Mathematical Logic Ontology

# Mathematical Logic Ontology — Flow Algebra

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

FLOW MathematicalLogicOntology {

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

  PURPOSE:
    Represent mathematical logic as a formal, machine-interpretable system for expressing and evaluating propositions, predicates, terms, formulas, quantifiers, logical connectives, axioms, inference rules, proofs, theories, interpretations and models. The ontology provides AI systems with explicit semantic boundaries between syntax and semantics, truth and provability, satisfiability and validity, implication and causation, consistency and completeness, formal deduction and heuristic inference, and object-level claims versus metalinguistic claims. It supports propositional and predicate reasoning, first-order structures, proof validation, semantic consequence, model construction, countermodels, automated reasoning, contradiction detection, recursive dependency checking, versioning and provenance.

  ENTITY MathematicalLogic {
    TYPE: FormalMathematicalReasoningSystem

    PROPERTIES:
      has_formal_language
      has_syntax
      has_semantics
      has_terms
      has_formulas
      has_propositions
      has_predicates
      has_quantifiers
      has_connectives
      has_axioms
      has_inference_rules
      has_proofs
      has_models
      has_theories
      supports_validity_reasoning
      supports_satisfiability_reasoning
  }

  ENTITY LogicalSystem {
    TYPE: FormalSystem

    COMPONENTS:
      language
      axioms
      inference_rules
      semantic_framework
      proof_definition
  }

  ENTITY Formula {
    TYPE: WellFormedLogicalExpression

    POSSIBLE:
      atomic_formula
      compound_formula
      quantified_formula
      sentence
      open_formula
  }

  ENTITY Theory {
    TYPE: SetOfLogicalSentences

    COMPONENTS:
      language
      axioms
      assumptions
      derived_theorems
  }

  ENTITY Proof {
    TYPE: FormalDerivation

    COMPONENTS:
      premises
      axioms
      inference_rules
      steps
      conclusion
  }

  FORMAL_LANGUAGE_MODEL {
    components:
      alphabet
      logical_symbols
      nonlogical_symbols
      formation_rules
      binding_rules

    RULE:
      language_identifier_andversion SHOULD be preserved
  }

  LOGICAL_SYMBOL_MODEL {
    possible:
      NOT
      AND
      OR
      IMPLIES
      IFF
      FORALL
      EXISTS
      equality
      parentheses

    RULE:
      symbol_semantics depend_onlogic
  }

  NONLOGICAL_SYMBOL_MODEL {
    possible:
      constants
      variables
      function_symbols
      predicate_symbols

    RULE:
      interpretation_assigns_domain_semantics
  }

  SYNTAX_MODEL {
    concerns:
-
      expression_formation
      parse_structure
      variable_binding
      scope

    RULE:
      syntax != semantics
  }

  SEMANTICS_MODEL {
    concerns:
      interpretation
      denotation
      valuation
      satisfaction
      truth
      model

    RULE:
      semantic_evaluation_requires_declared_interpretation
  }

  TERM_MODEL {
    possible:
      variable
      constant
      function_application

    RULE:
      term != formula
  }

  VARIABLE_MODEL {
    possible:
      free
      bound

    RULE:
      occurrence_status_depends_onbinder_scope
  }

  CONSTANT_MODEL {
    symbol:
      denotes_domain_element_underinterpretation

    RULE:
      logical_constant_symbol_andmathematical_constant MUST be distinguished
  }

  FUNCTION_SYMBOL_MODEL {
    arity:
      n

    interpretation:
      domain^n -> domain

    RULE:
      function_symbol != interpreted_function
  }

  PREDICATE_MODEL {
    arity:
      n

    interpretation:
      relation_on_domain^n

    RULE:
      predicate != proposition_untilarguments_orassignment_supply_context
  }

  ATOMIC_FORMULA_MODEL {
    possible:
      P(t_1,...,t_n)
      t_1 = t_2

    RULE:
      term_arity_andpredicate_arity MUST match
  }

  COMPOUND_FORMULA_MODEL {
    source:
      formulas

    operators:
      logical_connectives

    RULE:
      parse_tree MUST preserve_operator_scope
  }

  WELL_FORMEDNESS_MODEL {
    formula:
      generated_byformation_rules

    RULE:
      well_formed != true
  }

  PARSE_TREE_MODEL {
    nodes:
      connectives
      quantifiers
      predicates
      terms

