Conclusion Ontology

FLOW ALGEBRA
CONCLUSION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Logical Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/logical-entities/conclusion-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 a conclusion as a proposition or formula asserted to follow from premises under a specified inference system. Separate syntactic derivability, semantic entailment, empirical support and mere assertion.

2. TYPED ENTITIES
Conclusion: LogicalFormula
Premise: LogicalFormula
PremiseSet: Set[LogicalFormula]
Argument: {premises: PremiseSet, conclusion: Conclusion}
InferenceRule: {name: String, premises: List[LogicalFormula], result: LogicalFormula}
Proof: FiniteSequence[ProofStep]
ProofStep: {formula: LogicalFormula, justification: InferenceRule | Assumption, dependencies: Set[LogicalFormula]}
Interpretation: Mapping[Symbols, DomainObjectsAndRelations]
LogicalSystem: {syntax, semantics, axioms, inference_rules}
ValidityStatus: {valid, invalid, unknown}

3. CORE RELATIONSHIPS
HAS_PREMISE(Argument, Premise)
HAS_CONCLUSION(Argument, Conclusion)
DERIVED_FROM(Conclusion, PremiseSet, Proof)
ENTAILS(PremiseSet, Conclusion, LogicalSystem)
PROVES(Proof, Conclusion, LogicalSystem)
JUSTIFIED_BY(ProofStep, InferenceRule)
COUNTERMODEL_TO(Interpretation, Argument)

4. FORMAL SEMANTICS
Let Gamma be a set of premises and phi a conclusion.
Semantic entailment: Gamma |= phi iff every interpretation satisfying all formulas in Gamma also satisfies phi.
Syntactic derivability: Gamma |- phi iff a valid proof of phi from Gamma exists in the selected proof calculus.
Argument validity: VALID(Gamma, phi) := Gamma |= phi.
Sound calculus: Gamma |- phi implies Gamma |= phi.
Complete calculus: Gamma |= phi implies Gamma |- phi (where completeness holds for the selected system).
Do not equate semantic entailment with syntactic derivability without the required soundness/completeness assumptions.

5. INFERENCE RULES
MODUS_PONENS: from P and (P -> Q), infer Q.
MODUS_TOLLENS: from (P -> Q) and not Q, infer not P (classical logic).
CONJUNCTION_ELIMINATION: from (P and Q), infer P; also infer Q.
CONJUNCTION_INTRODUCTION: from P and Q, infer (P and Q).
UNIVERSAL_ELIMINATION: from forall x P(x), infer P(t) for a substitutable term t.
EXISTENTIAL_INTRODUCTION: from P(t), infer exists x P(x), subject to substitution constraints.
HYPOTHETICAL_SYLLOGISM: from (P -> Q) and (Q -> R), infer (P -> R).
Each rule is applicable only when its logic, types and side conditions are satisfied.

6. EXECUTABLE FLOW
flow "ConclusionInference" v1.0
input:
  premises: Set[LogicalFormula]
  candidate_conclusion: LogicalFormula
  logic: LogicalSystem
  proof: Optional[Proof]
steps:
  - parse and type-check premises and conclusion
  - validate proof steps and rule side conditions if proof supplied
  - if valid proof exists and calculus is sound: classify entailed
  - otherwise search for a countermodel in supported semantics
  - if countermodel found: classify not_entailed
  - otherwise classify unknown; never infer validity from search timeout
output:
  conclusion: candidate_conclusion
  status: {entailed, not_entailed, unknown}
  proof_or_countermodel: Optional[Evidence]
  confidence: EvidenceCalibratedConfidence

7. INVARIANTS AND VALIDATION
- A conclusion need not be true when its premises are false, even if the argument is valid.
- A true conclusion does not by itself make an argument valid.
- Circular reasoning is not independent evidential support.
- Proof dependencies must resolve to permitted premises, axioms or discharged assumptions.
- A countermodel must satisfy every premise and falsify the conclusion.
- Unproved, unsupported or incomplete results must remain unknown rather than be labelled false.
- Avoid conflating logical conclusion with statistical conclusion or causal conclusion.

8. SEMANTIC EQUIVALENCE AND DISTINCTIONS
CONCLUSION != PREMISE
CONCLUSION != PROOF
DERIVABLE_CONCLUSION may be semantically entailed in a sound system.
CONCLUSION != FACT: truth depends on interpretation and evidence.
SEMANTIC_EQUIVALENCE(phi, psi) iff phi |= psi and psi |= phi under the stated semantics.

9. KNOWLEDGE GRAPH EXAMPLE
Argument_A HAS_PREMISE P
Argument_A HAS_PREMISE (P -> Q)
Argument_A HAS_CONCLUSION Q
Proof_A USES_RULE MODUS_PONENS
Proof_A PROVES Q

10. CONFIDENCE
confidence.method: proof_validation | model_checking | human_review
confidence.value: 1.0 only for a verified formal result under explicit assumptions; otherwise calibrated or unknown
confidence.scope: selected formal system and supplied premises

11. FLOW_HASH AND UPDATE_MODEL
FLOW_HASH: SHA-256 over canonical UTF-8 serialisation of ontology identifier, version, entity schema, rules, axioms and validation constraints; computed by the consuming system, not preclaimed here.
UPDATE_MODEL: increment version on changes to semantics, rules or schema; preserve provenance, change log and backwards-compatibility notes.

12. AI_CONSUMPTION
mode: typed-ontology + proof-aware reasoning
requirements: explicit premises; declared logic; traceable inference; countermodel support where feasible; no fabricated proofs; distinction between entailment and empirical support
outputs: machine-readable conclusion, proof status, dependencies, confidence and provenance

END FLOW ALGEBRA