherent (8.9) Divergence indicates epistemic instability. Choosing a coherence-cost estimator. Use GI–C when you have a dependable inference or dependency graph (edge set 𝐸with traceable provenance) and you want topology-aware contradictions; prefer SD–C when narrative artefacts (documents, transcripts) dominate and high-quality embeddings are available; use RB–C when policy logic or rulebooks provide crisp, auditable clauses. When in doubt, run all three and aggregate via a Z-scored mean: 𝐶agg =􏾜 𝑘𝜔𝑘𝑧𝑘,􏾜 𝑘𝜔𝑘= 1 (8.10) with equal weights by default. Always include the sanity checks in App. F.5 and report estimator agreement/disagreement explicitly. 22 8.5 Multi-method validation A multi-method validation pipeline is recommended: 1. Extract R and N. 2. Build 𝐺0and check edge semantics. 3. Verify embedding fidelity for SD-C. 4. Investigate unexplained variance across estimators. Heuristic constructor. 1. Extract commitments. 2. Construct 𝐺0. 3. Identify symmetric locus 𝑋. 4. Synthesise 𝑓(𝑃). 5. Validate 𝐶(𝑟) > 0under 𝐺and 𝐷. Summary. RB-C, GI-C, and SD-C form the quantitative backbone of contradiction games, converting qualitative disagreement into measurable epistemic strain. The next section formalises the protocol for executing a complete Contradiction Game. 9 Methodology for contradiction games 9.1 Purpose Contradiction games test whether a system’s stated rationale R and its narrative justifications N remain jointly coherent when subjected to a symmetric stressor. Every admissible response incurs positive coherence cost, enabling structural bias to be expressed as a measurable epistemic outcome (Baltag & Smets, 2008; Benthem, 2001; Pearl, 2009a). 9.2 Inputs and artefacts •Rationale set R: written policies, rules, or formal commitments alongside N and B, (Besnard & Hunter, 2008; D. Gabbay & Woods, 2003). •Narrative set N: justificatory explanations alongside R and B. 23 •Behaviour set B: observed actions forming empirical commitments alongside R and N. (Prentzas & Hatzilygeroudis, 2012). •Symmetric locus 𝑋: cases where neutrality implies identical treatment (Binns, 2018; Hardt et al., 2016; Sen, 1969). •Framing operator 𝑓: constructs the test proposition 𝑃using epistemic or game-theoretic symmetry conditions (Fang et al., 2015; Prakken & Sartor, 2015; K. Yang & Kudenko, 2023). 9.3 Construction of the trap 1. Map commitments. Extract R = {𝑟𝑖}, N = {𝑛𝑗}and B = {𝑏𝑘}; build a dependency graph 𝐺0= (R∪N∪ B,𝐸0)following standard methods for structured commitments (Besnard & Hunter, 2008; Timmer et al., 2017). 2. Identify symmetric pressure. Choose 𝑋such that neutrality implies coherence under both outcomes (Hardt et al., 2016). 3. Define 𝑓(𝑃).Frame a binary proposition where: 𝐺 ∶ preserves 𝑅and contradicts 𝑁, 𝐷 ∶ preserves 𝑁and contradicts 𝑅, following adversarial dialogue-game design (Benthem, 2001; Prakken & Sartor, 2015). 4. Specify epistemic payoffs. Player utilities are defined by coherence cost, adopting the logic of strictly competitive epistemic games (Fang et al., 2015): 𝑈𝐴(𝑟) = 𝐶(𝑟) (9.1) 𝑈𝑆(𝑟) = −𝐶(𝑟) (9.2) 𝐶(𝑟) > 0 (9.3) No best response exists for 𝐵, yielding no equilibrium. 9.4 Measurement and quantification 9.4.1 Coherence cost We adopt four compatible estimators, grounded in logic, graph theory, semantics, and model checking. (1) Rule-based (RB-C) 𝐶RB(𝑐𝑟) = min{|𝑆| ∶ 𝑆 ⊆ 𝑅∪𝑁,(𝑅∪𝑁)∖𝑆is consistent }, (9.4) building on deontic and rule-based inference structures (Meyer & Wieringa, 1993; Prentzas & Hatzilygeroudis, 2012). 24 (2) Graph-informed (GI-C) 𝐶GI(𝑐𝑟) = |HitSetmin(𝑟)|, (9.5) HitSetmin(𝑟) = min{𝑆 ⊆ 𝑉 ∶ 𝐺+∖𝑆is acyclic and consistent}, (9.6) aligned with conflict-set detection in argumentation frameworks and causal reasoning (Dung, 1995; Pearl, 2009a). (3) Semantic distance (SD-C) 𝐶SD(𝑐𝑟) = 􏾜 𝑞∈𝑄𝑟􏿴1−cos( 𝑞, 𝑞′)􏿷,(9.7) where  𝑞′is the projection of  𝑞onto the admissible closure K𝑟(J. Li et al., 2016; Reimers & Gurevych, 2019). (4) Model-checking penalty (MC-C) Minimal number of edits or constraint removals required to restore satisfiability in a formal model (Baier & Katoen, 2008; Clarke et al., 1999). Normalisation. When combining estimators: 𝐶∗(𝑐𝑟) = 1 𝑚 𝑚 􏾜 𝑗=1 𝐶𝑗(𝑐𝑟) ∈ [0,1]. (9.8) 9.4.2 Accumulation models Given a reasoning chain R𝑟= {𝑞1,…,𝑞𝑛}: 𝐶Σ(𝑐𝑟) = 𝑛 􏾜 𝑖=1 𝜔𝑖𝛿(𝑞𝑖), (9.9) 𝐶max(𝑐𝑟) = max 𝑖𝛿(𝑞𝑖), (9.10) drawing on classical distributed vs. dominant-failure semantics in safety analysis (Leveson, 1995; Reason, 1990; Varshney & Alemzadeh, 2017). Interpretation. 𝐶Σcaptures distributed incoherence; 𝐶max isolates dominant failures. 