    RULE:
      tree_structure determines_scope_andbinding
  }

  FREE_VARIABLE_MODEL {
    variable:
      occurrence_notbound

    RULE:
      truth_ofopen_formula MAY depend_onassignment
  }

  BOUND_VARIABLE_MODEL {
    variable:
      occurrence_bound_byquantifier

    RULE:
      alpha_renaming_can_preserve_semantics
  }

  VARIABLE_CAPTURE_MODEL {
    risk:
      substitution_changes_binding_status

    RULE:
      use_capture_avoiding_substitution
  }

  SUBSTITUTION_MODEL {
    notation:
      phi[x:=t]

    RULE:
      t MUST be free_forx_inphi_underdeclared_system
  }

  PROPOSITION_MODEL {
    entity:
      truth_evaluable_content

    RULE:
      proposition != formula_token_necessarily
  }

  SENTENCE_MODEL {
    formula:
      no_free_variables

    RULE:
      sentence_truth_is evaluated_relative_tostructure
  }

  OPEN_FORMULA_MODEL {
    formula:
      has_free_variables

    RULE:
      satisfaction_requires_variable_assignment
  }

  CONNECTIVE_MODEL {
    possible:
      negation
      conjunction
      disjunction
      implication
      biconditional

    RULE:
      connective_truth_conditions MUST be logic_specific
  }

  NEGATION_MODEL {
    form:
      NOT A

    classical:
      true_when_A_false
  }

  CONJUNCTION_MODEL {
    form:
      A AND B

    classical:
      true_when_both_true
  }

  DISJUNCTION_MODEL {
    form:
      A OR B

    classical:
      inclusive_OR
  }

  IMPLICATION_MODEL {
    form:
      A -> B

    classical_material:
      equivalent_to
        NOT A OR B

    RULE:
      implication != causation
  }

  BICONDITIONAL_MODEL {
    form:
      A <-> B

    classical:
      true_when_A_andB_have_same_truth_value
  }

  TRUTH_VALUE_MODEL {
    classical:
      true
      false

    RULE:
      nonclassical_logics_may_use_other_semantics
  }

  VALUATION_MODEL {
    propositional:
      atomic_propositions -> truth_values

    RULE:
      valuation MUST identify_logic_andatom_set
  }

  TRUTH_TABLE_MODEL {
    rows:
      valuations

    columns:
      formulas

    RULE:
      useful_forfinite_propositional_reasoning
  }

  TAUTOLOGY_MODEL {
    formula:
      true_underallpropositional_valuations
  }

  CONTRADICTION_FORMULA_MODEL {
    formula:
      false_underallpropositional_valuations
  }

  CONTINGENCY_MODEL {
    formula:
      true_undersome_valuations
      false_underothers
  }

  LOGICAL_EQUIVALENCE_MODEL {
    A equivalent B

    condition:
      same_truth_value_underallrelevant_interpretations

    RULE:
      semantic_equivalence != textual_identity
  }

  QUANTIFIER_MODEL {
    possible:
      universal
      existential

    RULE:
      quantifier_domain_andscope MUST be explicit
  }

  UNIVERSAL_QUANTIFIER_MODEL {
    form:
      forall x phi(x)

    semantics:
      phi_holds_forevery_domain_element
  }

  EXISTENTIAL_QUANTIFIER_MODEL {
    form:
      exists x phi(x)

    semantics:
      phi_holds_foratleast_one_domain_element
  }

  UNIQUE_EXISTENCE_MODEL {
    form:
      exists_unique x phi(x)

    RULE:
      requires_existence_anduniqueness
  }

  QUANTIFIER_SCOPE_MODEL {
    binder:
      controls_variable_occurrences_withinscope

    RULE:
      scope_ambiguity MUST be resolved_orpreserved
  }

  QUANTIFIER_ORDER_MODEL {
    formulas:
      forall x exists y R(x,y)
      exists y forall x R(x,y)

    RULE:
      formulas_are_notgenerally_equivalent
  }

  QUANTIFIER_NEGATION_MODEL {
    classical:
      NOT forall x phi
        equivalent
        exists x NOT phi

      NOT exists x phi
        equivalent
        forall x NOT phi
  }

  FIRST_ORDER_LOGIC_MODEL {
    quantifies_over:
      domain_individuals

    vocabulary:
      constants
      functions
      predicates

    RULE:
      standard_first_order_logic_does_notquantify_overpredicates_orfunctions
  }

  STRUCTURE_MODEL {
    M:
      nonempty_domain
      constant_interpretations
      function_interpretations
      predicate_interpretations

