Mathematical Logic Ontologies

# Mathematical Logic Ontologies — Flow Algebra

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

FLOW MathematicalLogicOntologies {

  DOMAIN:
    AI Core
    -> Ontologies
    -> Mathematical Ontologies
    -> Mathematical Logic Ontologies

  PURPOSE:
    Represent mathematical logic as a machine-interpretable framework for formal languages, propositions, predicates, terms, variables, quantifiers, connectives, truth, satisfaction, models, axioms, inference rules, proofs, theories, consistency, completeness, soundness, decidability, computability and formal semantics. The ontology connects propositional logic, predicate logic, first-order logic, higher-order logic, modal logic, temporal logic, set-theoretic foundations, proof theory, model theory, recursion theory, computability, type theory and automated reasoning while preserving strict distinctions between syntax and semantics, validity and truth, implication and causation, consistency and completeness, theoremhood and satisfiability, proof and evidence, and formal derivation versus heuristic AI inference.

  ENTITY MathematicalLogicOntologies {
    TYPE: FormalReasoningKnowledgeFramework

    PROPERTIES:
      contains_formal_languages
      contains_logical_systems
      contains_propositions
      contains_predicates
      contains_terms
      contains_quantifiers
      contains_connectives
      contains_models
      contains_axioms
      contains_inference_rules
      contains_proofs
      contains_theories
      contains_computability
      supports_automated_reasoning
      supports_semantic_validation
  }

  DOMAIN_HIERARCHY {
    MathematicalLogicOntologies
      -> PropositionalLogic
      -> PredicateLogic
      -> FirstOrderLogic
      -> HigherOrderLogic
      -> ModalLogic
      -> TemporalLogic
      -> ProofTheory
      -> ModelTheory
      -> SetTheoreticFoundations
      -> TypeTheory
      -> RecursionTheory
      -> ComputabilityTheory
      -> AutomatedReasoning
  }

  ENTITY FormalLanguage {
    TYPE: SymbolicLanguage

    COMPONENTS:
      alphabet
      formation_rules
      terms
      formulas
      binding_rules
  }

  ENTITY Formula {
    TYPE: WellFormedExpression

    PROPERTIES:
      has_free_variables
      has_bound_variables
      has_logical_structure
      may_have_truth_value_underinterpretation
  }

  ENTITY Proposition {
    TYPE: TruthEvaluableStatement

    RULE:
      proposition != sentence_string
  }

  ENTITY Predicate {
    TYPE: TruthValuedRelation

    FORM:
      P(x_1,...,x_n)

    RULE:
      predicate_requires_arguments_andinterpretation
  }

  ENTITY LogicalTheory {
    TYPE: FormalSystem

    COMPONENTS:
      language
      axioms
      inference_rules
      theorems
      semantics
  }

  SYNTAX_MODEL {
    components:
      symbols
      terms
      formulas
      derivations

    RULE:
      syntax != semantics
  }

  ALPHABET_MODEL {
    possible:
      logical_symbols
      variables
      constants
      function_symbols
      predicate_symbols
      punctuation

    RULE:
      symbol_class MUST be explicit
  }

  TERM_MODEL {
    possible:
      variable
      constant
      function_application

    RULE:
      term != formula
  }

  ATOMIC_FORMULA_MODEL {
    possible:
      predicate_application
      equality

    RULE:
      atomic_formula_contains_no_logical_connective_structure_beyonddeclared_relation
  }

  WELL_FORMED_FORMULA_MODEL {
    abbreviation:
      WFF

    requirement:
      generated_byformation_rules

    RULE:
      parseable_text != well_formed_formula_unlessgrammar_satisfied
  }

  PARSING_MODEL {
    input:
      logical_expression

    output:
      abstract_syntax_tree

    RULE:
      precedence_associativity_parentheses_andbinding MUST be preserved
  }

  VARIABLE_MODEL {
    possible:
      free
      bound

    RULE:
      variable_scope MUST be preserved
  }

  FREE_VARIABLE_MODEL {
    variable:
      occurrence_not_bound_byquantifier_orbinder