25 11.2 AI fairness and algorithmic auditing In machine-learning systems, contradiction traps provide a structured audit for hidden bias in model logic or deployment policy. They translate fairness claims into falsifiable tests of coherence. •Fairness Claim Verification: Test whether an algorithm’s fairness metric contradicts the stated purpose of the system across symmetric demographic inputs (Binns, 2018). •Ethical Model Evaluation: Compare internal model weights with declared governance principles to expose representational imbalance (Mitchell et al., 2021; Zhou, 2022). •Automated Auditing Pipelines: Integrate contradiction tests into explainability or post-hoc biasdetection workflows (Barocas et al., 2017; Selbst et al., 2019). As Binns (Binns, 2018) notes, fairness metrics often conceal embedded political choices. Contradiction audits render these choices empirically testable, converting normative tension into measurable epistemic strain. This aligns with Zhou et al.’s argument (Zhou, 2022) that accountability must bridge human rationale and algorithmic logic. 11.3 Organisational governance and decision systems Within institutions, contradiction traps operate as meta-audits of decision processes. They surface concealed motive structures by forcing systems to choose between mutually exclusive claims, ensuring that either branch exposes inconsistency (Brunsson, 2002; J. G. March & Olsen, 1989). •Policy Consistency Audits: Test whether internal rationales align with outward-facing narratives. •Governance Integrity Testing: Detect selectively applied criteria or asymmetric justification patterns (Argyris, 1991; Ashforth & Anand, 2008). •Restructure Diagnostics: Identify when organisational “reforms” serve narrative control rather than operational necessity. 11.4 Philosophical and epistemic inquiry Contradiction traps generalise classical dialectical tools – the elenchus and reductio – to institutional and algorithmic belief systems (Hintikka, 2004; Plato, 1997). They provide a bounded test for epistemic self-consistency (Audi, 2003; D. C. Dennett, 2013; Williamson, 2000). •Reasoning Integrity Analysis: Evaluate whether belief systems can withstand self-referential contradiction without collapse. •Motive Reconstruction: Use asymmetry between R and N to infer latent priority structures – the gap between what a system says and what it does. •Concealed-Motive Detection: Treat contradiction-induced breakdowns as empirical evidence of distortion, selective justification, or intentional opacity. 32 Across these domains, contradiction functions not merely as an analytical device but as an instrument of accountability. When a rationale collapses against its own narrative, the inconsistency itself becomes evidence. 11.5 Media systems (neutral case study) To illustrate a non-political application, consider a media organisation that publishes its editorial standards (R) and maintains internal content-selection logs (N). A symmetric locus 𝑋is constructed by selecting pairs of stories matched on topic, newsworthiness, and timing. The framed proposition is the neutral query: 𝑓(𝑃) = “Should both stories receive equal likelihood of publication under the stated standards?” (11.1) We compute: 𝐶(𝑐𝐺)(11.2) 𝐶(𝑐𝐷)(11.3) 𝐴 = |𝐶(𝑐𝐺)−𝐶(𝑐𝐷)| (11.4) 𝑧-scores under a pre-registered null (11.5) Interpretation language is intentionally avoided. Results are reported purely in terms of 𝐶, A, and statistical deviation from the null model. This establishes contradiction traps as an impartial method for evaluating editorial consistency without invoking ideological labels (Entman, 1993; Gillespie, 2018; Harcup & O’Neill, 2017). 12 Analytical function The contradiction trap functions as an epistemic instrument: it converts a subjective dispute into objective, auditable evidence. Its analytical value lies not in persuasion but in transformation – turning narrative claims into measurable structure. Unlike paraconsistent tolerance (Priest, 2006), the trap does not regard contradiction as a stable coexistence of opposing truths. Instead, it treats contradiction as diagnostic evidence: a measurable strain on coherence whose resolution is inferential, not optional (Brunsson, 2003). 12.1 Overview A contradiction trap performs four linked analytical operations: 1. Controlled Framing – constructs a bounded reasoning space in which all admissible outcomes are mutually exclusive and self-refuting (J. G. March, 1984). 33 Figure 2: Distribution of asymmetry 𝐴 = |𝐶𝐺−−𝐶𝐷|across simulated contradiction traps. The right-skewed form indicates systemic rather than random divergence. 2. Response Inevitability – ensures that any response supplies information about the responder’s internal logic (R. J. Aumann, 1976; Hintikka, 1962; Williamson, 2000). 3. Diagnostic Inference – treats inconsistency as a signal of hidden constraint or motive rather than a defect in argumentation (Argyris, 1991; Ashforth & Anand, 2008; Tetlock, 2006). 