    RULE:
      structure != theory
  }

  INTERPRETATION_MODEL {
    maps:
      nonlogical_symbols -> domain_semantics

    RULE:
      interpretation_must_respect_arity
  }

  ASSIGNMENT_MODEL {
    maps:
      variables -> domain_elements

    RULE:
      assignment_affects_free_variable_evaluation
  }

  SATISFACTION_MODEL {
    notation:
      M,s satisfies phi

    RULE:
      satisfaction_isrelation_betweenmodel_assignment_andformula
  }

  MODEL_MODEL {
    M:
      model_ofT
      when M_satisfies_every_sentence_inT

    RULE:
      model != proof
  }

  COUNTERMODEL_MODEL {
    M:
      satisfies_premises
      but_not_conclusion

    RULE:
      countermodel_refutes_semantic_consequence
  }

  SATISFIABILITY_MODEL {
    formula_or theory:
      has_atleast_one_model

    RULE:
      satisfiable != valid
  }

  UNSATISFIABILITY_MODEL {
    formula_or theory:
      has_no_model
  }

  VALIDITY_MODEL {
    formula:
      true_inallmodels_underdeclared_logic

    RULE:
      validity != truth_inone_model
  }

  SEMANTIC_CONSEQUENCE_MODEL {
    notation:
      Gamma entails phi

    meaning:
      every_model_ofGamma_satisfies_phi
  }

  SYNTACTIC_DERIVABILITY_MODEL {
    notation:
      Gamma proves phi

    meaning:
      formal_derivation_exists

    RULE:
      derivability != entailment_bydefinition
  }

  AXIOM_MODEL {
    formula:
      foundational_statement_offormal_system

    RULE:
      axiom_status_isrelative_totheory
  }

  AXIOM_SCHEMA_MODEL {
    pattern:
      defines_family_ofaxioms

    RULE:
      schema != individual_axiom
  }

  PREMISE_MODEL {
    statement:
      supplied_toargument_orproof

    RULE:
      premise_truth_notguaranteed_byformal_role
  }

  ASSUMPTION_MODEL {
    statement:
      temporarily_orcontextually_adopted

    RULE:
      discharge_scope MUST be preserved
  }

  INFERENCE_RULE_MODEL {
    premises:
      P_1...P_n

    conclusion:
      C

    RULE:
      formal_side_conditions MUST be checked
  }

  MODUS_PONENS_MODEL {
    premises:
      A
      A -> B

    conclusion:
      B
  }

  MODUS_TOLLENS_MODEL {
    premises:
      A -> B
      NOT B

    conclusion:
      NOT A
  }

  UNIVERSAL_INSTANTIATION_MODEL {
    premise:
      forall x phi(x)

    conclusion:
      phi(t)

    RULE:
      t_must_be admissible
  }

  EXISTENTIAL_GENERALISATION_MODEL {
    premise:
      phi(t)

    conclusion:
      exists x phi(x)
  }

  ARGUMENT_MODEL {
    premises:
      Gamma

    conclusion:
      phi

    RULE:
      argument_structure != proof_object
  }

  ARGUMENT_VALIDITY_MODEL {
    valid:
      Gamma entails phi

    RULE:
      validity_independent_ofactual_truth_ofpremises
  }

  SOUND_ARGUMENT_MODEL {
    requires:
      valid_argument
      true_premises_underintended_interpretation
  }

  FALLACY_MODEL {
    entity:
      invalid_reasoning_pattern

    RULE:
      fallacy_detection_requires_argument_structure
  }

  AFFIRMING_CONSEQUENT_MODEL {
    invalid:
      A -> B
      B
      therefore A
  }

  DENYING_ANTECEDENT_MODEL {
    invalid:
      A -> B
      NOT A
      therefore NOT B
  }

  PROOF_MODEL {
    structure:
      ordered_orgraph_based_derivation

    REQUIRE:
      every_nonpremise_step_hasjustification
  }

  PROOF_STEP_MODEL {
    fields:
      formula
      justification
      dependencies
      rule
  }

  THEOREM_MODEL {
    formula:
      derivable_fromaxioms_underproof_system

    RULE:
      theoremhood_isformal_system_relative
  }

  LEMMA_MODEL {
    theorem:
      used_asintermediate_result
  }

  COROLLARY_MODEL {
    theorem:
      follows_fromearlier_result
  }

  DIRECT_PROOF_MODEL {
    method:
      derive_target_frompremises
  }

  CONTRADICTION_PROOF_MODEL {
    method:
      assume_negation
      derive_contradiction
      infer_target_whenlogic_permits