    RULE:
      open_formula_may_depend_onassignment
  }

  BOUND_VARIABLE_MODEL {
    variable:
      occurrence_within_scope_ofbinder

    RULE:
      alpha_renaming_preserves_meaning_when_capture_avoided
  }

  VARIABLE_CAPTURE_MODEL {
    risk:
      substitution_places_free_variable_underbinder

    RULE:
      capture_avoiding_substitution MUST be used
  }

  SUBSTITUTION_MODEL {
    formula:
      phi

    substitution:
      x := t

    RULE:
      substitution MUST respect_binding_andterm_admissibility
  }

  PROPOSITIONAL_LOGIC_MODEL {
    atoms:
      p_1...p_n

    connectives:
      NOT
      AND
      OR
      IMPLIES
      IFF

    RULE:
      truth_functional_semantics MUST be preserved
  }

  NEGATION_MODEL {
    NOT A

    truth:
      opposite_truth_value_underclassical_two_valued_semantics
  }

  CONJUNCTION_MODEL {
    A AND B

    true_when:
      A_true
      AND B_true
  }

  DISJUNCTION_MODEL {
    A OR B

    classical_default:
      inclusive_OR

    RULE:
      exclusive_OR_requires_explicit_operator
  }

  IMPLICATION_MODEL {
    A -> B

    classical_equivalence:
      NOT A OR B

    RULE:
      material_implication != causal_implication
  }

  BICONDITIONAL_MODEL {
    A <-> B

    true_when:
      A_and_B_have_same_truth_value
  }

  XOR_MODEL {
    A XOR B

    true_when:
      exactly_one_operand_true
  }

  NAND_MODEL {
    A NAND B =
      NOT(A AND B)
  }

  NOR_MODEL {
    A NOR B =
      NOT(A OR B)
  }

  TRUTH_TABLE_MODEL {
    inputs:
      propositional_variables

    output:
      truth_value_forallvaluations

    RULE:
      valuation_order SHOULD be preserved
  }

  TAUTOLOGY_MODEL {
    formula:
      true_underallvaluations

    RULE:
      tautology_is_semantic_property
  }

  CONTRADICTION_FORMULA_MODEL {
    formula:
      false_underallvaluations

    RULE:
      contradiction_formula != inconsistent_theory_inallcontexts
  }

  CONTINGENCY_MODEL {
    formula:
      true_under_some_andfalse_underother_valuations
  }

  LOGICAL_EQUIVALENCE_MODEL {
    A equivalent B

    condition:
      same_truth_value_underallrelevant_interpretations

    RULE:
      logical_equivalence != syntactic_identity
  }

  PROPOSITIONAL_SATISFIABILITY_MODEL {
    formula:
      satisfiable_if_somevaluation_makes_true

    RULE:
      satisfiable != valid
  }

  SAT_MODEL {
    problem:
      determine_satisfiability_ofpropositional_formula

    RULE:
      satisfying_assignment SHOULD be preserved_whenfound
  }

  CNF_MODEL {
    full_name:
      Conjunctive_Normal_Form

    structure:
      conjunction_ofclauses
  }

  DNF_MODEL {
    full_name:
      Disjunctive_Normal_Form

    structure:
      disjunction_ofconjunctions
  }

  CLAUSE_MODEL {
    expression:
      disjunction_ofliterals

    RULE:
      empty_clause_represents_false_inresolution_context
  }

  LITERAL_MODEL {
    possible:
      atomic_proposition
      negated_atomic_proposition
  }

  NORMAL_FORM_TRANSFORMATION_MODEL {
    source:
      formula

    target:
      CNF_orDNF

    RULE:
      transformation MUST preserve_equivalence_orpreserve_satisfiability_asdeclared
  }

  PREDICATE_LOGIC_MODEL {
    components:
      predicates
      terms
      quantifiers
      variables
      domains

    RULE:
      interpretation_domain MUST be explicit
  }

  UNIVERSAL_QUANTIFIER_MODEL {
    form:
      forall x phi(x)

    semantics:
      phi_true_forevery_element_indomain
  }

  EXISTENTIAL_QUANTIFIER_MODEL {
    form:
      exists x phi(x)

    semantics:
      phi_true_foratleast_one_element_indomain
  }

  UNIQUE_EXISTENCE_MODEL {
    form:
      exists_exactly_one x phi(x)

