Axiom Ontology

# Axiom Ontology — Flow Algebra

URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/foundations-of-logic/axiom-ontology.html

FLOW AxiomOntology {

  DOMAIN:
    AI Core
    -> Ontologies
    -> Mathematical Ontologies
    -> Mathematical Logic Ontologies
    -> Foundations of Logic
    -> Axiom Ontology

  PURPOSE:
    Represent an axiom as a formally designated starting statement, formula, judgment or schema within an explicitly identified mathematical or logical system. The ontology models axiom identity, language, syntax, semantic interpretation, axiom schemata, instances, independence, consistency, redundancy, derivability, theory extension, foundational role, logical versus nonlogical axioms, equality axioms, arithmetic and set-theoretic axioms, modal axiom schemata, versioning, dependency graphs, proof usage, metatheoretic assessment and provenance. It enables AI systems to distinguish axioms from definitions, premises, assumptions, postulates, inference rules and derived theorems, while preventing an axiom's formal status from being confused with empirical truth, universal truth or proof-independent certainty.

  ENTITY Axiom {
    TYPE: FormallyDesignatedStartingStatement

    PROPERTIES:
      has_identifier
      belongs_to_formal_system
      belongs_to_theory
      has_language
      has_formula
      may_be_instance_of_schema
      may_have_semantic_interpretation
      may_be_independent
      may_be_redundant
      may_be_used_in_proof
      has_version
      has_provenance
  }

  ENTITY AxiomSchema {
    TYPE: FormulaGeneratingMetalinguisticPattern

    COMPONENTS:
      schema_identifier
      metavariables
      pattern
      admissible_substitutions
      side_conditions
  }

  ENTITY AxiomInstance {
    TYPE: ConcreteFormulaGeneratedBySchema

    COMPONENTS:
      schema_identifier
      substitution
      resulting_formula
  }

  ENTITY AxiomSet {
    TYPE: FormalTheoryBasis

    COMPONENTS:
      axiom_identifiers
      formal_language
      theory_identifier
      version
  }

  AXIOM_IDENTITY_MODEL {
    fields:
      axiom_identifier
      canonical_name
      theory_identifier
      formal_system_identifier
      version

    RULE:
      axiom_identity MUST be system_andversion_specific
  }

  AXIOM_STATUS_MODEL {
    status:
      designated_asaxiom_withinformal_system

    RULE:
      axiom_status_isrelational_notintrinsic_toformula
  }

  FORMULA_MODEL {
    axiom:
      well_formed_formula_orjudgment

    RULE:
      axiom MUST satisfy_language_formation_rules
  }

  LANGUAGE_MODEL {
    fields:
      language_identifier
      signature
      syntax
      binding

    RULE:
      axiom_formula MUST be interpreted_underdeclared_language
  }

  THEORY_MODEL {
    theory:
      collection_ofaxioms_andderived_consequences

    RULE:
      same_formula MAY beaxiom_inone_theory_andtheorem_inanother
  }

  FORMAL_SYSTEM_MODEL {
    components:
      language
      axioms
      inference_rules

    RULE:
      axiom MUST not_be confused_withinference_rule
  }

  AXIOM_VS_PREMISE_MODEL {
    axiom:
      designated_byformal_system_or theory

    premise:
      supplied_forparticular_argument_orproof

    RULE:
      premise != axiom
  }

  AXIOM_VS_ASSUMPTION_MODEL {
    axiom:
      globally_orformally_designated_starting_statement

    assumption:
      local_orconditional_proof_context

    RULE:
      assumption != axiom
  }

  AXIOM_VS_DEFINITION_MODEL {
    axiom:
      proposition_orjudgment_asserted_informal system

    definition:
      introduces_orfixes_meaning_orconstruction

    RULE:
      definition != axiom_unlessformalisation_explicitly_encodes_definition_asaxiom
  }

  AXIOM_VS_THEOREM_MODEL {
    axiom:
      accepted_withoutderivation_withinsystem

    theorem:
      derived_byformal_rules

    RULE:
      formula_role_depends_onsystem
  }

  AXIOM_VS_RULE_MODEL {
    axiom:
      formula_orjudgment

    inference_rule:
      transition_schema_frompremises_toconclusion

    RULE:
      axiom_schema != inference_rule
  }

  POSTULATE_MODEL {
    term:
      often_used_asnear_synonym_foraxiom_inparticular_domains

    RULE:
      terminology_context SHOULD be preserved
  }

  LOGICAL_AXIOM_MODEL {
    axiom:
      intended_toexpress_logic_independent_ofspecific_nonlogical_theory