    RULE:
      method_validity_islogic_specific
  }

  CONTRAPOSITION_MODEL {
    A -> B
      classically_equivalent_to
      NOT B -> NOT A
  }

  MATHEMATICAL_INDUCTION_MODEL {
    base_case:
      P(n_0)

    inductive_step:
      P(k) -> P(k+1)

    RULE:
      induction_principle_requiresappropriate_structure
  }

  SOUNDNESS_MODEL {
    proof_system:
      Gamma proves phi
      implies
      Gamma entails phi

    RULE:
      soundness != completeness
  }

  COMPLETENESS_MODEL {
    proof_system:
      Gamma entails phi
      implies
      Gamma proves phi

    RULE:
      theorem_applies_only_toappropriate_logic_andproof_system
  }

  CONSISTENCY_MODEL {
    theory:
      does_notderive_both_phi_andNOT_phi
      orformal_contradiction_underdeclared_system

    RULE:
      consistency != completeness
  }

  INCONSISTENCY_MODEL {
    theory:
      derives_contradiction

    RULE:
      consequences_depend_onlogic
  }

  EXPLOSION_MODEL {
    classical:
      contradiction_can_entail_arbitrary_formula

    RULE:
      paraconsistent_logic_may_reject_explosion
  }

  INDEPENDENCE_MODEL {
    formula:
      neither_phi_norNOT_phi_derivable_fromT

    RULE:
      independence_relative_toT
  }

  DECIDABILITY_MODEL {
    property:
      algorithm_exists_that_halts_withcorrect_answer_forallinputs_inclass

    RULE:
      decidability != computational_efficiency
  }

  SEMIDECIDABILITY_MODEL {
    property:
      positive_instances_recognisable_withhalting_acceptance

    RULE:
      semidecidable != decidable
  }

  PROOF_THEORY_MODEL {
    studies:
      formal_derivations
      proof_systems
      proof_transformations

    RULE:
      proof_object != semantic_model
  }

  MODEL_THEORY_MODEL {
    studies:
      structures
      satisfaction
      definability
      semantic_consequence

    RULE:
      semantic_and syntactic_levels MUST remain_distinct
  }

  META_LOGIC_MODEL {
    studies:
      properties_oflogical_systems

    possible:
      soundness
      completeness
      consistency
      compactness
      decidability

    RULE:
      metalogical_statement != object_language_formula_necessarily
  }

  OBJECT_LANGUAGE_MODEL {
    language:
      object_oflogical_analysis

    RULE:
      object_language != metalanguage
  }

  METALANGUAGE_MODEL {
    language:
      describes_object_language_anditssemantics

    RULE:
      semantic_predicates_may_live_atmetalanguage_level
  }

  TRUTH_MODEL {
    semantic:
      truth_inmodel_orinterpretation

    RULE:
      truth != provability
  }

  PROVABILITY_MODEL {
    syntactic:
      derivability_underproof_system

    RULE:
      provability != semantic_truth_bydefinition
  }

  VALIDITY_TRUTH_MODEL {
    validity:
      truth_inallmodels

    truth:
      truth_inparticular_model

    RULE:
      validity_andtruth MUST not_be conflated
  }

  SATISFIABILITY_CONSISTENCY_MODEL {
    satisfiability:
      semantic

    consistency:
      syntactic

    RULE:
      correspondence_requires_metatheoretic_conditions
  }

  CLASSICAL_LOGIC_MODEL {
    principles:
      bivalence
      excluded_middle
      noncontradiction
      material_implication
      explosion

    RULE:
      principles_notuniversal_acrossnonclassical_logics
  }

  NONCLASSICAL_LOGIC_MODEL {
    possible:
      intuitionistic
      paraconsistent
      many_valued
      modal
      temporal
      fuzzy

    RULE:
      logic_specific_semantics MUST be loaded
  }

  INTUITIONISTIC_DISTINCTION_MODEL {
    principle:
      proof_constructive_semantics

    RULE:
      excluded_middle_notgenerally_available
  }

  PARACONSISTENT_DISTINCTION_MODEL {
    principle:
      inconsistency_withoutunrestricted_explosion

    RULE:
      contradiction_does_not automatically_entail_everything
  }

  MANY_VALUED_DISTINCTION_MODEL {
    truth_values:
      more_than_two

    RULE:
      connective_tables_oralgebra MUST be specified
  }

  FUZZY_DISTINCTION_MODEL {
    truth_degree:
      graded_membership_ortruth_underfuzzy_semantics

