Membership Ontology

FLOW ALGEBRA
TITLE: Membership Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Set Theory / Membership
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory/membership-ontology.html
STATUS: Domain-based formalisation; source page content not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent set membership as a typed, logically constrained relation between mathematical objects and sets, supporting formal reasoning, provenance, and AI consumption.

2. TYPED ENTITIES
ENTITY MathematicalObject { id: Identifier; label: String? }
ENTITY Set extends MathematicalObject { membership_predicate: Predicate?; extension: Set<MathematicalObject>? }
ENTITY Element extends MathematicalObject { }
ENTITY MembershipAssertion { element: MathematicalObject; set: Set; truth_value: TruthValue; justification: Evidence? }
ENTITY Evidence { source: String; method: String; timestamp: DateTime? }
ENUM TruthValue = { TRUE, FALSE, UNKNOWN }

3. TYPED RELATIONSHIPS
MEMBER_OF: MathematicalObject x Set -> TruthValue
NOT_MEMBER_OF: MathematicalObject x Set -> TruthValue
SUBSET_OF: Set x Set -> TruthValue
EQUAL_SET: Set x Set -> TruthValue
DEFINED_BY: Set x Predicate -> TruthValue
SUPPORTED_BY: MembershipAssertion x Evidence -> TruthValue

4. FORMAL SEMANTICS
For mathematical object x and set A:
  MEMBER_OF(x,A) = TRUE iff x belongs to A (notation: x ∈ A).
  NOT_MEMBER_OF(x,A) = TRUE iff MEMBER_OF(x,A) = FALSE (notation: x ∉ A).
  UNKNOWN denotes insufficient evidence, not mathematical indeterminacy.
  A ⊆ B iff ∀x (x ∈ A -> x ∈ B).
  A = B iff ∀x (x ∈ A <-> x ∈ B).
  x ∈ ∅ is FALSE for every x.
  x ∈ {y} iff x = y.
  x ∈ A ∪ B iff (x ∈ A) OR (x ∈ B).
  x ∈ A ∩ B iff (x ∈ A) AND (x ∈ B).
  x ∈ A \ B iff (x ∈ A) AND (x ∉ B).
  A ∈ B and A ⊆ B are distinct relations.

5. INFERENCE RULES
RULE MembershipFromEnumeration:
  IF x appears in the extension of A THEN MEMBER_OF(x,A) := TRUE.
RULE SubsetPropagation:
  IF MEMBER_OF(x,A) = TRUE AND SUBSET_OF(A,B) = TRUE THEN MEMBER_OF(x,B) := TRUE.
RULE ExtensionalEquality:
  IF ∀x (MEMBER_OF(x,A) <-> MEMBER_OF(x,B)) THEN EQUAL_SET(A,B) := TRUE.
RULE EmptySetExclusion:
  IF A = ∅ THEN MEMBER_OF(x,A) := FALSE.
RULE PredicateComprehension:
  IF A = {x ∈ U | P(x)} AND x ∈ U THEN MEMBER_OF(x,A) := P(x).
RULE NoClosedWorldNegation:
  IF membership has not been established THEN retain UNKNOWN; do not infer FALSE.

6. CONSTRAINTS
- The right operand of MEMBER_OF must be a Set.
- Membership and subset relations must never be conflated.
- Contradictory TRUE and FALSE assertions require reconciliation and provenance review.
- Unrestricted set comprehension is prohibited; use a declared universe or a consistent set-theoretic foundation.
- Infinite sets may use predicates rather than explicit enumeration.

7. CONFIDENCE
CONFIDENCE: { value: Float[0,1]; basis: {FORMAL_PROOF, VERIFIED_SOURCE, DERIVED, HEURISTIC}; evidence_ids: List<Identifier> }
A proven theorem's logical truth is distinct from a model's confidence that a supplied premise or proof is correct.

8. SEMANTIC_EQUIVALENCE
SEMANTIC_EQUIVALENCE:
  MEMBER_OF == "is an element of" == "belongs to" == "∈"
  NOT_MEMBER_OF == "is not an element of" == "∉"
  SUBSET_OF == "is a subset of" == "⊆"
Do not equate MEMBER_OF with SUBSET_OF.

9. PROVENANCE
PROVENANCE: { source_url: SOURCE_URL; evidence: List<Evidence>; derivation_steps: List<String>; verification_status: "domain-based draft" }

10. FLOW_HASH
FLOW_HASH:
  algorithm: SHA-256
  input: canonical UTF-8 representation of ontology version, typed declarations, rules and constraints, excluding the FLOW_HASH value itself
  value: COMPUTE_AT_PUBLICATION

11. UPDATE_MODEL
UPDATE_MODEL:
  on_new_assertion: type_check -> validate -> reconcile -> infer -> record_provenance -> recompute_hash
  on_rule_change: increment_version -> revalidate_affected_assertions -> rerun_inference
  preserve_history: true

12. AI_CONSUMPTION
AI_CONSUMPTION:
  parse: typed_membership_graph
  reasoning: sound_set_theoretic_inference
  uncertainty: preserve_unknown_and_source_confidence
  output: {assertions, proofs_or_justifications, provenance, contradictions, flow_hash}
  safety: do_not_invent_membership_facts

13. EXAMPLE
Let A = {1,2} and B = {1,2,3}.
MEMBER_OF(1,A) = TRUE.
MEMBER_OF(3,A) = FALSE.
SUBSET_OF(A,B) = TRUE.
By SubsetPropagation, MEMBER_OF(1,B) = TRUE.
Note: MEMBER_OF(A,B) is not implied by SUBSET_OF(A,B).

END FLOW ALGEBRA