    RULE:
      logical_axiom_classification depends_onproof_system
  }

  NONLOGICAL_AXIOM_MODEL {
    axiom:
      constrains_domain_specific_theory

    examples:
      arithmetic
      set_theory
      geometry
      algebra

    RULE:
      nonlogical_axiom_interpretation depends_onsignature
  }

  AXIOM_SCHEMA_MODEL {
    schema:
      metalinguistic_pattern

    RULE:
      schema != object_language_formula_necessarily
  }

  SCHEMA_METAVARIABLE_MODEL {
    metavariable:
      ranges_overformulas_terms_orother syntactic_objects

    RULE:
      metavariable != object_language_variable
  }

  SCHEMA_INSTANCE_MODEL {
    source:
      axiom_schema

    substitution:
      metavariables -> expressions

    output:
      axiom_instance

    RULE:
      substitution MUST satisfy_schema_side_conditions
  }

  FINITE_AXIOM_SET_MODEL {
    theory:
      finitely_many_axioms

    RULE:
      finite_axiom_set != finite_theorem_set
  }

  INFINITE_AXIOM_SET_MODEL {
    theory:
      infinitely_many_axioms

    RULE:
      infinite_axiom_set MAY be represented_byfinite_schema
  }

  FINITELY_AXIOMATISABLE_MODEL {
    theory:
      equivalent_tofinite_axiom_set_underdeclared_language

    RULE:
      finitely_axiomatisable != decidable
  }

  RECURSIVELY_ENUMERABLE_AXIOM_MODEL {
    axiom_set:
      effectively_enumerable

    RULE:
      recursively_enumerable != decidable_membership
  }

  DECIDABLE_AXIOM_SET_MODEL {
    axiom_membership:
      mechanically_decidable

    RULE:
      decidable_axiom_set != decidable_theoremhood
  }

  INDEPENDENCE_MODEL {
    axiom_A:
      notderivable_fromremaining_axioms

    RULE:
      independence_isrelative_toaxiom_set_andlogic
  }

  AXIOM_INDEPENDENCE_PROOF_MODEL {
    common_method:
      construct_model_ofremaining_axioms_whereA_false
      whensemantic_framework_supports

    RULE:
      proof_method_andmetatheoretic_assumptions MUST be preserved
  }

  REDUNDANT_AXIOM_MODEL {
    axiom:
      derivable_fromother_axioms

    RULE:
      redundant_axiom MAY remain_useful_forpresentation
  }

  MINIMAL_AXIOM_SET_MODEL {
    axiom_set:
      no_member_redundant_underdeclared criterion

    RULE:
      minimal != unique
  }

  EQUIVALENT_AXIOMATISATION_MODEL {
    axiom_sets:
      generate_same_theorems_undercommon_language_ortranslation

    RULE:
      equivalence_criterion MUST be explicit
  }

  CONSISTENCY_MODEL {
    axiom_set:
      does_notlead_todesignated_contradiction_underproof_system

    RULE:
      consistency != truth
  }

  SYNTACTIC_CONSISTENCY_MODEL {
    theory:
      no_formula_anditsnegation_bothderivable
      underdeclared_definition

    RULE:
      exact_definition depends_onlogic
  }

  SEMANTIC_SATISFIABILITY_MODEL {
    axiom_set:
      hasatleastone_model

    RULE:
      satisfiability_andconsistency_arelinked_onlyunderappropriate_metatheory
  }

  RELATIVE_CONSISTENCY_MODEL {
    claim:
      consistency_ofA_implies_consistency_ofB

    RULE:
      relative_consistency != absolute_consistency
  }

  CONSISTENCY_STRENGTH_MODEL {
    theories:
      compared_byrelative_consistency_orproof_theoretic_strength

    RULE:
      strength_relation MUST be defined
  }

  COMPLETENESS_DISTINCTION_MODEL {
    axiom_set:
      may_be syntactically_complete_orincomplete

    RULE:
      completeness != consistency
  }

  SYNTACTIC_COMPLETENESS_MODEL {
    theory:
      forrelevant_sentence_phi
      either_phi_ornegation_phi_derivable

    RULE:
      syntactic_completeness != semantic_completeness_oflogic
  }

  CATEGORICITY_MODEL {
    theory:
      models_unique_uptoisomorphism_underdeclared_cardinality_orsemantics

    RULE:
      categoricity != completeness
  }

  MODEL_MODEL {
    structure:
      satisfies_allaxioms

    RULE:
      model != proof
  }

  MODEL_CLASS_MODEL {
    theory:
      determines_class_ofmodels

    RULE:
      axioms_constrain_models_semantically
  }

  COUNTERMODEL_MODEL {
    forcandidate_consequence:
      model_ofaxioms_wherecandidate_false