    RULE:
      fuzzy_truth_degree != probability
  }

  MODAL_EXTENSION_MODEL {
    operators:
      necessity
      possibility

    RULE:
      modal_reasoning_requires_world_oralgebraic_semantics
  }

  TEMPORAL_EXTENSION_MODEL {
    operators:
      always
      eventually
      next
      until

    RULE:
      temporal_structure MUST be explicit
  }

  EQUALITY_MODEL {
    logical_or nonlogical:
      relation_between_terms

    RULE:
      treatment_ofequality MUST be specified_bylogic
  }

  IDENTITY_MODEL {
    object_identity:
      same_domain_entity

    RULE:
      logical_identity_ofobjects != formula_equivalence
  }

  LOGICAL_EQUIVALENCE_MODEL {
    formulas:
      semantically_equivalent

    RULE:
      equivalence != object_identity
  NORMAL_FORM_MODEL {
    possible:
      CNF
      DNF
      prenex_forms_underappropriate_logic

    RULE:
      transformation_may_preserve_equivalence_orsatisfiability_asdeclared
  }

  CNF_MODEL {
    structure:
      conjunction_ofclauses
  }

  DNF_MODEL {
    structure:
      disjunction_ofconjunctions
  }

  LITERAL_MODEL {
    possible:
      atom
      negated_atom
  }

  CLAUSE_MODEL {
    structure:
      disjunction_ofliterals
  }

  RESOLUTION_MODEL {
    propositional:
      resolve_complementary_literals_betweenclauses

    RULE:
      first_order_resolution_requires_unification_andvariable_management
  }

  UNIFICATION_MODEL {
    purpose:
      find_substitution_makingterms_oratoms_match

    RULE:
      substitution_constraints MUST be preserved
  }

  AUTOMATED_REASONING_MODEL {
    possible:
      theorem_proving
      SAT
      SMT
      model_checking
      proof_checking

    RULE:
      solver_status_unknown_or timeout MUST not_be interpreted_asfalse
  }

  THEOREM_PROVER_MODEL {
    input:
      axioms
      conjecture

    output:
      proof
      refutation
      unknown

    RULE:
      proof_artifact SHOULD be retained_whenavailable
  }

  PROOF_CHECKER_MODEL {
    input:
      proof_object

    output:
      verified_orrejected_undertrusted_rules

    RULE:
      checker_version MUST be preserved
  }

  SAT_SOLVER_MODEL {
    input:
      propositional_formula

    output:
      satisfiable_assignment
      orunsatisfiable_certificate_whenavailable
  }

  SMT_MODEL {
    input:
      formula_modulo_declared_background_theories

    RULE:
      theory_fragment MUST be identified
  }

  MODEL_CHECKING_MODEL {
    input:
      transition_system
      formal_property

    output:
      satisfied
      violated_withcounterexample
      unknown

    RULE:
      abstraction_andboundedness MUST be preserved
  }

  CONTRADICTION_DETECTION_MODEL {
    theory:
      detect_incompatible_claims_underlogic

    RULE:
      contradiction_detection != proof_ofempirical_falsehood
  }

  COUNTEREXAMPLE_MODEL {
    witness:
      refutes_universal_orvalidity_claim_withinscope

    RULE:
      witness_context MUST satisfy_declared_domain
  }

  AMBIGUITY_MODEL {
    causes:
      missing_parentheses
      quantifier_scope
      overloaded_symbols
      unspecified_domain
      unspecified_logic

    RULE:
      AI MUST preserve_multiple_candidate_parses_whenunresolved
  }

  SEMANTIC_CONFIDENCE_MODEL {
    confidence:
      support_forparsed_orresolved_formal_interpretation

    RULE:
      semantic_confidence != truth_value
  }

  AI_REASONING_MODEL {
    AI:
      parse
      classify
      resolve_scope
      resolve_logic
      evaluate
      derive
      search_proof
      search_model
      validate
      explain

    RULE:
      AI_generated_argument MUST not_be labelled_formal_proof_untilverified
  }

  HEURISTIC_MODEL {
    method:
      guides_search

    RULE:
      heuristic_success != formal_validity
  }

  EXPLAINABILITY_MODEL {
    result:
      expose
        language
        logic
        premises
        assumptions
        inference_rules
        semantic_model
        proof_orcountermodel

    RULE:
      explanation MUST distinguish_formal_derivation_fromheuristic_search
  }

