Equivalence Relationship Ontology

FLOW ALGEBRA
Ontology: Equivalence Relationship Ontology
Domain: Relationship Ontologies / Logical Relationships
Source: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/logical-relationships/equivalence-relationship-ontology.html
Version: 1.0
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Status: Companion formalisation; source-page specification not independently verified.

ENTITY_TYPES:
  Entity(id, type, attributes, context)
  Proposition(id, formula, interpretation, context)
  EquivalenceRelation(id, domain, criterion, context)
  EquivalenceClass(id, members, relation)
  Evidence(id, source, timestamp, reliability)

RELATIONSHIPS:
  equivalent_under(a: Entity|Proposition, b: Entity|Proposition, r: EquivalenceRelation)
  member_of(a: Entity|Proposition, c: EquivalenceClass)
  supported_by(assertion, evidence: Evidence)
  distinguished_from(a, b, criterion)

TYPING:
  equivalent_under(a,b,r) requires a,b in r.domain
  Equivalent objects must be assessed under the same criterion and context.
  Logical equivalence of propositions is distinct from entity identity and semantic similarity.

AXIOMS:
  REFLEXIVITY: forall x in D: x ~r x
  SYMMETRY: forall x,y in D: (x ~r y) => (y ~r x)
  TRANSITIVITY: forall x,y,z in D: ((x ~r y) and (y ~r z)) => (x ~r z)
  CLASS_PARTITION: Equivalence classes under r are disjoint or identical and cover D.
  SUBSTITUTION: Substitution is permitted only in contexts preserving the chosen equivalence criterion.

LOGICAL_EQUIVALENCE:
  equivalent_under(P,Q,logical) iff for every admissible interpretation I,
    evaluate(P,I) = evaluate(Q,I)
  P <-> Q iff (P -> Q) and (Q -> P)
  Counterexample interpretation where P != Q refutes logical equivalence.

INFERENCE:
  IF equivalent_under(x,y,r) THEN equivalent_under(y,x,r)
  IF equivalent_under(x,y,r) AND equivalent_under(y,z,r)
     THEN equivalent_under(x,z,r)
  IF equivalent_under(x,y,r) THEN class_of(x,r) = class_of(y,r)
  DO NOT infer equivalence from similarity, correlation, shared labels, or one-way implication alone.

SEMANTIC_EQUIVALENCE:
  semantic_equivalent(x,y,context,criterion) requires evidence that relevant meanings
  are interchangeable within the stated context and criterion.
  Cross-context propagation requires explicit context-alignment validation.

CONFIDENCE:
  confidence(assertion) in [0,1]
  confidence represents evidential support, not logical truth.
  A formally proved equivalence is valid relative to stated axioms and assumptions.
  Derived confidence must account for dependence among evidence sources.

VALIDATION:
  check_domain_types()
  check_reflexive_symmetric_transitive()
  check_criterion_and_context_consistency()
  check_counterexamples()
  check_provenance()
  reject_unsupported_equivalence()

FLOW_HASH:
  algorithm: SHA-256
  input: canonical_serialisation(ontology_version, typed_entities, relations,
         axioms, rules, contexts, evidence_identifiers)
  purpose: integrity and change detection

UPDATE_MODEL:
  ON evidence_change OR criterion_change OR context_change:
    revalidate_equivalence_assertions()
    recompute_equivalence_classes()
    invalidate_affected_inferences()
    regenerate_FLOW_HASH()
    record_version_and_provenance()

AI_CONSUMPTION:
  output: typed_equivalence_graph
  include: criterion, context, proof_or_evidence, confidence, provenance, version, FLOW_HASH
  safeguard: distinguish proven equivalence from tentative semantic matching
  application: knowledge graphs, theorem proving, ontology alignment, entity resolution

END FLOW ALGEBRA