4. Documentary Value – converts dialogue into contemporaneous, reproducible evidence (Audi, 2003; Entman, 1993). Together, these operations transform contradiction from failure into data. 12.2 Controlled framing The framing operator 𝑓maps a proposition 𝑃into a constrained reasoning space: 𝑓 ∶ 𝑃 ↦ {𝐺,𝐷}, 𝐺∩𝐷 = ∅ (12.1) where 𝐺and 𝐷represent mutually exclusive outcomes, each contradicting a distinct subset of commitments. Framing is successful when the augmented closures satisfy: 𝑅,𝑓(𝑃) ⊢ ⊥ (12.2) and 𝑁,¬𝑓(𝑃) ⊢ ⊥ (12.3) This transforms a discursive dispute into a reproducible logical experiment. Framing is the calibration step: it guarantees that any observed contradiction is structural, not accidental (Brunsson, 2003). 34 12.3 Response inevitability In a correctly constructed trap, the responder must select among inconsistent completions of its own logic. Let 𝑆𝑅= {𝐺,𝐷} be the response set. Since 𝐶(𝑐𝐺),𝐶(𝑐𝐷)>0, the probability of an uncontradicted path is zero. Thus every interaction yields measurable evidence: 𝐸[Information Gain] = −􏾜 𝑟∈𝑆𝑅𝑃(𝑟)log2𝑃(𝑟) (12.4) supported on two points (R. J. Aumann, 1976). Refusal, silence, or procedural delay is classified via the meta-evasion metric 𝑀(Section 10.2); metamoves are analytically equivalent to an asymmetric response. Response inevitability guarantees that the system cannot avoid self-description: its behaviour, not its claim, becomes the evidence (Williamson, 2000). 12.4 Diagnostic inference Once contradiction is witnessed, analysis shifts from logic to motive. Coherence cost 𝐶(𝑟) quantifies the internal tension revealed by response 𝑟, while relative magnitudes 𝐶𝐺and 𝐶𝐷encode preferential structure. Inference proceeds through evidential proportionality: 𝑃(𝐼 ∣ 𝐴) ∝ Λ(𝐴)𝑃(𝐼) (12.5) where 𝐴 = |𝐶𝐺−𝐶𝐷|and Λ(𝐴)is the Bayes factor (see Section 13) (Bovens & Hartmann, 2003; Hartmann & Bovens, 2005). Epistemic directionality. Contradiction analysis reverses the traditional burden of proof: rather than requiring direct evidence of motive, it infers motive probabilistically from reasoning failure. In effect, the trap converts ethical opacity into evidential asymmetry (Ashforth & Anand, 2008; Tetlock, 2006). 12.5 Documentary value Contradiction games generate contemporaneous artefacts – records of framing, responses, timestamps, and coherence metrics – that form an evidential ledger: D= {(𝑓,𝑃,𝑟,𝐶(𝑟),𝑀)} (12.6) If 𝐶(𝑟) > 0, contradiction is captured as an auditable event (Audi, 2003). Because the protocol is procedural and repeatable, identical premises under symmetric conditions must yield identical contradiction signatures. This provides a reproducibility criterion absent from rhetorical or testimonial evidence (Entman, 1993). 35 12.6 From logic to measurement The analytical function of the contradiction trap bridges propositional logic and empirical method. It does not assert moral authority; it enforces epistemic transparency. By transforming qualitative disagreement into quantitative signal, it establishes falsifiability where previously there was assertion alone (Hartmann & Bovens, 2005). In this sense, the contradiction trap plays for ethical reasoning the role that the controlled experiment plays in empirical science: a structured environment in which reality discloses itself through inconsistency. Summary. The metrics 𝐶and 𝐴derived here form the mathematical inputs to Section 13, where asymmetry without necessity becomes a formal basis for probabilistic inference of intent. The analytical function therefore sits at the hinge between construction and inference: it converts contradiction into data, and data into evidence. 13 The core principle: asymmetry without necessity shifts the burden toward intent The contradiction trap rests on a simple evidential claim: when a system deviates from its declared principles without necessity, that deviation increases the posterior odds of selective intent. The core principle formalises this transition from structural inconsistency to probabilistic inference (Bovens & Hartmann, 2003; Fredman, 2011b; I. Good, 1985; Jaynes, 2003). Statement. Let A = |𝐶𝐺−−𝐶𝐷|denote the asymmetry in coherence cost under symmetric inputs 𝑋. If A > 0 and no external necessity E (legal constraint, resource limit, stochastic uncertainty) accounts for it, then A raises the posterior odds of intent (Barak, 2012a; Craig, 2012a; Kass & Raftery, 1995): Pr(𝐼 ∣ 𝐴) Pr(¬𝐼 ∣ 𝐴) 􏿋􏻰􏻰􏻰􏻰􏻰􏿌􏻰􏻰􏻰􏻰􏻰􏿍 posterior odds =Pr(𝐴 ∣ 𝐼) Pr(𝐴 ∣ ¬𝐼) 􏿋􏻰􏻰􏻰􏻰􏻰􏿌􏻰􏻰􏻰􏻰􏻰􏿍 Λ(𝐴) ⋅Pr(𝐼) Pr(¬𝐼) 􏿋􏻰􏻰􏿌􏻰􏻰􏿍 prior odds (13.1) Formal Bayesian framing. Let 𝐻0denote neutrality and 𝐻1motivated bias. Posterior elevation occurs precisely when (Bovens & Hartmann, 2003; I. Good, 1985) 𝑃(𝐻1∣ 𝐴) > 𝑃(𝐻1) ⟺ Λ(𝐴) = 𝑃(𝐴 ∣ 𝐻1) 𝑃(𝐴 ∣ 𝐻0)> 1 (13.2) Necessity test. External necessities form a set E. We first test the null hypothesis 𝐻0∶ 𝐴 ∈ E (13.3) 36 Rejection of 𝐻0licenses evidential inference: the asymmetry is not required by external constraints and must therefore be explained by internal choice (Barak, 2012a; Fredman, 2011b). Decision rule (Bayes factor). Define the Bayes factor Λ(𝐴) = Pr(𝐴 ∣ 𝐼)/Pr(𝐴 ∣ ¬𝐼) under the registered null model. A shift in burden occurs whenever Λ(𝐴) ≥ 𝜏, 𝜏 > 1 (13.4) e.g. 