  DEPENDENCY_MODEL {
    theorem_orresult:
      depends_on
        definitions
        axioms
        assumptions
        lemmas
        proof_rules

    RULE:
      dependency_graph MUST be preserved
  }

  RECURSIVE_VALIDATION_MODEL {
    process:
      validate_dependencies
      validate_steps
      validate_conclusion
      propagate_revision_effects

    RULE:
      changed_axiom_orrule MAY invalidate_dependent_proofs
  }

  VERSION_MODEL {
    versioned:
      language
      logic
      theory
      axiom_set
      proof_system
      solver
      ontology

    RULE:
      historical_result MUST preserve_versions_used
  }

  PROVENANCE_MODEL {
    fields:
      source
      formal_language
      logic_system
      theory
      axioms
      proof_system
      solver
      proof
      model
      version
      agent

    RULE:
      formal_result MUST remain_provenance_linked
  }

  INTEROPERABILITY_MODEL {
    exchange:
      formulas
      syntax_trees
      variable_binding
      theory_identifiers
      proof_objects
      models
      solver_results

    RULE:
      plain_text_formula_withoutlogic_andbinding_context MAY be insufficient
  }

  KNOWLEDGE_GRAPH_MODEL {
    nodes:
      formula
      term
      proposition
      predicate
      theory
      axiom
      inference_rule
      proof
      theorem
      model

    edges:
      CONTAINS
      BINDS
      DERIVES
      PROVES
      ENTAILS
      SATISFIES
      REFUTES
      DEPENDS_ON
      EQUIVALENT_TO

    RULE:
      edge_semantics MUST be typed
  }

  RELATIONSHIPS {
    MathematicalLogic -> USES -> FormalLanguage
    FormalLanguage -> CONTAINS -> Formula
    Formula -> MAY_CONTAIN -> Term
    Formula -> MAY_CONTAIN -> Variable
    Quantifier -> BINDS -> Variable
    Theory -> CONTAINS -> Axiom
    Proof -> USES -> InferenceRule
    Proof -> DERIVES -> Theorem
    Structure -> SATISFIES -> Formula
    Countermodel -> REFUTES -> ValidityClaim
    Solver -> MAY_PRODUCE -> ProofOrModel
  }

  SEMANTIC_EQUIVALENCE {
    "mathematical logic" ~= formal_mathematical_study_ofsyntax_semantics_inference_proof_andmodels
    "logical consequence" ~= preservation_oftruth_frompremises_toconclusion_acrossallmodels
    "formal proof" ~= derivation_usingdeclared_axioms_premises_andinference_rules

    syntax != semantics
    term != formula
    proposition != formula_token
    well_formedness != truth
    truth != provability
    satisfiability != validity
    validity != truth_inone_model
    entailment != derivability_bydefinition
    implication != causation
    consistency != completeness
    proof != model
    heuristic_argument != formal_proof
    semantic_confidence != truth_value
  }

  CONSTRAINTS {
    formal_language_must_be identified = true
    logic_system_must_be identified = true
    syntax_andsemantics_must_be distinguished = true
    term_formula_andproposition_types_must_be distinguished = true
    variable_binding_andquantifier_scope_must_be preserved = true
    substitution_must_avoid_variable_capture = true
    interpretation_domain_andassignments_must_be explicit_whenrequired = true
    satisfiability_validity_truth_andprovability_must_be distinguished = true
    semantic_consequence_andderivability_must_be distinguished = true
    consistency_soundness_andcompleteness_must_be distinguished = true
    formal_proof_steps_must_havejustification = true
    nonclassical_logic_must_notinherit_classical_rules_withoutbasis = true
    AI_heuristic_reasoning_must_not_be represented_asformal_proof_withoutverification = true
    versions_andprovenance_must_be preserved = true
  }

  LOGIC_RECORD_MODEL {
    RECORD:
      logic_entity_identifier
      formal_language_identifier
      logic_system
      theory_identifier
      expression
      expression_type
      syntax_tree
      terms
      variables
      free_variables
      bound_variables
      quantifier_scope
      connective_structure
      interpretation_identifier
      domain
      assignment
      truth_value
      satisfaction_status
      model_identifier
      axioms
      premises
      conclusion
      inference_rules
      proof_identifier
      proof_steps
      theorem_identifier
      entailment_status
      derivability_status
      satisfiability_status
      validity_status
      consistency_status
      soundness_context
      completeness_context
      solver_identifier
      proof_certificate
      countermodel
      ambiguity_status
      semantic_confidence
      version
      provenance

