Contradiction Proof Ontology

FLOW ALGEBRA
CONTRADICTION PROOF ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Proof Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/proof-entities/contradiction-proof-ontology.html
Version: 1.0 | Date: 2026-10-14 | Locale: en-GB
Authors: Mark Higgins — Sky (Recursive AI)
Status: Proposed formalisation; source page text not independently verified.

1. PURPOSE
Represent proof by contradiction (reductio ad absurdum) as a typed, auditable derivation in a specified formal logic. Distinguish classical from intuitionistic conclusions.

2. ENTITY TYPES
Proof: {id, logic_system, assumptions, goal, steps, status}
Proposition: {id, formula, free_variables, language}
Assumption: {id, proposition, scope_id, temporary:boolean}
InferenceStep: {id, premises:[step_id], rule, conclusion, scope_id}
Contradiction: {id, positive_formula, negative_formula, witness_steps}
Discharge: {id, assumption_id, conclusion_step_id, rule}
ProofContext: {id, available_assumptions, logic_system}
Countermodel: {id, interpretation, refuted_formula}

3. TYPED RELATIONSHIPS
Proof HAS_GOAL Proposition
Proof USES_ASSUMPTION Assumption
InferenceStep DEPENDS_ON InferenceStep
InferenceStep DERIVES Proposition
Contradiction WITNESSED_BY InferenceStep
Discharge CLOSES Assumption
Proof OPERATES_IN ProofContext
Countermodel REFUTES Proposition

4. FORMAL CORE
Let Γ denote undischarged assumptions and φ a well-formed formula.
Classical reductio:
  If Γ, ¬φ ⊢ ⊥, then Γ ⊢ φ.
Intuitionistic negation introduction:
  If Γ, φ ⊢ ⊥, then Γ ⊢ ¬φ.
Intuitionistic derivation from Γ, ¬φ ⊢ ⊥ gives Γ ⊢ ¬¬φ;
  deriving φ from ¬¬φ additionally requires a classical principle.
Contradiction witness: Γ ⊢ ψ and Γ ⊢ ¬ψ implies Γ ⊢ ⊥.
Scope discipline: the temporary assumption must be discharged by the rule;
  no conclusion may depend on an improperly escaped local assumption.

5. OPERATIONAL FLOW
flow "ContradictionProof" v1.0:
  input: Γ:AssumptionSet, goal:Proposition, logic:LogicSystem
  require: well_formed(Γ,goal) and consistent_rule_registry(logic)
  open_scope: assume(not(goal)) as temporary_assumption
  derive: steps using sound rules with explicit dependencies
  require: derives_bottom(steps) and validates_scope(steps)
  if logic == CLASSICAL:
    discharge temporary_assumption using classical_reductio
    output: derivation(Γ |- goal)
  else:
    discharge temporary_assumption using negation_introduction
    output: derivation(Γ |- not(not(goal)))
  validate: each inference, assumption discharge, provenance

6. INFERENCE RULES
R1: ψ, ¬ψ -> ⊥ (contradiction introduction).
R2: [φ] ... ⊥ -> ¬φ (negation introduction, discharge φ).
R3: [¬φ] ... ⊥ -> φ (classical reductio only, discharge ¬φ).
R4: [¬φ] ... ⊥ -> ¬¬φ (intuitionistically valid).
R5: ⊥ -> χ (explosion, in standard classical/intuitionistic logic; not paraconsistent systems).

7. VALIDATION
V1: All formulas are syntactically well-formed.
V2: Every step cites an applicable rule and accessible premises.
V3: The contradiction is actually derived, not merely asserted.
V4: Assumptions and scopes are explicitly tracked and correctly discharged.
V5: Classical-only steps are rejected under intuitionistic semantics.
V6: No proof is marked valid solely because a search timed out or failed.
V7: Proof validity is distinct from the truth of premises or empirical claims.
V8: Any inconsistency in Γ is flagged for review; in explosive logics it permits trivial derivation.

8. SEMANTIC EQUIVALENCE AND DISTINCTIONS
SEMANTIC_EQUIVALENCE: proof_by_contradiction ~ reductio_ad_absurdum (within declared logic context).
SEMANTIC_DIFFERENCE: direct_proof != contradiction_proof.
SEMANTIC_DIFFERENCE: contradiction_proof != proof_by_contrapositive.
SEMANTIC_DIFFERENCE: classical_reductio != intuitionistic_negation_introduction.
CONTEXT: proof method depends on logic system, axioms and allowed inference rules.

9. EXAMPLE (CLASSICAL)
Goal: There is no smallest positive rational number.
Assume: q is the smallest positive rational number.
Then q/2 is positive and rational, and q/2 < q.
Contradiction: q is smallest, but q/2 is a smaller positive rational.
Discharge assumption; conclude no smallest positive rational exists.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: mathematical_rule_confidence = HIGH for standard classical/intuitionistic logic distinctions.
CONFIDENCE: source_page_alignment = UNVERIFIED.
PROVENANCE: standard mathematical logic; formalisation proposed for Kireth Semantic Library.
UPDATE_MODEL: increment version when rules, ontology mappings or semantics change; preserve change history.

11. FLOW_HASH
FLOW_HASH algorithm: SHA-256 over canonical UTF-8 serialisation of the flow body,
excluding the FLOW_HASH output field itself; normalise LF newlines and stable key ordering.
FLOW_HASH value: COMPUTE_AT_INGESTION (not asserted here).

12. AI_CONSUMPTION
AI_CONSUMPTION:
  entity: ContradictionProof
  parse: typed propositions, assumptions, rules, steps, scope and logic system
  reason: validate each derivation step and discharge
  output: {status: VALID|INVALID|UNVERIFIED, conclusion, proof_trace, logic_system, warnings}
  safeguards: never conflate proof-search failure with disproof; distinguish logic systems
  interoperability: proof assistants, theorem ontologies, logical entity graphs

END FLOW ALGEBRA