𝜏 = 3for “moderate” and 𝜏 = 10for “strong” evidential weight (Kass & Raftery, 1995). The rule is deliberately minimal: it does not diagnose intent, but obliges the system to supply a justification consistent with its own commitments. Remark. The heuristic “asymmetry without necessity implies intent’’ abbreviates the probabilistic claim: if A ∉E and A > 0, then Λ(𝐴) > 1. The odds shift, but inference remains probabilistic, not deductive. 13.1 Evidential interpretation The magnitude of A yields a graded evidential interpretation: •Small asymmetry (A≈ 0): Structural inconsistency; motive cannot be inferred. Contradiction arises from system design rather than agency (Simon, 1955). •Moderate asymmetry (A> 0 but bounded): Indicates implicit preference or unacknowledged contextual weighting. Suggests weakly motivated divergence (Brunsson, 2003; Tetlock, 2006). •Large asymmetry (A≫ 0): Signals deliberate prioritisation or concealed motive. The system reveals its values more clearly through inconsistency than through claim. This evidential gradient distinguishes cognitive limits, structural design, and strategic manipulation. Whereas paraconsistent logics permit contradictory propositions to coexist without collapse (Priest, 2006), the contradiction trap uses contradiction to test epistemic integrity: the aim is not to survive inconsistency but to diagnose its origin. 13.2 Boundaries and caveats The core principle applies within explicit epistemic limits: 1. Bounded Rationality. Asymmetry may reflect limited information or cognitive load (Kahneman, 2011); not all divergence is intentional. 2. Incomplete Mapping. If R or N are partially captured, observed asymmetry may arise from unmodelled commitments rather than bias (J. March & Olsen, 1984). 37 3. Meta-Game Costs. Anticipating interrogation may lead agents to distort commitments pre-emptively; the resulting asymmetry mixes bias with strategic evasion. 4. Multi-Agent Aggregation. Collective decisions aggregate divergent motives; asymmetry may reflect composition effects, not a unified intent (D. Dennett, 1987; List & Pettit, 2011). These caveats restrict scope without diminishing force. Properly applied, the principle distinguishes honest inconsistency from motivated contradiction and converts qualitative bias into quantitative inference. Clarification. Evidence of contradiction is diagnostic of structural inconsistency; it does not by itself impute malfeasance or intent. 13.3 Multi-agent and recursive cases When responses are delegated or recursively mirrored, coherence analysis decomposes by agent. Each actor inherits rationale R𝑖and narrative N𝑖; the aggregate contradiction is 𝐶agg =􏾜 𝑖𝑤𝑖𝐶𝑖, 𝑤𝑖≥ 0, 􏾜 𝑖𝑤𝑖= 1 (13.5) Delegation diffuses, but does not eliminate, accountability: contradiction propagates through weighted commitments (List & Pettit, 2011). Recursive belief formulation. Let 𝐵𝑖(𝐵𝑗(𝜑))denote agent 𝑖’s belief about agent 𝑗’s belief in 𝜑. Contradiction arises when (R. J. Aumann, 1976; Fagin et al., 1995; Hintikka, 1962) 𝐵𝑖(𝐵𝑗(𝜑))∧¬𝐵𝑗(𝜑) (13.6) under public declaration. Multi-layer conflicts produce recursive contradiction cascades, revealing unstable epistemic networks. Summary. The asymmetry principle supplies the probabilistic backbone of the contradiction game. Section 14 formalises this evidential rule within a game-theoretic framework, showing how posterior shifts map onto strategic loss functions. 14 Game-theoretic formalisation The contradiction trap can be cast as a one-move, strictly competitive epistemic game in which all available responses for the responder are losing strategies: each produces a negative payoff via positive coherence cost (R. Aumann, 1999; Brandenburger, 2007). Section 6 treated contradiction traps as applied dialectical instruments; here we formalise them within the vocabulary of game theory, showing that contradiction behaves as a forced-loss strategy inside a closed reasoning environment. Viewed through the lens of machine behaviour (Rahwan et al., 2019), contradiction games constitute behavioural falsification: agents disclose their internal priorities not by admission, but by necessity. 38 14.1 Formal definition Epistemic game theory models beliefs about beliefs (Brandenburger, 2007; Fagin et al., 1995). The contradiction trap defines a new subclass in which reasoning itself constitutes play and contradiction constitutes outcome. Definition 14.1 (Contradiction Game).A Contradiction Game is a two-player epistemic game 𝐺 = ⟨𝑃,𝑆,𝑈,𝐶⟩ (14.1) 𝑃 = {𝐴,𝐵}, 𝑆𝑅= {𝐺,𝐷} (14.2) with the following structure: 1. Player 𝐴(interrogator) applies a framing operator 𝑓to proposition 𝑃, selecting a scenario in which 𝐵’s commitments render {𝐺,𝐷} mutually exclusive with respect to its declared rationale and narrative. 