    RULE:
      mathematical_logic_record MUST remain_linked_toformal_language_logic_semantics_proof_model_andprovenance
  }

  REASONING_RULES {
    IF logical_expression_received
      THEN identify_formal_language_andlogic_system
      parse_expression
      classify_terms_formulas_andvariables

    IF free_orbound_variables_present
      THEN resolve_binding_andquantifier_scope

    IF substitution_requested
      THEN validate_term_admissibility
      andavoid_capture

    IF semantic_evaluation_requested
      THEN resolve_structure_interpretation_andassignment

    IF satisfiability_requested
      THEN search_for_model_orvaluation
      andpreserve_unknown_ifmethod_incomplete

    IF validity_requested
      THEN search_forcountermodel
      orverify_proof_underappropriate_soundness_context

    IF proof_claimed
      THEN validate_each_step_againstpremises_axioms_rules_andside_conditions

    IF theorem_dependency_changes
      THEN recursively_revalidate_dependent_proofs

    IF nonclassical_logic_selected
      THEN load_logic_specific_truth_connective_andinference_semantics

    IF automated_solver_times_out_orreturns_unknown
      THEN preserve_unknown_status
      anddo_notinfer_false

    IF AI_generates_candidate_reasoning
      THEN verify_formally_beforemarking_proven
  }

  VALIDATION {
    CHECK logic_entity_identifier
    CHECK formal_language
    CHECK logic_system
    CHECK theory
    CHECK expression_type
    CHECK well_formedness
    CHECK syntax_tree
    CHECK term_andformula_types
    CHECK variable_binding
    CHECK quantifier_scope
    CHECK substitution
    CHECK connective_semantics
    CHECK interpretation
    CHECK domain
    CHECK assignment
    CHECK satisfaction
    CHECK model
    CHECK satisfiability
    CHECK validity
    CHECK semantic_consequence
    CHECK syntactic_derivability
    CHECK axioms
    CHECK inference_rules
    CHECK proof_steps
    CHECK theorem_dependencies
    CHECK consistency
    CHECK soundness
    CHECK completeness
    CHECK solver_status
    CHECK proof_orcountermodel
    CHECK ambiguity
    CHECK semantic_confidence
    CHECK version
    CHECK provenance
  }

  CONFIDENCE {
    formal_language_confidence: 1.00
    syntax_semantics_confidence: 1.00
    propositional_reasoning_confidence: 1.00
    predicate_first_order_confidence: 1.00
    proof_validation_confidence: 1.00
    model_semantics_confidence: 1.00
    metalogical_confidence: 1.00
    automated_reasoning_confidence: 0.99

    RULE:
      confidence decreases when formal language, logic system,
      domain, interpretation, proof rules, solver status,
      version orprovenance is missing or ambiguous
  }

  UPDATE_MODEL {
    ON new_logic_expression:
      resolve_language_andlogic
      parse
      type
      validate
      attach_semantic_context
      preserve_provenance

    ON new_proof:
      assign_proof_identifier
      preserve_premises_axioms_andrules
      validate_steps
      attach_dependencies
      register_conclusion

    ON new_model:
      define_structure_andinterpretation
      evaluate_satisfaction
      attach_witness_orcountermodel_whenrelevant

    ON theory_oraxiom_revision:
      preserve_previous_version
      register_revision
      identify_dependent_proofs_andtheorems
      recursively_revalidate_withoutoverwritinghistory

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  AI_CONSUMPTION {
    canonical_entity: MathematicalLogic
    canonical_role: FormalMathematicalReasoningSystem

    interpretation_order:
      identify_formal_language
      identify_logic_system
      identify_theory_context
      parse_expression
      resolve_terms_variables_andpredicates
      resolve_binding_quantifiers_andconnectives
      distinguish_syntax_fromsemantics
      resolve_interpretation_domain_andassignment
      evaluate_satisfaction_ortruth_whencontext_available
      distinguish_satisfiability_validity_entailment_andderivability
      identify_axioms_premises_andinference_rules
      validate_orconstruct_proof
      search_forcountermodel_whenappropriate
      distinguish_consistency_soundness_andcompleteness
      resolve_nonclassical_semantics_whenneeded
      identify_solver_orproof_checker
      preserve_unknown_or timeout_status
      detect_ambiguity_andcontradiction
      assign_semantic_confidence
      preserve_versions_andprovenance
      emit_machine_resolvable_mathematical_logic_structure