    RULE:
      countermodel_refutes_semantic_consequence
  }

  SEMANTIC_CONSEQUENCE_MODEL {
    axiom_set:
      Gamma

    formula:
      phi

    relation:
      Gamma entails phi

    RULE:
      entailment != syntactic_derivability_bydefinition
  }

  SYNTACTIC_DERIVABILITY_MODEL {
    relation:
      Gamma proves phi

    RULE:
      derivation_usesaxioms_andinference_rules
  }

  SOUNDNESS_MODEL {
    proof_system:
      derivability_fromaxioms_implies_semantic_consequence

    RULE:
      soundness_requires_declared_semantics
  }

  COMPLETENESS_MODEL {
    proof_system:
      semantic_consequence_implies_derivability

    RULE:
      completeness_islogic_specific
  }

  PROOF_USAGE_MODEL {
    proof:
      may_cite_axiom_directly

    RULE:
      cited_axiom_identifier_andversion SHOULD be preserved
  }

  AXIOM_DEPENDENCY_MODEL {
    theorem:
      depends_on subset_ofaxioms

    RULE:
      dependency SHOULD be computed_transitively
  }

  AXIOM_FOOTPRINT_MODEL {
    theorem:
      set_ofaxioms_required_byproof_orallknownproofs_underdeclared criterion

    RULE:
      proof_specific_footprint != necessary_axiom_basis_unlessminimality_proven
  }

  UNUSED_AXIOM_MODEL {
    axiom:
      notused_inparticular_proof

    RULE:
      unused_inoneproof != globally_redundant
  }

  AXIOM_REVISION_MODEL {
    change:
      add
      remove
      replace
      modify

    RULE:
      revision creates_newtheory_orversion
  }

  AXIOM_ADDITION_MODEL {
    theory:
      T

    extended:
      T_plusA

    RULE:
      theorem_set_can_onlyexpand_undermonotonic_deductive_system
  }

  AXIOM_REMOVAL_MODEL {
    theory:
      T_minusA

    RULE:
      dependent_theorems MAY lose_derivability
  }

  CONSERVATIVE_EXTENSION_MODEL {
    extension:
      adds_no_new_theorems_inbase_language

    RULE:
      conservativity_requires_formal_proof
  }

  DEFINITIONAL_EXTENSION_MODEL {
    new_symbols:
      introduced_byexplicit_definitions

    RULE:
      definitional_extension MAY be conservative_underconditions
  }

  THEORY_STRENGTH_MODEL {
    stronger:
      proves_alltheorems_ofweaker_system_undercommon_translation_pluspossiblymore

    RULE:
      comparison_language_ortranslation MUST be declared
  }

  FOUNDATION_MODEL {
    axiom_set:
      provides_starting_basis_forformal theory

    RULE:
      foundational_role != epistemic_certainty
  }

  EMPIRICAL_DISTINCTION_MODEL {
    axiom:
      formal_status

    empirical_claim:
      supported_orrefuted_byobservation

    RULE:
      formal_axiom_status does_not establish_empirical_truth
  }

  SELF_EVIDENCE_DISTINCTION_MODEL {
    historical_orphilosophical:
      some_axioms_described_asself_evident

    RULE:
      ontology MUST notdefine_axiom_asnecessarilyself_evident
  }

  TRUTH_DISTINCTION_MODEL {
    axiom:
      adopted_withinsystem

    truth:
      semantic_property_underinterpretation

    RULE:
      axiom != universally_true_statement_bydefinition
  }

  ANALYTIC_DISTINCTION_MODEL {
    axiom:
      may_haveformal_role_independent_ofphilosophical_analytic_status

    RULE:
      logical_role MUST notencodecontested_epistemological_classification_asfact
  }

  ARITHMETIC_AXIOM_MODEL {
    domains:
      natural_numbers
      integers
      arithmetic_structures

    RULE:
      arithmetic_language_andtheory MUST be identified
  }

  PEANO_AXIOM_MODEL {
    theory:
      Peano_arithmetic_orrelated_formalisation

    possible:
      zero_axioms
      successor_axioms
      induction_schema

    RULE:
      exact_axiomatisation MUST be version_resolved
  }

  INDUCTION_AXIOM_SCHEMA_MODEL {
    schema:
      P(0)
      AND
      forall n(P(n)->P(S(n)))
      implies
      forall n P(n)
      underrepresentative_form