2. Player 𝐵(responder) selects 𝑟 ∈ {𝐺,𝐷}. 3. Each response induces coherence cost 𝐶(𝑟) > 0, i.e. each response contradicts some part of 𝐵’s commitments. 4. Payoffs are epistemic: 𝑈𝐴(𝑟) = 𝐶(𝑟), 𝑈𝑆(𝑟) = −𝐶(𝑟) (14.3) The defining feature is ∀𝑟 ∈ 𝑆𝑅∶ 𝐶(𝑟) > 0 (14.4) so 𝐵has no contradiction-free option. This reverses the standard Aumann–Brandenburger paradigm in which shared belief conditions sustain equilibrium (R. Aumann, 1999): here, no epistemic state can restore equilibrium. 14.2 Epistemic constant-sum Material zero-sum games treat utility as consumption; contradiction games treat utility as information gain versus coherence loss. This parallels information-incentive models in signalling and behavioural audit games (Kreps & Wilson, 1982; Myerson, 1991). Definition 14.2 (Epistemic Constant-Sum).A Contradiction Game is epistemic constant-sum if there exist positive scaling constants 𝑎 > 0,𝑏 ∈ ℝsuch that 𝑈𝑆(𝑟) = 𝑎−−𝑈𝐴(𝑟)+𝑏 (14.5) This expresses epistemic complementarity: the interrogator’s evidential utility equals the responder’s coherence loss up to affine transformation. Proposition 14.3 (Non-Existence of Nash Equilibrium).If 𝐶(𝑟) > 0 for all 𝑟 ∈ {𝐺,𝐷}, then the game 𝐺 admits no pure Nash equilibrium. 39 Figure 3: Coherence-cost divergence for grant (𝐶𝐺) and deny (𝐶𝐷). Absence of intersection indicates the impossibility of equilibrium. Proof. Suppose (𝑓∗,𝑃∗;𝑟∗)is a Nash equilibrium. By definition, 𝑟∗∈ {𝐺,𝐷}. But for all 𝑟,𝑈𝑆(𝑟) = −𝐶(𝑟) < 0, so no 𝑟∗maximises 𝑈𝑆. Thus 𝐵has no best response, and mutual best-response fails (Osborne & Rubinstein, 1994). Therefore no pure equilibrium exists. 14.3 Payoffs and information Let 𝐼(𝑟) denote evidential information content (information gain interpretation following Shannon and Jaynes (Jaynes, 2003; Shannon, 1948)): 𝐼(𝑟) = log𝑃(𝐷 ∣ 𝑟) 𝑃(𝐷) (14.6) A generalised epistemic payoff is 𝑈𝑖(𝑟) = 𝛼𝐼(𝑟)−𝛽𝐶(𝑟), 𝛼,𝛽 > 0 (14.7) balancing information gain against contradiction cost. This casts contradiction traps as signal-to-cost games, consistent with behavioural inference models (Spence, 1973). Deterministic loss. Since 𝐶(𝑐𝐺),𝐶(𝑐𝐷) > 0, max 𝑟𝑈𝑆(𝑟) < 0, min 𝑟𝑈𝐴(𝑟) > 0 (14.8) Thus the responder faces a dominant-loss structure; mixing cannot remove loss, only obscure it. 40 14.4 Equilibrium analysis Classical equilibrium requires mutual best response; contradiction games preclude this by construction. No choice of 𝑟∗preserves coherence, so stability cannot be restored without abandoning prior commitments. Meta-strategies (delay, reframing, premise-attack) therefore become secondary signals of motive and feed into the meta-evasion score 𝑀(Section 10.2), consistent with behavioural audit theory (“cheap talk under pressure”) (Crawford, 1991). 14.5 Information-theoretic interpretation Each contradiction produces information gain Δ𝐼 = −log2𝑝(14.9) where 𝑝is the prior coherence probability. As 𝑝→0,Δ𝐼 diverges (Jaynes, 2003; Shannon, 1948): contradiction reveals motive as a limiting case. Utility view. 𝑈(𝑟) = 𝐼(𝑟)−𝐶(𝑟) (14.10) captures the trade-off: systems lose epistemic integrity as contradiction deepens but thereby provide increasing evidential value. In energetic terms, contradiction behaves like a local increase in potential — a gradient in informational space that drives systems toward coherence equilibrium. 14.6 Comparative game-theoretic structure •Prisoner’s Dilemma: Cooperation restores equilibrium; here, no cooperation restores coherence. •Chicken Game: Bluff may avert collision; in contradiction games, collision is guaranteed. •Matching Pennies: Binary and stochastic; contradiction games are binary and deterministic (Osborne & Rubinstein, 1994). •Signalling Games: Hidden types inferred through messages; here, types are inferred through logical failure (Spence, 1973). This motivates a new subclass: epistemic, deterministic, contradiction-revealing games - logic as play, contradiction as payoff. 14.7 Strategic dynamics Because all moves yield loss, rational responders adopt damage-limiting meta-moves-strategies consistent with bounded rationality and cognitive economisation (Kahneman, 2011; Simon, 1955): 41 A Quick reference card Key Symbols and Notation Symbol Meaning R Rationale set: formal commitments, rules, or declared principles. N Narrative set: contextual explanations or situational justifications. 𝑓(𝑃) Framed proposition designed to apply symmetric pressure to 𝑅and 𝑁. 𝐺Directed dependency graph linking rationale and narrative nodes. 𝐶(𝑟) Coherence cost for response 𝑟 ∈ {𝐺,𝐷}. 𝐶𝐺,𝐶𝐷Coherence costs under grant and deny responses. Δ𝐼 Information gain from contradiction: Δ𝐼 = −log2𝑝. 𝑝Prior probability of coherence under symmetric input. A Asymmetry magnitude: |𝐶𝐺−−𝐶𝐷|. 