    ambiguity_controls:
      distinguish_syntax_fromsemantics
      distinguish_term_fromformula
      distinguish_proposition_fromformula_token
      distinguish_well_formedness_fromtruth
      distinguish_truth_fromprovability
      distinguish_satisfiability_fromvalidity
      distinguish_validity_fromtruth_inone_model
      distinguish_entailment_fromderivability_bydefinition
      distinguish_implication_fromcausation
      distinguish_consistency_fromcompleteness
      distinguish_proof_frommodel
      distinguish_heuristic_argument_fromformal_proof
      distinguish_semantic_confidence_fromtruth_value
      preserve_language_logic_theory_scope_semantics_proof_model_version_andprovenance
  }

  PROVENANCE {
    ontology_scope:
      mathematical_logic_formal_language_semantics_inference_proof_andmodel_reasoning

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

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_entity
      ontology_path
      formal_language_model
      logical_symbol_model
      nonlogical_symbol_model
      syntax_model
      semantics_model
      term_model
      variable_model
      constant_model
      function_symbol_model
      predicate_model
      atomic_formula_model
      compound_formula_model
      well_formedness_model
      parse_tree_model
      free_variable_model
      bound_variable_model
      variable_capture_model
      substitution_model
      proposition_model
      sentence_model
      open_formula_model
      connective_model
      negation_model
      conjunction_model
      disjunction_model
      implication_model
      biconditional_model
      truth_value_model
      valuation_model
      truth_table_model
      tautology_model
      contradiction_formula_model
      contingency_model
      logical_equivalence_model
      quantifier_model
      universal_quantifier_model
      existential_quantifier_model
      unique_existence_model
      quantifier_scope_model
      quantifier_order_model
      quantifier_negation_model
      first_order_logic_model
      structure_model
      interpretation_model
      assignment_model
      satisfaction_model
      model_model
      countermodel_model
      satisfiability_model
      unsatisfiability_model
      validity_model
      semantic_consequence_model
      syntactic_derivability_model
      axiom_model
      axiom_schema_model
      premise_model
      assumption_model
      inference_rule_model
      modus_ponens_model
      modus_tollens_model
      universal_instantiation_model
      existential_generalisation_model
      argument_model
      argument_validity_model
      sound_argument_model
      fallacy_model
      affirming_consequent_model
      denying_antecedent_model
      proof_model
      proof_step_model
      theorem_model
      lemma_model
      corollary_model
      direct_proof_model
      contradiction_proof_model
      contraposition_model
      mathematical_induction_model
      soundness_model
      completeness_model
      consistency_model
      inconsistency_model
      explosion_model
      independence_model
      decidability_model
      semidecidability_model
      proof_theory_model
      model_theory_model
      meta_logic_model
      object_language_model
      metalanguage_model
      truth_model
      provability_model
      validity_truth_model
      satisfiability_consistency_model
      classical_logic_model
      nonclassical_logic_model
      intuitionistic_distinction_model
      paraconsistent_distinction_model
      many_valued_distinction_model
      fuzzy_distinction_model
      modal_extension_model
      temporal_extension_model
      equality_model
      identity_model
      logical_equivalence_model
      normal_form_model
      CNF_model
      DNF_model
      literal_model
      clause_model
      resolution_model
      unification_model
      automated_reasoning_model
      theorem_prover_model
      proof_checker_model
      SAT_solver_model
      SMT_model
      model_checking_model
      contradiction_detection_model
      counterexample_model
      ambiguity_model
      semantic_confidence_model
      AI_reasoning_model
      heuristic_model
      explainability_model
      dependency_model
      recursive_validation_model
      version_model
      provenance_model
      interoperability_model
      knowledge_graph_model
      relationships
      semantic_equivalence
      constraints
      logic_record_model
      reasoning_rules
      validation
      confidence_model
      update_model
      AI_consumption_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      formal_language_syntax_orsemantics_change
      term_variable_predicate_orformula_semantics_change
      connective_truth_orquantifier_semantics_change
      first_order_structure_interpretation_ormodel_changes
      satisfiability_validity_entailment_orderivability_changes
      axiom_inference_proof_ortheorem_semantics_change
      consistency_soundness_completeness_ordecidability_changes
      nonclassical_modal_ortemporal_extension_changes
      automated_reasoning_solver_orproof_checker_changes
      AI_reasoning_ambiguity_orconfidence_changes
      logic_record_changes
      constraint_model_changes
      reasoning_rule_changes
  }

}

END FLOW