    RULE:
      exact_language_andformula_schema MUST be preserved
  }

  SET_THEORY_AXIOM_MODEL {
    theories:
      ZF
      ZFC
      alternatives

    RULE:
      theory_identifier MUST be explicit
  }

  EXTENSIONALITY_AXIOM_MODEL {
    representative:
      sets_withsame_members_areequal

    RULE:
      formal_formula depends_onlanguage
  }

  EMPTY_SET_AXIOM_MODEL {
    representative:
      existence_ofset_withnoelements

    RULE:
      may_be theorem_oraxiom_indifferent_axiomatisations
  }

  PAIRING_AXIOM_MODEL {
    representative:
      forobjects_a_b_exists_set_containing_exactly_a_andb

    RULE:
      formalisation MUST be theory_specific
  }

  UNION_AXIOM_MODEL {
    representative:
      union_ofmembers_ofset_exists

    RULE:
      exact_formula MUST be preserved_ifused
  }

  POWER_SET_AXIOM_MODEL {
    representative:
      set_ofallsubsets_exists

    RULE:
      theory_context MUST be preserved
  }

  INFINITY_AXIOM_MODEL {
    representative:
      asserts_existence_ofinductive_infinite_set

    RULE:
      formal_variant MUST be identified
  }

  REPLACEMENT_SCHEMA_MODEL {
    schema:
      functional_definable_images_ofsets_formsets_underZF_style_theory

    RULE:
      schema_instance_andformula_complexity MUST be preserved
  }

  FOUNDATION_AXIOM_MODEL {
    aliases:
      regularity

    RULE:
      exact_form_androle depend_onset_theory
  }

  CHOICE_AXIOM_MODEL {
    theory:
      axiom_ofchoice_or equivalent_forms_underappropriate_background_theory

    RULE:
      equivalence_claim MUST identify_background_axioms
  }

  GEOMETRIC_AXIOM_MODEL {
    domain:
      geometry

    examples:
      incidence
      order
      congruence
      parallel

    RULE:
      geometric_axiom_system MUST be identified
  }

  PARALLEL_POSTULATE_MODEL {
    Euclidean_geometry:
      characteristic_parallel_axiom_or equivalent_form

    RULE:
      independence_fromotherEuclidean_axioms isrelative_tochosen_axiomatisation
  }

  ALGEBRAIC_AXIOM_MODEL {
    structures:
      group
      ring
      field
      lattice

    RULE:
      structural_axioms defineclass_ofmodels
  }

  GROUP_AXIOM_MODEL {
    properties:
      associativity
      identity
      inverses

    RULE:
      operation_signature MUST be explicit
  }

  RING_AXIOM_MODEL {
    properties:
      additive_group
      multiplication
      distributivity
      optional_identity_orcommutativity_conventions

    RULE:
      ring_definition_variant MUST be preserved
  }

  FIELD_AXIOM_MODEL {
    structure:
      commutative_field_axioms

    RULE:
      nonzero_multiplicative_inverse_condition MUST excludezero
  }

  ORDER_AXIOM_MODEL {
    possible:
      reflexive
      antisymmetric
      transitive
      total

    RULE:
      partial_andtotal_order_axioms MUST be distinguished
  }

  LOGICAL_AXIOM_SCHEMA_MODEL {
    examples:
      propositional_axiom_schemata
      quantifier_axioms

    RULE:
      proof_calculus MUST be identified
  }

  MODAL_AXIOM_MODEL {
    examples:
      K
      T
      4
      5
      B

    RULE:
      modal_axiom_name MUST map_toexplicit_formula_schema
  }

  MODAL_K_MODEL {
    schema:
      box(A -> B) -> (box A -> box B)

    RULE:
      notation_andmodal_logic_context MUST be preserved
  }

  MODAL_T_MODEL {
    schema:
      box A -> A

    RULE:
      frame_correspondence_requires_standard_Kripke_semantics
  }

  MODAL_4_MODEL {
    schema:
      box A -> box box A
  }

  MODAL_5_MODEL {
    representative:
      diamond A -> box diamond A

    RULE:
      exact_notational_variant SHOULD be preserved
  }

  EQUALITY_AXIOM_MODEL {
    possible:
      reflexivity
      substitutivity

    RULE:
      equality_may_be logical_oraxiomatised_relation
  }

  REFLEXIVITY_AXIOM_MODEL {
    schema:
      forall x x=x
      underrepresentative_first_order_form
  }

  SUBSTITUTIVITY_AXIOM_MODEL {
    principle:
      identical_objects_may_be substituted_inappropriate_formulas

    RULE:
      formal_schema_andrestrictions MUST be preserved
  }

  TYPE_THEORY_AXIOM_MODEL {
    possible:
      universes
      extensionality
      choice
      proof_irrelevance
      univalence_inparticularsystems