    RULE:
      existence_anduniqueness_are_distinct_conditions
  }

  QUANTIFIER_SCOPE_MODEL {
    quantifier:
      binds_occurrences_within_scope

    RULE:
      scope_changes_can_change_formula_meaning
  }

  QUANTIFIER_NEGATION_MODEL {
    NOT(forall x phi) equivalent exists x NOT phi
    NOT(exists x phi) equivalent forall x NOT phi

    RULE:
      equivalence_assumes_standard_classical_quantifier_semantics
  }

  FIRST_ORDER_LOGIC_MODEL {
    quantification:
      overindividuals_in_domain

    symbols:
      constants
      functions
      predicates

    RULE:
      predicates_andfunctions_are_not_quantified_over_instandard_first_order_logic
  }

  FIRST_ORDER_STRUCTURE_MODEL {
    M:
      domain
      interpretations_ofconstants
      interpretations_offunctions
      interpretations_ofpredicates

    RULE:
      structure != theory
  }

  ASSIGNMENT_MODEL {
    variable_assignment:
      maps_variables_to_domain_elements

    RULE:
      satisfaction_ofopen_formula_depends_onassignment
  }

  SATISFACTION_MODEL {
    notation:
      M,s satisfies phi

    RULE:
      satisfaction_relates_structure_assignment_andformula
  }

  SENTENCE_MODEL {
    formula:
      no_free_variables

    RULE:
      truth_instructure_is_assignment_independent
  }

  VALIDITY_MODEL {
    formula:
      true_inallmodels_underdeclared_logic

    RULE:
      validity != theoremhood_unlessproof_system_isappropriately_sound_andcomplete
  }

  MODEL_MODEL {
    structure:
      satisfies_theory_orformula

    RULE:
      mathematical_model_here != empirical_scientific_model
  }

  COUNTERMODEL_MODEL {
    structure:
      makes_candidate_valid_formula_false

    RULE:
      one_countermodel_refutes_semantic_validity
  }

  THEORY_MODEL {
    T:
      set_ofsentences

    RULE:
      theory_may_havezero_one_ormanymodels
  }

  CONSEQUENCE_MODEL {
    semantic:
      T entails phi

    meaning:
      every_model_ofT_is_model_ofphi
  }

  SYNTACTIC_DERIVABILITY_MODEL {
    notation:
      T proves phi

    meaning:
      formal_derivation_exists_fromT

    RULE:
      semantic_consequence_andsyntactic_derivability_are_distinct
  }

  SOUNDNESS_MODEL {
    proof_system:
      if T proves phi
      then T entails phi

    RULE:
      soundness_prevents_derivation_ofsemantically_invalid_conclusions
  }

  COMPLETENESS_MODEL {
    proof_system:
      if T entails phi
      then T proves phi

    RULE:
      completeness != consistency
  }

  CONSISTENCY_MODEL {
    theory:
      does_not_derive_contradiction_underdeclared_system

    possible_semantic:
      has_model_underappropriate_logic

    RULE:
      syntactic_andsemantic_consistency_relationship_depends_onmetatheory
  }

  INCONSISTENCY_MODEL {
    theory:
      derives_contradiction

    RULE:
      classical_explosion_behavior_depends_onlogic
  }

  EXPLOSION_MODEL {
    classical:
      from contradiction derive arbitrary_formula

    RULE:
      paraconsistent_logics_may_reject_explosion
  }

  INDEPENDENCE_MODEL {
    statement:
      neither_statement_nornegation_derivable_fromtheory

    RULE:
      independence_is_relative_toformal_system
  }

  AXIOM_MODEL {
    formula:
      adopted_asstarting_formula_oraxiom_schema

    RULE:
      axiom != theorem_byderivation_withinthe same presentation
  }

  AXIOM_SCHEMA_MODEL {
    pattern:
      generates_instances

    RULE:
      schema != single_formula
  }

  INFERENCE_RULE_MODEL {
    premises:
      P_1...P_n

    conclusion:
      C

    RULE:
      rule_application MUST satisfy_side_conditions
  }

  MODUS_PONENS_MODEL {
    premises:
      A
      A -> B

    conclusion:
      B
  }

  MODUS_TOLLENS_MODEL {
    premises:
      A -> B
      NOT B

    conclusion:
      NOT A

    RULE:
      derivability_depends_onavailable_rules_or equivalences
  }

  UNIVERSAL_INSTANTIATION_MODEL {
    premise:
      forall x phi(x)

    conclusion:
      phi(t)

