Implication Ontology

FLOW ALGEBRA
TITLE: Implication Ontology
VERSION: 1.0
SOURCE: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/logical-relationships/implication-ontology-2.html
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source-page specification not independently verified.

ONTOLOGY:
  id: kireth:ImplicationOntology
  domain: Logical Relationships
  purpose: Represent conditional logical consequence with explicit scope, evidence and validation.

TYPES:
  Proposition := {id: URI, expression: Formula, context: Context}
  Formula := Atom | Not(Formula) | And(Formula, Formula) | Or(Formula, Formula) | Implies(Formula, Formula)
  Implication := {antecedent: Formula, consequent: Formula, interpretation: Interpretation, context: Context}
  Interpretation := Material | Entailment | Rule
  Context := {logic: LogicSystem, assumptions: Set[Formula], scope: String}
  LogicSystem := Classical | Intuitionistic | Custom
  Evidence := {source: URI?, method: String, timestamp: ISO8601?}
  Confidence := Real[0,1]

RELATIONSHIPS:
  antecedent_of(A, I) : Formula × Implication
  consequent_of(B, I) : Formula × Implication
  implies(A, B, C) : Formula × Formula × Context
  entails(Gamma, B, C) : Set[Formula] × Formula × Context
  equivalent_to(A, B, C) : Formula × Formula × Context
  contradicts(A, B, C) : Formula × Formula × Context

FLOW RULES:
  R1 MATERIAL_IMPLICATION:
    classical_truth(Implies(A,B)) = (not classical_truth(A)) or classical_truth(B)
  R2 MODUS_PONENS:
    IF verified(A,C) AND verified(Implies(A,B),C) THEN derive(B,C)
  R3 HYPOTHETICAL_SYLLOGISM:
    IF entails({A},B,C) AND entails({B},D,C) THEN entails({A},D,C)
  R4 CONTRAPOSITION_CLASSICAL:
    IF C.logic = Classical AND entails({A},B,C) THEN entails({Not(B)},Not(A),C)
  R5 EQUIVALENCE:
    IF entails({A},B,C) AND entails({B},A,C) THEN equivalent_to(A,B,C)
  R6 NO_CONVERSE_INFERENCE:
    implies(A,B,C) DOES_NOT_ENTAIL implies(B,A,C)
  R7 NO_CAUSALITY_INFERENCE:
    implies(A,B,C) DOES_NOT_ENTAIL causes(A,B)
  R8 CONTEXT_ISOLATION:
    Inference across contexts requires an explicit context-mapping justification.

CONSTRAINTS:
  - Every implication has exactly one antecedent and one consequent formula.
  - Logic-system semantics must be declared before applying rules.
  - Rule-based implications are not automatically material implications.
  - Vacuous truth applies to classical material implication, not arbitrary causal or normative rules.
  - Avoid treating correlation, temporal succession or co-occurrence as logical implication.
  - Reject unsupported cross-context substitutions and circular justification.

SEMANTIC_EQUIVALENCE:
  strategy: bidirectional_entailment
  conditions: [same_logic, compatible_context, verified_proof]
  note: Lexical similarity alone is insufficient for logical equivalence.

CONFIDENCE:
  representation: [0.0,1.0]
  policy: Confidence records evidence quality, not classical truth value.
  combination: Rule-specific and provenance-aware; do not assume independence.

PROVENANCE:
  capture: [source, author, method, assumptions, logic_system, timestamp, revision]
  derived_claim: retain premise identifiers and applied inference rule

VALIDATION:
  - Check formula well-formedness and type compatibility.
  - Verify each inference against the declared logic system.
  - Check premises and context consistency when required.
  - Distinguish satisfiability, validity, derivability and empirical support.
  - Test material implication truth table: TT=T, TF=F, FT=T, FF=T.
  - Flag converse fallacy and affirming-the-consequent errors.

FLOW_HASH:
  algorithm: SHA-256
  input: canonical_serialisation(ontology_id, version, types, relationships, rules, constraints)
  output: lowercase_hex_digest
  note: Compute at publication; no hash value is asserted here.

UPDATE_MODEL:
  trigger: [new_evidence, rule_revision, logic_change, context_change, correction]
  steps: [validate_change, preserve_previous_version, recompute_derivations, revalidate, recalculate_FLOW_HASH, publish]

AI_CONSUMPTION:
  format: UTF-8 plain text
  mode: typed semantic and logical inference
  safeguards: [logic_awareness, context_awareness, provenance_tracking, fallacy_detection]
  outputs: [verified_implications, derivation_traces, validation_reports, uncertainty_annotations]

END FLOW ALGEBRA