    RULE:
      type_theory_variant MUST be explicit
  }

  COMPUTATIONAL_AXIOM_MODEL {
    axiom:
      may_affect_normalisation_decidability_orcomputational_content

    RULE:
      computational_consequences SHOULD be recorded_whenknown
  }

  AXIOM_AS_REWRITE_MODEL {
    equation_or equivalence:
      oriented_asrewrite_rule_forcomputation

    RULE:
      rewrite_orientation != logical_axiom_semantics
  }

  AXIOM_AS_CONSTRAINT_MODEL {
    axiom:
      constrains_admissible_models
    RULE:
      model_theoretic_andproof_theoretic_roles MUST be distinguished
  }

  MODEL_CLASS_MODEL {
    axioms:
      define_orrestrict_class_ofstructures

    RULE:
      two_axiomatisations MAY define_same_model_class_underdeclared_semantics
  }

  AXIOM_TRANSLATION_MODEL {
    source_axiom:
      translated_betweenformal_languages_or theories

    RULE:
      translation MUST preserve_declared_semantic_orproof_relation
  }

  AXIOM_MAPPING_MODEL {
    relation:
      exact
      equivalent_underbackground_theory
      stronger
      weaker
      related
      incompatible

    RULE:
      mapping_type MUST be explicit
  }

  EQUIVALENT_AXIOM_MODEL {
    A:
      equivalent_toB_relative_to background_theory_T

    RULE:
      equivalence_context MUST be preserved
  }

  STRONGER_AXIOM_MODEL {
    A:
      entails_orprovesB_underdeclared background

    RULE:
      strength_relation_isdirectional
  }

  WEAKER_AXIOM_MODEL {
    B:
      follows_fromA_underdeclared background

    RULE:
      weaker != less_useful
  }

  AXIOM_INTERACTION_MODEL {
    combined_axioms:
      may_haveconsequences_notderivable_fromindividual_axioms_alone

    RULE:
      dependency_reasoning MUST consider_axiom_sets
  }

  INCOMPATIBLE_AXIOM_MODEL {
    axiom_sets:
      jointly_unsatisfiable_orinconsistent_underdeclared framework

    RULE:
      incompatibility_context MUST be explicit
  }

  AXIOM_PROVENANCE_MODEL {
    fields:
      source
      author_ortradition
      formalisation
      theory
      version
      publication_orstandard
      transformation_history

    RULE:
      historical_source_andformal_current_definition SHOULD be distinguishable
  }

  VERSION_MODEL {
    versioned:
      axiom_formula
      axiom_schema
      theory
      formal_system
      language
      ontology

    RULE:
      proof_dependency MUST reference_axiom_version
  }

  DEPRECATION_MODEL {
    axiom:
      superseded_removed_orreformulated

    RULE:
      deprecated_axiom MUST remain_available_forhistorical_proof_reconstruction
  }

  REVISION_IMPACT_MODEL {
    changed_axiom:
      affects
        theorem_dependencies
        consistency_analysis
        model_class
        proof_certificates

    RULE:
      impact SHOULD be recursively_traversed
  }

  DEPENDENCY_GRAPH_MODEL {
    nodes:
      axioms
      definitions
      theorems
      proofs
      models
      theories

    edges:
      USES
      DERIVED_FROM
      ENTAILS
      INDEPENDENT_OF
      EQUIVALENT_TO
      EXTENDS
      CONFLICTS_WITH

    RULE:
      dependency_edges MUST be typed_andversioned
  }

  AXIOM_USAGE_MODEL {
    proof:
      cites_axiom_orinstance

    RULE:
      direct_andtransitive_axiom_usage SHOULD be distinguishable
  }

  PROOF_CERTIFICATE_MODEL {
    certificate:
      may_include_axiom_identifiers_andinstances

    RULE:
      checker MUST resolve_exact_axiom_set
  }

  AUTOMATED_REASONING_MODEL {
    AI_or theorem_prover:
      identify_axioms
      instantiate_schemata
      search_consequences
      test_independence
      compare_axiomatisations

    RULE:
      generated_axiom_instance MUST be validated_againstschema
  }

  AXIOM_DISCOVERY_DISTINCTION_MODEL {
    AI:
      may_propose_candidate_axiom

    RULE:
      proposed_statement != accepted_axiom_untilformal_system_revision_designates_it
  }

  CANDIDATE_AXIOM_MODEL {
    statement:
      proposed_foraddition_totheory

    metadata:
      rationale
      expected_consequences
      consistency_tests
      provenance