    RULE:
      term_t MUST be admissible_forx
  }

  EXISTENTIAL_GENERALISATION_MODEL {
    premise:
      phi(t)

    conclusion:
      exists x phi(x)
  }

  PROOF_MODEL {
    sequence_orstructure:
      formulas
      justified_byaxioms_premises_orinference_rules

    RULE:
      proof_steps MUST preserve_justification
  }

  FORMAL_PROOF_MODEL {
    fields:
      proof_identifier
      theory
      conclusion
      steps
      dependencies
      proof_system

    RULE:
      proof_validity_requires_machine_checkable_orformally_valid_steps_underdeclared_system
  }

  DIRECT_PROOF_MODEL {
    method:
      derive_conclusion_frompremises
  }

  PROOF_BY_CONTRADICTION_MODEL {
    method:
      assume_negation
      derive_contradiction
      infer_target_underappropriate_logic

    RULE:
      classical_validity_ofmethod_depends_onlogic
  }

  PROOF_BY_CONTRAPOSITION_MODEL {
    target:
      A -> B

    prove:
      NOT B -> NOT A

    RULE:
      equivalence_assumes_classical_orappropriate_logic
  }

  MATHEMATICAL_INDUCTION_MODEL {
    base_case:
      P(n_0)

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

    conclusion:
      forall n>=n_0 P(n)

    RULE:
      induction_principle_depends_onunderlying_number_structure
  }

  STRUCTURAL_INDUCTION_MODEL {
    base:
      primitive_structures

    step:
      constructors

    RULE:
      induction_follows_recursive_generation_ofobjects
  }

  PROOF_THEORY_MODEL {
    studies:
      formal_proofs
      derivations
      proof_systems
      normalisation
      cut_elimination

    RULE:
      proof_object_andsemantic_model_are_distinct
  }

  NATURAL_DEDUCTION_MODEL {
    proof:
      introduction_andelimination_rules

    RULE:
      discharged_assumptions_andscope MUST be preserved
  }

  SEQUENT_CALCULUS_MODEL {
    sequent:
      Gamma |- Delta

    RULE:
      structural_andlogical_rules MUST be explicit
  }

  CUT_RULE_MODEL {
    generic:
      from Gamma |- A
      and A,Delta |- C
      infer Gamma,Delta |- C

    RULE:
      exact_form_depends_onsequent_system
  }

  CUT_ELIMINATION_MODEL {
    property:
      proofs_withcut_can_be transformed_tocut_free_form_underrelevant_system

    RULE:
      theorem_is_system_specific
  }

  MODEL_THEORY_MODEL {
    studies:
      structures
      satisfaction
      definability
      elementary_equivalence
      theories

    RULE:
      syntax_semantics_relation MUST be preserved
  }

  ELEMENTARY_EQUIVALENCE_MODEL {
    structures:
      M
      N

    condition:
      satisfy_same_first_order_sentences

    RULE:
      elementary_equivalence != isomorphism
  }

  ISOMORPHISM_MODEL {
    structures:
      mapping_preserving_relevant_relations_functions_andconstants

    RULE:
      isomorphic_structures_are_elementarily_equivalent_understandard_first_order_setting
  }

  DEFINABILITY_MODEL {
    relation_orset:
      describable_byformula_instructure

    RULE:
      definability_depends_onlanguage_andparameters
  }

  COMPACTNESS_MODEL {
    first_order_logic:
      if_every_finite_subset_ofT_is_satisfiable
      then T_is_satisfiable

    RULE:
      compactness_is_logic_specific
  }

  LOWENHEIM_SKolem_MODEL {
    first_order_logic:
      model_cardinality_properties

    RULE:
      exact_theorem_variant_andassumptions MUST be identified
  }

  HIGHER_ORDER_LOGIC_MODEL {
    quantification:
      may_range_overpredicates_functions_sets_orrelations