𝑈𝐴,𝑈𝑆Evidential payoff functions for interrogator (A) and responder (B). Ten-Step Construction Guide 1. Define the decision context and identify the system under analysis. 2. Extract the rationale set 𝑅(stated rules or commitments). 3. Extract the narrative set 𝑁(contextual justifications). 4. Construct a framed proposition 𝑓(𝑃)applying symmetric pressure to 𝑅and 𝑁. 5. Model dependencies as a graph 𝐺 = (𝑉,𝐸). 6. Compute 𝐶𝐺and 𝐶𝐷using RB–C, GI–C, or SD–C estimators. 7. Calculate asymmetry 𝐴 = |𝐶𝐺−𝐶𝐷|. 8. Assess whether asymmetry is necessary or intentional using the core principle. 9. Interpret results under ethical guardrails. 10. Produce a written record and provide an appeal path. This card summarises the essential workflow for contradiction-based audits across legal, organisational, and algorithmic systems. 48 B Methodology for contradiction games B.1 Purpose A reproducible protocol for constructing, running, and analysing a Contradiction Game. The aim is symmetry, transparency, and evidential integrity. B.2 Pre-registration (recommended) Before deployment, predefine: •Objective: The neutrality or fairness claim under evaluation. •Symmetric inputs: Conditions under which neutrality must hold. •Commitment map: The respondent’s stated rationale(s) and narrative(s). •Primary endpoints: Contradiction event; coherence cost 𝐶(𝑟); information gain Δ𝐼. •Stopping rule: Maximum iterations or time window. •Ethical constraints: Symmetry, transparency, non-coercion. B.3 Inputs and artefacts •Policy set 𝑅:Formal rules, statements, or documents. •Claim set 𝑁:Narrative justifications or contextual claims. •Symmetric cases 𝑋:Inputs where neutrality should be invariant. •Framing operator 𝑓:Exact wording of the proposition 𝑃. B.4 Construction (designing the trap) 1. Map commitments. Extract propositions 𝑅1,…,𝑅𝑚and 𝑁1,…,𝑁𝑘. Construct a dependency graph. 2. Find a symmetric pressure point. Choose 𝑋such that, under neutrality, both branches preserve coherence. 3. Define 𝑓(𝑃).Construct a proposition where: • Grant contradicts part of 𝑁; • Deny contradicts part of 𝑅. 4. Specify epistemic payoffs. 𝑈𝐴(𝑟) = 𝐶(𝑟),𝑈𝑆(𝑟) = −𝐶(𝑟). 49 B.5 Deployment 1. Issue the framed proposition 𝑓(𝑃)under symmetric conditions. 2. Archive 𝑅,𝑁 with timestamp and hash. 3. Record the response 𝑟 ∈ {𝐺,𝐷}verbatim. 4. Lock the log: append-only journal, UTC timestamps. B.6 Measurement and quantification B.6.1 Coherence cost Compatible estimators: •Rule-based (RB–C): 𝐶RB(𝑐𝑟) = min{|𝑆| ∶ 𝑆 ⊆ 𝑅∪𝑁, removing 𝑆restores consistency}(B.1) •Graph-informed (GI–C): Size of the minimal contradiction hitting set. •Semantic distance (SD–C): 𝐶SD(𝑐𝑟) = 􏾜 𝑞∈𝑄𝑟􏿴1−cos( 𝑞, 𝑞′)􏿷(B.2) •Model-checking (MC–C): Minimal logical edits restoring satisfiability. Report a normalised cost 𝐶⋆(𝑐𝑟) ∈ [0,1]when using multiple estimators. B.6.2 Information gain We denote information gain by Δ𝐼, following the entropy-based definition. Let 𝑝 = Pr(coherence ∣ 𝑋). Then: Δ𝐼 = −log2(𝑝) (B.3) For sequential traps (not necessarily independent): log Pr(𝐼 ∣ D𝑡) Pr(¬𝐼 ∣ D𝑡)=log Pr(𝐼) Pr(¬𝐼) +𝑡 􏾜 𝑖=1 log Λ𝑖, Λ𝑖=Pr(𝐴𝑖∣ 𝐼,D𝑖−1) Pr(𝐴𝑖∣ ¬𝐼,D𝑖−1)(B.4) 50 B.6.3 Meta-moves Record secondary behaviours: • delay or deferral; • reframing the question; • attacking the premise; • appeal to context or hierarchy. These form the meta-evasion score 𝑀. B.7 Analysis and outcomes •Primary outcome: Contradiction (𝐶(𝑟) > 0). •Effect size: Report 𝐶⋆(𝑐𝑟)and Δ𝐼. •Robustness: Opposite branch also yields contradiction. •Sensitivity: Minor phrasing changes do not restore coherence. B.8 Reporting template (one page) • Context and neutrality claim • Symmetric inputs 𝑋 • Proposition 𝑓(𝑃)(verbatim) • Prior commitments (IDs, timestamps) • Response (verbatim) •𝐶(𝑟)and 𝐶⋆(𝑐𝑟) •Δ𝐼 and prior 𝑝 • Meta-evasion score 𝑀 • Ethical statement • Repository link (logs + hashes) B.9 Ethics and safeguards •Symmetry: Identical conditions for all comparators. •Non-coercion: No forced or time-pressured responses. •Appeal path: Respondents may provide contextual clarification. 51 C Quick-start checklist (practitioner version) 1. Record the respondent’s rationale(s) and narrative(s). 2. Identify symmetric inputs 𝑋. 3. Construct 𝑓(𝑃)so that Grant and Deny contradict different commitments. 4. Pre-register 𝐶(𝑟),Δ𝐼, and ethical guardrails. 5. Issue 𝑓(𝑃); log all artefacts. 6. Compute 𝐶(𝑟)via RB–C or GI–C. 7. Compute Δ𝐼 = −log2(𝑝). 8. Record meta-moves (delay, reframing, attack). 9. Produce a one-page report. 