    RULE:
      candidate_status MUST remain_explicit
  }

  AXIOM_SELECTION_MODEL {
    criteria:
      independence
      simplicity
      expressive_adequacy
      tradition
      computational_properties

    RULE:
      selection_criteria_aremetatheoretic_orpragmatic_notproof_oftruth
  }

  AMBIGUITY_MODEL {
    possible:
      theory_unspecified
      version_unspecified
      schema_instance_confusion
      axiom_definition_confusion
      historical_name_collision

    RULE:
      ambiguity MUST be resolved_orpreserved
  }

  SEMANTIC_CONFIDENCE_MODEL {
    confidence:
      support_forresolved_axiom_identity_role_orinterpretation

    RULE:
      semantic_confidence != axiom_truth
  }

  VALIDATION_MODEL {
    checks:
      axiom_identity
      system
      theory
      language
      well_formedness
      schema_membership
      instance_substitution
      side_conditions
      dependency
      version
      provenance

    RULE:
      validation MUST use_exact_axiomatisation
  }

  INTEROPERABILITY_MODEL {
    exchange:
      axiom_identifier
      formula
      schema
      theory
      language
      version
      semantic_links
      provenance

    RULE:
      axiom_name_alone_isnotlossless
  }

  KNOWLEDGE_GRAPH_MODEL {
    nodes:
      axiom
      axiom_schema
      axiom_instance
      theory
      formal_system
      theorem
      proof
      model

    edges:
      BELONGS_TO
      INSTANCE_OF
      DERIVES
      USES
      ENTAILS
      INDEPENDENT_OF
      EQUIVALENT_TO
      CONFLICTS_WITH
      SUPERSEDES

    RULE:
      relation_scope_andversion MUST be preserved
  }

  RELATIONSHIPS {
    Axiom -> BELONGS_TO -> FormalTheory
    Axiom -> BELONGS_TO -> FormalSystem
    AxiomInstance -> INSTANCE_OF -> AxiomSchema
    Proof -> USES -> Axiom
    Theorem -> DERIVED_FROM -> AxiomSet
    Model -> SATISFIES -> Axiom
    Axiom -> MAY_BE_INDEPENDENT_OF -> OtherAxioms
    AxiomSet -> DEFINES -> ModelClass
    AxiomVersion -> SUPERSEDES -> PriorAxiomVersion
  }

  SEMANTIC_EQUIVALENCE {
    "axiom" ~= formally_designated_starting_statement_withinidentified_theory_orformal_system
    "axiom schema" ~= metalinguistic_pattern_generating_axiom_instances
    "axiom independence" ~= nonderivability_ofaxiom_fromremaining_axioms_underidentified_logic

    axiom != premise
    axiom != assumption
    axiom != definition
    axiom != theorem
    axiom != inference_rule
    axiom_schema != axiom_instance
    axiom_status != empirical_truth
    consistency != completeness
    satisfiability != syntactic_consistency_bydefinition
    redundant_axiom != unused_axiom
    candidate_axiom != accepted_axiom
    semantic_confidence != axiom_truth
  }

  CONSTRAINTS {
    axiom_identifier_system_theory_andversion_must_be preserved = true
    axiom_formula_must_be well_formed_underdeclared_language = true
    axiom_premise_assumption_definition_theorem_andrule_roles_must_be distinguished = true
    axiom_schema_andinstance_must_be distinguished = true
    schema_metavariables_andobject_variables_must_be distinguished = true
    schema_instantiation_side_conditions_must_be checked = true
    independence_andredundancy_claims_must_be relative_toidentified_axiom_set = true
    consistency_andcompleteness_must_be distinguished = true
    formal_axiom_status_must_not_be interpreted_asempirical_oruniversal_truth = true
    theory_extensions_andaxiom_revisions_must_be versioned = true
    theorem_andproof_dependencies_must_reference_axiom_versions = true
    candidate_AI_proposed_axioms_must_remain_candidates_untilformally_adopted = true
    provenance_must_be preserved = true
  }

  AXIOM_RECORD_MODEL {
    RECORD:
      axiom_identifier
      canonical_name
      axiom_type
      formal_system_identifier
      formal_system_version
      theory_identifier
      theory_version
      language_identifier
      formula
      syntax_tree
      schema_identifier
      schema_metavariables
      schema_instance_status
      schema_substitution
      side_conditions
      logical_or nonlogical_status
      definition_oraxiom_distinction
      independence_status
      redundancy_status
      consistency_context
      satisfiability_context
      model_class
      equivalent_axioms
      stronger_weaker_relations
      proof_usage
      theorem_dependencies
      version
      deprecation_status
      ambiguity_status
      semantic_confidence
      provenance