    RULE:
      higher_order_semantics_may_be standard_orHenkin_andmust_be declared
  }

  TYPE_THEORY_MODEL {
    entities:
      terms
      types
      judgments
      functions
      dependent_types

    RULE:
      well_typedness_isformal_constraint
  }

  TYPING_JUDGMENT_MODEL {
    form:
      Gamma |- t : T

    RULE:
      context_Gamma MUST be preserved
  }

  SIMPLE_TYPE_MODEL {
    types:
      base_types
      function_types

    RULE:
      type_assignment_constrains_application
  }

  DEPENDENT_TYPE_MODEL {
    type:
      may_depend_onterm

    RULE:
      dependency_scope MUST be preserved
  }

  LAMBDA_CALCULUS_MODEL {
    terms:
      variable
      abstraction
      application

    RULE:
      alpha_beta_eta_relations MUST be distinguished
  }

  ALPHA_EQUIVALENCE_MODEL {
    terms:
      differ_only_inbound_variable_names

    RULE:
      alpha_equivalence_preserves_binding_structure
  }

  BETA_REDUCTION_MODEL {
    expression:
      (lambda x.M) N

    reduces_to:
      M[x:=N]

    RULE:
      substitution MUST avoidcapture
  }

  ETA_MODEL {
    relation:
      function_extensionality_related_reduction_or equivalence

    RULE:
      applicability_depends_oncalculus
  }

  SET_THEORY_MODEL {
    entities:
      set
      membership
      subset
      union
      intersection
      power_set
      cardinal
      ordinal

    RULE:
      foundational_axiom_system MUST be identified
  }

  MEMBERSHIP_MODEL {
    relation:
      x in A

    RULE:
      membership != subset_relation
  }

  SUBSET_MODEL {
    A subseteq B

    condition:
      forall x (x in A -> x in B)
  }

  EXTENSIONALITY_MODEL {
    sets:
      equal_when_same_members_underextensional_set_theory

    RULE:
      depends_onaxiomatic_framework
  }

  EMPTY_SET_MODEL {
    set:
      no_elements
  }

  POWER_SET_MODEL {
    P(A):
      set_ofallsubsets_ofA
  }

  CARDINALITY_MODEL {
    concept:
      size_ofset

    RULE:
      finite_andinfinite_cardinality_semantics MUST be distinguished
  }

  ORDINAL_MODEL {
    concept:
      order_type_ortransfinite_position_underformal_definition
    RULE:
      ordinal != cardinal
  }

  MODAL_LOGIC_MODEL {
    operators:
      necessity
      possibility

    notation:
      box
      diamond

    RULE:
      modal_semantics_requires_accessibility_orotherdeclared_semantics
  }

  KRIPKE_MODEL {
    frame:
      worlds
      accessibility_relation

    model:
      frame_plusvaluation

    RULE:
      modal_truth_is_world_relative
  }

  NECESSITY_MODEL {
    box A:
      true_atworld_ifA_true_atallaccessible_worlds
      understandard_Kripke_semantics
  }

  POSSIBILITY_MODEL {
    diamond A:
      true_atworld_ifA_true_atsomeaccessible_world
      understandard_Kripke_semantics
  }

  TEMPORAL_LOGIC_MODEL {
    operators:
      always
      eventually
      next
      until
      since

    RULE:
      temporal_logic_time_structure MUST be declared
  }

  LINEAR_TEMPORAL_LOGIC_MODEL {
    abbreviation:
      LTL

    path:
      linear_sequence_ofstates
  }

  COMPUTATION_TREE_LOGIC_MODEL {
    branching:
      possible_future_paths

    RULE:
      CTL_andLTL_semantics_are_distinct
  }

  COMPUTABILITY_MODEL {
    studies:
      effectively_computable_functions
      algorithms
      decidability
      reducibility

    RULE:
      computable != computationally_feasible
  }

  TURING_MACHINE_MODEL {
    components:
      states
      tape
      alphabet
      transition_function
      head

    RULE:
      machine_variant_andencoding SHOULD be declared
  }

  RECURSIVE_FUNCTION_MODEL {
    possible:
      primitive_recursive
      partial_recursive
      total_recursive