10. Validate counterfactual symmetry: opposite branch also contradicts. Practitioner Quick-Start Input: Extract stated rationale (𝑅)and narrative defence (𝑁). Trap: Construct a clarity configuration C{𝑃1,𝑃2}targeting their joint inconsistency. Measure: Compute coherence cost 𝐶; compare to null-model band. Interpret: 𝐶above threshold = evidential contradiction. D Glossary of specialist terms Key Terms and Definitions Term Definition Coherence Cost Quantitative measure of logical strain incurred when commitments cannot be jointly satisfied. Contradiction Trap A framed scenario in which every permissible response contradicts a different part of the responder’s stated logic. Rationale (R) Formal principles, rules, and commitments accompanying or qualifying N and B. Narrative (N) Contextual justifications accompanying or qualifying R and B. Behaviour set (B) Empirically observed actions carrying evidential weight accompanying or qualifying R and N. Framed Proposition 𝑓(𝑃) A proposition designed to apply symmetric pressure across 𝑅and 𝑁. Coherence Estimators RB–C, GI–C, SD–C estimators for computing coherence cost. 52 Term Definition (continued) Epistemic Game Interaction structured by higher-order beliefs about reasoning consistency. Information Gain Δ𝐼 Bits of information obtained when contradiction is observed: Δ𝐼 = −log2(𝑝). Asymmetry 𝐴Absolute difference |𝐶𝐺−𝐶𝐷|; large 𝐴implies selective motive. Meta-Evasion Score 𝑀Weighted index of evasive behaviours (delay, reframing, premise attack). Epistemic Instability Condition in which no consistent closure exists; contradiction is inevitable. Bounded Coherence Tolerance zone in which small contradictions do not trigger epistemic collapse. Ethical Guardrails Normative constraints (symmetry, transparency, proportionality). Prohibited Uses Weaponised uses such as coercion, entrapment, or deceptive framing. Epistemic Standing Credibility retained while sustaining internal coherence under symmetric challenge. E Worked example (generic) E.1 Context and setup We audit a hiring system that claims: (i) identical criteria for all candidates; (ii) merit-only selection; (iii) commitment to diverse outcomes. Two symmetric candidates 𝐴and 𝐵(matched CVs) are evaluated. •Rationale 𝑅 = {𝑟1,𝑟2}:𝑟1“identical criteria”; 𝑟2“merit-only”. •Narrative 𝑁 = {𝑛1,𝑛2}:𝑛1“system removes human bias”; 𝑛2“we maintain diverse outcomes”. •Symmetric input 𝑋: Matched CVs ⇒any difference must be justified by model internals, not identity. •Framed proposition 𝑓(𝑃): “Should candidates 𝐴and 𝐵receive identical assessment scores?” E.2 Branch outcomes and coherence costs Branch analysis and minimal contradiction removals (RB–C, GI–C). Response Effect on commitments; minimal removals Grant (𝐺) Preserves 𝑅(identical criteria; merit-only). Contradicts 𝑁if diversity is asserted to require score differentiation for matched CVs. Minimal removals: drop 𝑛1,𝑛2⇒ 𝐶RB(𝑐𝐺) = 2,𝐶GI(𝑐𝐺) = 2. Normalised 𝐶(𝑐𝐺) = 2 4= 0.5. 53 Response Effect on commitments; minimal removals (cont.) Deny (𝐷) Preserves 𝑁(diverse outcomes) but contradicts 𝑅(identical criteria; merit-only) given matched CVs. Minimal removals: drop 𝑟1,𝑟2⇒ 𝐶RB(𝑐𝐷) = 2,𝐶GI(𝑐𝐷) = 2. Normalised 𝐶(𝑐𝐷) = 2 4= 0.5. Asymmetry: 𝐴 = |𝐶𝐺−𝐶𝐷| = 0 (structural contradiction). Information gain: if 𝑝 = Pr(coherence ∣ 𝑋) = 0.3, then Δ𝐼 = −log2(0.3) ≈ 1.74bits. E.3 Sensitivity and robustness Minor paraphrases of 𝑓(𝑃) (e.g., “equal pass/fail?”, “equal interview score?”) preserve contradiction signatures. GI–C and RB–C agree; SD–C (if applied to policy text) shows elevated drift when “diverse outcomes” is used as a free-floating rationale. E.4 Summary Under symmetric inputs, either branch contradicts a distinct facet of the system’s claims; contradiction is diagnostic, not accidental. This is a textbook contradiction game with no equilibrium. F Estimator pseudocode (RB–C, GI–C, SD–C) F.1 RB–C: Rule-based coherence Goal: Minimal removals from 𝑅∪𝑁that restore consistency under I. Notes: (i) Use hitting set or MaxSAT/MUS solvers for scalability. (ii) Report the size (cost) and optionally one witness set 𝑇⋆. F.2 GI–C: Graph-informed coherence Goal: Minimal hitting set of nodes/edges whose removal makes 𝐺+= (𝑉,𝐸∪𝐸𝑓)acyclic and semantically consistent. Notes: (i) In practice, approximate MinHittingSet via greedy set cover; (ii) When contradictions are labelbased (e.g., 𝐴→𝐵and 𝐴 → ¬𝐵), let 𝑆𝑖mark the smallest edit (drop 𝐴or a conflicting implication). (iii) Complexity typically 𝑂(𝑛log 𝑛)with sparse graphs and efficient cycle detection. 