    RULE:
      axiom_record MUST support_reconstruction_offormal_role_formula_theory_dependencies_version_andprovenance
  }

  REASONING_RULES {
    IF statement_claimed_asaxiom
      THEN identify_formal_system_theory_language_andversion
      verify_designated_axiom_status

    IF axiom_schema_claimed
      THEN identify_metavariables_pattern_andside_conditions
      distinguish_schema_fromobject_language_formula

    IF candidate_instance_provided
      THEN compute_orverify_schema_substitution
      check_side_conditions
      confirm_instance_status

    IF axiom_independence_claimed
      THEN identify_remaining_axiom_set
      require_formal_derivation_analysis_orappropriate_model_evidence

    IF axiom_redundancy_claimed
      THEN search_forvalid_derivation_fromremaining_axioms

    IF theory_consistency_claimed
      THEN preserve_metatheoretic_framework
      anddistinguish_syntactic_consistency_fromsemantic_satisfiability

    IF axiom_added_removed_orrevised
      THEN create_newtheory_version
      preserve_previous_axiom_set
      recursively_identify_affected_proofs_theorems_andmodels

    IF AI_proposes_new_axiom
      THEN mark_as_candidate
      attach_rationale_andprovenance
      do_notassign_formal_axiom_status_withoutadoption

    IF same_axiom_name_maps_todifferent_formulas
      THEN preserve_allcandidates
      resolve_bytheory_version_andsource
  }

  VALIDATION {
    CHECK axiom_identifier
    CHECK axiom_type
    CHECK formal_system
    CHECK formal_system_version
    CHECK theory
    CHECK theory_version
    CHECK language
    CHECK formula_well_formedness
    CHECK schema_status
    CHECK schema_metavariables
    CHECK schema_instance
    CHECK substitution
    CHECK side_conditions
    CHECK logical_nonlogical_classification
    CHECK axiom_definition_premise_assumption_theorem_distinctions
    CHECK independence
    CHECK redundancy
    CHECK consistency_context
    CHECK satisfiability_context
    CHECK model_class
    CHECK equivalence_strength_relations
    CHECK proof_usage
    CHECK theorem_dependencies
    CHECK version
    CHECK deprecation
    CHECK ambiguity
    CHECK semantic_confidence
    CHECK provenance
  }

  CONFIDENCE {
    axiom_identity_confidence: 1.00
    formal_role_confidence: 1.00
    schema_instance_confidence: 1.00
    independence_confidence: 0.99
    consistency_context_confidence: 0.99
    model_theoretic_confidence: 0.99
    dependency_confidence: 1.00
    provenance_confidence: 1.00

    RULE:
      confidence decreases when formal system, theory, version,
      language, schema, dependency, metatheoretic context
      orprovenance is missing or ambiguous
  }

  UPDATE_MODEL {
    ON new_axiom:
      assign_orresolve_axiom_identifier
      attach_formal_system_theory_language_andversion
      validate_formula_or schema
      update_axiom_set
      preserve_provenance
      compute_FLOW_HASH

    ON axiom_revision:
      preserve_previous_axiom_version
      create_new_version
      update_theory_version
      identify_affected_dependencies
      recursively_revalidate_current_results

    ON schema_revision:
      preserve_previous_schema
      register_new_pattern_orconditions
      identify_affected_instances
      revalidate_instances

    ON theory_revision:
      preserve_previous_axiom_set
      register_added_removed_orreplaced_axioms
      recompute_dependency_andmodel_relations

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  AI_CONSUMPTION {
    canonical_entity: Axiom
    canonical_role: FormallyDesignatedStartingStatement

    interpretation_order:
      identify_axiom_candidate
      identify_formal_system_andversion
      identify_theory_andversion
      identify_formal_language
      validate_formula_well_formedness
      determine_axiom_schema_orinstance_status
      resolve_metavariables_substitutions_andside_conditions
      classify_logical_or nonlogical_role
      distinguish_axiom_frompremise_assumption_definition_theorem_andrule
      identify_model_theoretic_role
      evaluate_independence_orredundancy_onlywhenrequested_andcontext_available
      reconstruct_proof_andtheorem_dependencies
      resolve_equivalent_stronger_orweaker_axiom_relations
      identify_revision_deprecation_orcandidate_status
      detect_ambiguity
      assign_semantic_confidence
      preserve_versions_andprovenance
      emit_machine_resolvable_axiom_structure

    ambiguity_controls:
      distinguish_axiom_frompremise
      distinguish_axiom_fromassumption
      distinguish_axiom_fromdefinition
      distinguish_axiom_fromtheorem
      distinguish_axiom_frominference_rule
      distinguish_axiom_schema_fromaxiom_instance
      distinguish_axiom_status_fromempirical_truth
      distinguish_consistency_fromcompleteness
      distinguish_satisfiability_fromsyntactic_consistency_bydefinition
      distinguish_redundant_axiom_fromunused_axiom
      distinguish_candidate_axiom_fromaccepted_axiom
      distinguish_semantic_confidence_fromaxiom_truth
      preserve_system_theory_formula_schema_dependencies_version_andprovenance
  }