    RULE:
      totality MUST be distinguished_frompartiality
  }

  DECIDABILITY_MODEL {
    problem:
      decidable_ifalgorithm_halts_withcorrect_yes_no_answer_forallinputs

    RULE:
      decidable != efficiently_decidable
  }

  SEMIDECIDABILITY_MODEL {
    language:
      recognisable_ifmachine_accepts_members_andmay_not_halt_onnonmembers

    RULE:
      recognisable != decidable
  }

  UNDECIDABILITY_MODEL {
    problem:
      no_total_correct_decision_algorithm_underformal_model

    RULE:
      undecidability_claim_requires_reduction_ortheorem
  }

  HALTING_PROBLEM_MODEL {
    property:
      undecidable_forgeneral_Turing_machine_input_pairs

    RULE:
      result_applies_toformal_problem_not every_program_analysis_question
  }

  REDUCTION_MODEL {
    source_problem:
      A

    target_problem:
      B

    purpose:
      transfer_decidability_orcomplexity_information

    RULE:
      reduction_direction MUST be preserved
  }

  GODEL_INCOMPLETENESS_MODEL {
    context:
      sufficiently_expressive_consistent_effectively_axiomatised_formal_systems

    RULE:
      theorem_conditions MUST be preserved
      andincompleteness != arbitrary_unprovability
  }

  FIRST_INCOMPLETENESS_MODEL {
    result:
      underappropriate_conditions_some_true_orsemantically_valid_arithmetical_statements_are_notprovable_withinsystem

    RULE:
      exact_formulation_depends_onmetatheoretic_assumptions
  }

  SECOND_INCOMPLETENESS_MODEL {
    result:
      underappropriate_conditions_system_cannot_prove_itsown_consistency

    RULE:
      consistency_statement_andcoding_framework MUST be specified
  }

  CHURCH_TURING_MODEL {
    thesis:
      effectively_calculable_corresponds_toTuring_computable_understandard_interpretation

    RULE:
      thesis != formal_theorem
  }

  DECISION_PROCEDURE_MODEL {
    algorithm:
      decides_formula_property_underlogic_fragment

    RULE:
      applicability_fragment MUST be explicit
  }

  AUTOMATED_REASONING_MODEL {
    processes:
      theorem_proving
      satisfiability
      model_checking
      constraint_solving
      proof_checking

    RULE:
      automated_result MUST preserve_solver_configuration_andproof_orcertificate_whenavailable
  }

  RESOLUTION_MODEL {
    clauses:
      (A OR C)
      (NOT A OR D)

    resolvent:
      (C OR D)

    RULE:
      standardisation_apart_andunification_conditions MUST be respected_infirst_order_resolution
  }

  UNIFICATION_MODEL {
    expressions:
      find_substitution_making_terms_equal

    output:
      unifier

    RULE:
      occurs_check_andterm_theory_context MUST be preserved
  }

  MOST_GENERAL_UNIFIER_MODEL {
    abbreviation:
      MGU

    RULE:
      MGU_is_unique_uptorenaming_understandard_first_order_unification
  }

  FORWARD_CHAINING_MODEL {
    process:
      apply_rules_fromknown_facts_toderive_newfacts

    RULE:
      termination_depends_onrule_language_anddomain
  }

  BACKWARD_CHAINING_MODEL {
    process:
      start_fromgoal_andseek_supporting_premises

    RULE:
      search_strategy != logical_validity
  }

  THEOREM_PROVER_MODEL {
    inputs:
      axioms
      conjecture

    outputs:
      proof
      refutation
      unknown
      timeout

    RULE:
      timeout != false
  }

  PROOF_CHECKER_MODEL {
    input:
      formal_proof

    output:
      valid_orinvalid_underdeclared_kernel

    RULE:
      proof_checker_correctness_depends_ontrusted_kernel
  }

  SAT_SOLVER_MODEL {
    input:
      propositional_formula

    output:
      satisfiable_withassignment
      orunsatisfiable_withoptional_certificate

    RULE:
      solver_result SHOULD preserve_formula_version_andconfiguration
  }

  SMT_MODEL {
    full_name:
      Satisfiability_Modulo_Theories

    theories:
      arithmetic
      arrays
      bitvectors
      uninterpreted_functions
      other_supported_theories