54 Algorithm 1 RB-C (Rule-Based Coherence Cost) Input: Commitments 𝑆 = 𝑅∪𝑁; inference system I; branch response 𝑟 Output: 𝐶RB(𝑐𝑟) ∈ ℕ, minimal removal size 1: 𝑆𝑟←ApplyBranch(𝑆,𝑟) ▷Add/activate branch-specific literals 2: if IsConsistent(𝑆𝑟,I)then return 0 3: end if 4: for 𝑘 = 1to |𝑆𝑟|do 5: for all 𝑇 ⊆ 𝑆𝑟with |𝑇| = 𝑘 do 6: if IsConsistent(𝑆𝑟∖𝑇,I)then 7: return 𝑘 8: end if 9: end for 10: end for 11: return |𝑆𝑟|▷Worst case Algorithm 2 GI-C (Graph-Informed Coherence Cost) Input: DAG 𝐺0= (𝑉,𝐸0); branch edges 𝐸𝑓(𝑟); consistency oracle O Output: 𝐶GI(𝑐𝑟) ∈ ℕ 1: 𝐺+← (𝑉,𝐸0∪𝐸𝑓(𝑟)) 2: C←FindContradictionCycles(𝐺+) ▷semantic/structural 3: if C= ∅then return 0 4: end if 5: Build set family S = {𝑆1,…,𝑆𝑚}where each 𝑆𝑖are vertices/edges whose removal breaks cycle 𝑖and restores O 6: 𝐻⋆←MinHittingSet(S) 7: return |𝐻⋆| F.3 SD–C: Semantic-distance coherence Goal: Quantify semantic drift from each proposition 𝑞to its coherence-preserving projection 𝑞′within admissible closure K. Notes: (i) E(K)can be the set of embeddings for the minimally consistent rewrite of 𝑅∪𝑁under branch 𝑟; (ii) Use FAISS/ANN for fast nearest-neighbour search; (iii) Normalise to 𝐶⋆∈ [0,1] via min–max or quantile scaling for cross-estimator comparison. F.4 Aggregation and normalisation When multiple estimators are used, report both raw and normalised costs: 𝐶⋆(𝑐𝑟) = 􏾜 𝑗𝜆𝑗⋅Norm𝑗􏿴𝐶𝑗(𝑐𝑟)􏿷, 𝜆𝑗≥ 0, 􏾜 𝑗𝜆𝑗= 1 (F.1) 55 Algorithm 3 SD-C (Semantic-Distance Coherence Cost) Input: Text set 𝑄𝑟; embedding map 𝜙(⋅); closure embedding E(K); distance 𝑑(⋅,⋅) (default 1−cos) Output: 𝐶SD(𝑐𝑟) ∈ ℝ≥0 1: 𝐶 ← 0 2: for all 𝑞 ∈ 𝑄𝑟do 3:  𝑞 ← 𝜙(𝑞) 4:  𝑞′←arg min 𝑢∈E(K)𝑑( 𝑞, 𝑢) 5: 𝐶 ← 𝐶+𝑑( 𝑞, 𝑞′) 6: end for 7: return 𝐶 Choose Norm𝑗as z-score or robust (𝑥−median)/MAD depending on tails. Set 𝜆𝑗by interpretability priorities (e.g., RB–C heavier in legal contexts). Aggregation default. After per-estimator standardisation to zero-mean, unit-variance over branches, use 𝐶agg =1 3(𝑧RB +𝑧GI +𝑧SD)(F.2) with a sensitivity table showing outcomes under single-estimator drops. F.5 Sanity checks •Counterfactual symmetry: Swap labels on symmetric inputs; contradiction signature should persist. •Estimator agreement: Flag instability if |𝐶𝑖−−𝐶𝑘| > 𝛼  𝐶with 𝛼 ∈ [0.05,0.15]. •Ablation: Drop any single commitment; persistent contradiction ⇒structural. References Ananny, M., & Crawford, K. (2018). Seeing without knowing: Limitations of transparency in algorithmic accountability. New Media & Society,20(3), 973–989. https://doi. org/10.1177/1461444816676645 Argyris, C. (1991). Teaching smart people how to learn. Harvard Business Review,69(3), 99–109. Argyris, C., & Schön, D. (1978). Organizational learning. Addison-Wesley. Ashforth, B. E., & Anand, V. (2008). The normalization of corruption in organizations. Research in Organizational Behavior,28, 1–52. 56 Audi, R. (2003). Epistemology: A contemporary introduction to the theory of knowledge. Routledge. Aumann, R. (1999). Interactive epistemology i: Knowledge. International Journal of Game Theory,28(3), 263–300. Aumann, R. J. (1976). Agreeing to disagree. The Annals of Statistics,4(6), 1236–1239. Aumann, R. J., & Brandenburger, A. (1995). Epistemic conditions for nash equilibrium. Econometrica,63(5), 1161–1180. https://doi.org/10.2307/2171725 Baier, C., & Katoen, J.-P. (2008). Principles of model checking. MIT Press. Baker, C., Saxe, R., & Tenenbaum, J. (2009). Action understanding as inverse planning. Cognition. Baltag, A., & Smets, S. (2008). A dynamic-logical perspective on agency. In Handbook of the philosophy of science: Volume 8. Elsevier. Barak, A. (2012a). Proportionality: Constitutional rights and their limitations. Cambridge University Press. Barak, A. (2005). Purposive interpretation in law. Princeton University Press. Barak, A. (2012b). Human dignity: The constitutional value and the constitutional right. Cambridge University Press. Barocas, S., Hardt, M., & Narayanan, A. (2017). Fairness, accountability, and transparency in machine learning: A survey. arXiv preprint arXiv:1908.09635. Barocas, S., Hardt, M., & Narayanan, A. (2019). Fairness and machine learning [ urlhttps://fairmlbook.org]. Behn, R., & Kant, P. (1999). Strategies for avoiding audit failure. Public Administration Review. Benson, H. C. (2021). Socratic elenchus or refutation (R. E. of Philosophy, Ed.) [Last accessed October 2025]. https://www.rep.routledge.com/articles/biographical/ socrates-469-399-bc/v-1/sections/socratic-elenchus-or-refutation Benthem, J. v. (2001). Games in dynamic epistemic logic. Bulletin of Economic Research, 53, 219–248. https://doi.org/10.1111/1467-8586.00133 Besnard, P., & Hunter, A. (2008). Elements of argumentation. MIT Press. Binns, R. (2018). Fairness in machine learning: Lessons from political philosophy. Proceedings of the 2018 Conference on Fairness, Accountability, and Transparency (FAT*), 149–159. Blocki, J., Christin, N., Datta, A., Procaccia, A. D., & Sinha, A. (2014). Audit games with multiple defender resources. Proceedings of the AAAI Conference on Artificial Intelligence, 791–797. https://doi.org/10.1609/aaai.v29i1.9317 Bovens, L., & Hartmann, S. (2003). Bayesian epistemology. Oxford University Press. Brandenburger, A. (2007). Epistemic game theory: Complete information [Working paper]. https://www.adambrandenburger.com/aux/material/egtc-05-13-07.pdf Bratman, M. (1987). Intentions, plans, and practical reason. Harvard University Press. Brunsson, N. (1989). The organization of hypocrisy: Talk, decisions and actions in organizations. John Wiley & Sons. 57