  PROVENANCE {
    ontology_scope:
      axioms_axiom_schemata_formal_theories_independence_consistency_andAI_axiom_reasoning

    parent_path:
      AI Core
      -> Ontologies
      -> Mathematical Ontologies
      -> Mathematical Logic Ontologies
      -> Foundations of Logic
      -> Axiom Ontology

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_entity
      ontology_path
      axiom_identity_model
      axiom_status_model
      formula_model
      language_model
      theory_model
      formal_system_model
      axiom_vs_premise_model
      axiom_vs_assumption_model
      axiom_vs_definition_model
      axiom_vs_theorem_model
      axiom_vs_rule_model
      postulate_model
      logical_axiom_model
      nonlogical_axiom_model
      axiom_schema_model
      schema_metavariable_model
      schema_instance_model
      finite_axiom_set_model
      infinite_axiom_set_model
      finitely_axiomatisable_model
      recursively_enumerable_axiom_model
      decidable_axiom_set_model
      independence_model
      axiom_independence_proof_model
      redundant_axiom_model
      minimal_axiom_set_model
      equivalent_axiomatisation_model
      consistency_model
      syntactic_consistency_model
      semantic_satisfiability_model
      relative_consistency_model
      consistency_strength_model
      completeness_distinction_model
      syntactic_completeness_model
      categoricity_model
      model_model
      model_class_model
      countermodel_model
      semantic_consequence_model
      syntactic_derivability_model
      soundness_model
      completeness_model
      proof_usage_model
      axiom_dependency_model
      axiom_footprint_model
      unused_axiom_model
      axiom_revision_model
      axiom_addition_model
      axiom_removal_model
      conservative_extension_model
      definitional_extension_model
      theory_strength_model
      foundation_model
      empirical_distinction_model
      self_evidence_distinction_model
      truth_distinction_model
      analytic_distinction_model
      arithmetic_axiom_model
      Peano_axiom_model
      induction_axiom_schema_model
      set_theory_axiom_model
      extensionality_axiom_model
      empty_set_axiom_model
      pairing_axiom_model
      union_axiom_model
      power_set_axiom_model
      infinity_axiom_model
      replacement_schema_model
      foundation_axiom_model
      choice_axiom_model
      geometric_axiom_model
      parallel_postulate_model
      algebraic_axiom_model
      group_axiom_model
      ring_axiom_model
      field_axiom_model
      order_axiom_model
      logical_axiom_schema_model
      modal_axiom_model
      modal_K_model
      modal_T_model
      modal_4_model
      modal_5_model
      equality_axiom_model
      reflexivity_axiom_model
      substitutivity_axiom_model
      type_theory_axiom_model
      computational_axiom_model
      axiom_as_rewrite_model
      axiom_as_constraint_model
      axiom_translation_model
      axiom_mapping_model
      equivalent_axiom_model
      stronger_axiom_model
      weaker_axiom_model
      axiom_interaction_model
      incompatible_axiom_model
      axiom_provenance_model
      version_model
      deprecation_model
      revision_impact_model
      dependency_graph_model
      axiom_usage_model
      proof_certificate_model
      automated_reasoning_model
      axiom_discovery_distinction_model
      candidate_axiom_model
      axiom_selection_model
      ambiguity_model
      semantic_confidence_model
      validation_model
      interoperability_model
      knowledge_graph_model
      relationships
      semantic_equivalence
      constraints
      axiom_record_model
      reasoning_rules
      validation
      confidence_model
      update_model
      AI_consumption_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      axiom_identity_formula_system_theory_orversion_changes
      axiom_schema_metavariable_instance_orside_condition_changes
      independence_redundancy_consistency_orcompleteness_semantics_change
      model_class_entailment_orderivability_context_changes
      axiom_addition_removal_extension_orstrength_relations_change
      arithmetic_set_theory_geometry_algebra_modal_ortype_axiom_models_change
      dependency_proof_usage_orcertificate_changes
      AI_candidate_axiom_orselection_semantics_change
      axiom_record_changes
      constraint_model_changes
      reasoning_rule_changes
      version_orprovenance_changes
  }

}

END FLOW