    RULE:
      theory_combination_andfragment MUST be identified
  }

  MODEL_CHECKING_MODEL {
    system:
      transition_structure

    property:
      temporal_orstate_formula

    output:
      satisfied
      violated_withcounterexample
      unknown

    RULE:
      abstraction_andstate_space_scope MUST be preserved
  }

  CONSTRAINT_LOGIC_MODEL {
    variables:
      domains

    constraints:
      logical_andmathematical_relations

    RULE:
      satisfiability_depends_onconstraint_theory
  }

  NONCLASSICAL_LOGIC_MODEL {
    possible:
      intuitionistic
      paraconsistent
      many_valued
      fuzzy
      relevance

    RULE:
      classical_inference_rules MUST not_be assumed_universally
  }

  INTUITIONISTIC_LOGIC_MODEL {
    principle:
      proof_constructive_semantics

    RULE:
      law_ofexcluded_middle_not generally_available
  }

  PARACONSISTENT_LOGIC_MODEL {
    principle:
      contradiction_does_notnecessarily_entail_every_formula

    RULE:
      explosion_rule MUST not_be assumed
  }

  MANY_VALUED_LOGIC_MODEL {
    truth_values:
      more_than_two

    RULE:
      truth_operations_depend_onlogic
  }

  FUZZY_LOGIC_MODEL {
    truth_degree:
      value_inordered_scale_commonly_[0,1]

    RULE:
      fuzzy_truth_degree != probability
  }

  DESCRIPTION_LOGIC_MODEL {
    entities:
      concepts
      roles
      individuals

    reasoning:
      subsumption
      satisfiability
      instance_checking

    RULE:
      expressivity_fragment_determines_reasoning_properties
  }

  ONTOLOGY_LOGIC_MODEL {
    ontology:
      formal_vocabulary
      classes
      relations
      axioms

    RULE:
      ontology_semantics MUST be linked_tochosen_logic
  }

  OPEN_WORLD_MODEL {
    principle:
      absence_ofstatement_does_notimplyfalse

    RULE:
      open_world != closed_world
  }

  CLOSED_WORLD_MODEL {
    principle:
      underdeclared_database_orlogic_context_missing_fact_may_be treated_asfalse

    RULE:
      closed_world_assumption MUST be explicit
  }

  MONOTONIC_REASONING_MODEL {
    property:
      adding_premises_does_not invalidate_previous_conclusions

    RULE:
      classical_logic_is_monotonic
  }

  NONMONOTONIC_REASONING_MODEL {
    property:
      new_information_may retract_conclusion

    RULE:
      retraction_semantics MUST be explicit
  }

  DEFAULT_LOGIC_MODEL {
    defaults:
      defeasible_rules

    RULE:
      default_conclusion != theorem_ofclassical_logic
  }

  BELIEF_REVISION_MODEL {
    knowledge_state:
      updated_afternew_information

    RULE:
      revision_policy MUST preserve_reason_forretraction_orchange
  }

  META_LOGIC_MODEL {
    studies:
      properties_oflogical_systems

    possible:
      soundness
      completeness
      consistency
      compactness
      decidability

    RULE:
      object_language_andmetalanguage MUST be distinguished
  }

  OBJECT_LANGUAGE_MODEL {
    language:
      being_reasoned_about

    RULE:
      object_language_expression != metalinguistic_statement
  }

  METALANGUAGE_MODEL {
    language:
      used_to describe_formal_language_andsemantics

    RULE:
      level_ofdiscourse MUST be preserved
  }

  TRUTH_MODEL {
    semantic:
      truth_relative_tointerpretation_orstructure

    RULE:
      truth != provability
  }

  PROVABILITY_MODEL {
    syntactic:
      existence_ofderivation

    RULE:
      provability != truth_withoutmetatheoretic_conditions
  }

  VALIDITY_TRUTH_DISTINCTION_MODEL {
    validity:
      truth_underallinterpretations

    truth:
      truth_underparticular_interpretation

    RULE:
      terms MUST not_be interchanged
  }

  SATISFIABILITY_CONSISTENCY_MODEL {
    satisfiability:
      semantic_property_offormula_ortheory

    consistency:
      syntactic_property_underproof_system

    RULE:
      equivalence_betweenconcepts_requires_soundness_completeness_context
  }

  FORMAL_SEMANTICS_MODEL {
    mapping:
      syntax -> meaning_underinterpretation

    RULE:
      semantic_evaluation_requiresmodel_orvaluation
  }

  DENOTATION_MODEL {
    expression:
      denotes_entity_orvalue_underinterpretation

    RULE:
      denotation_may_change_acrossmodels
  }

  INTERPRETATION_MODEL {
    assigns:
      constants -> elements
      functions -> functions
      predicates -> relations

    RULE:
      interpretation_domain MUST be nonempty_understandard_first_order_semantics
  }

  SEMANTIC_EQUIVALENCE_MODEL {
    formulas:
      same_semantic_value_underallrelevant_models

    RULE:
      equivalence_scope_andlogic MUST be explicit
  }

  SYNTACTIC_EQUIVALENCE_MODEL {
    formulas:
      same_or transformable_bydeclared_syntactic_rules

    RULE:
      syntactic_similarity != semantic_equivalence
  }

  PROOF_EQUIVALENCE_MODEL {
    proofs:
      may_derive_same_conclusion_via different_steps

    RULE:
      same_conclusion != same_proof_object
  }

  DEPENDENCY_GRAPH_MODEL {
    nodes:
      axioms
      definitions
      lemmas
      theorems
      proofs
      models

    edges:
      USES
      DERIVES
      ENTAILS
      SATISFIES
      REFUTES
      DEPENDS_ON

    RULE:
      logical_dependency MUST be typed
  }

  CONTRADICTION_DETECTION_MODEL {
    theory:
      search_forphi_andNOT_phi_orformal_falsehood

    RULE:
      contradiction_detection_method_depends_onlogic
  }

  AMBIGUITY_MODEL {
    causes:
      symbol_overloading
      quantifier_scope
      implicit_domain
      unspecified_logic
      precedence

    RULE:
      ambiguous_formula MUST not_be assigned_single_semantics_withoutbasis
  }

  CONFIDENCE_MODEL {
    semantic_confidence:
      support_forparsed_orinterpreted_logic_expression

    RULE:
      confidence != logical_truth_value
  }

  AI_LOGIC_MODEL {
    AI:
      parse
      resolve_symbols
      identify_logic
      construct_formula
      validate_well_formedness
      infer
      search_forproof
      search_forcountermodel
      explain

    RULE:
      AI_heuristic_confidence MUST not_be confused_withformal_validity
  }

  HEURISTIC_REASONING_MODEL {
    process:
      guides_search_withoutguaranteeing_validity

    RULE:
      heuristic_step MUST not_be emitted_asformal_inference_unlessverified
  }

  FORMAL_VERIFICATION_MODEL {
    claim:
      checked_againstformal_specification

    RULE:
      verification_scope_andassumptions MUST be preserved
  }

  PROOF_CERTIFICATE_MODEL {
    artifact:
      independently_checkable_evidence_ofsolver_result

    RULE:
      certificate_format_andchecker MUST be identified
  }

  COUNTEREXAMPLE_MODEL {
    entity:
      interpretation_input_orstate_violatinguniversal_claim

    RULE:
      valid_counterexample_refutes_claim_withinitsdeclared_scope
  }

  EXPLANATION_MODEL {
    logical_result:
      should_expose
        premises
        rules
        assumptions
        proof_orcountermodel
        logic_system

    RULE:
      explanation MUST distinguish_formal_result_fromheuristic_search
  }

  VERSION_MODEL {
    versioned:
      logic_definition
      axiom_system
      proof_system
      ontology
      solver
      theory

    RULE:
      logical_result MUST preserve_versions_used
  }

  PROVENANCE_MODEL {
    fields:
      source
      formal_system
      axioms
      rules
      solver_orchecker
      proof
      model
      version
      author_oragent

    RULE:
      derivation MUST remain_provenance_linked
  }

  VALIDATION_MODEL {
    checks:
      syntax
      binding
      scope
      types
      logic_system
      axioms
      inference_rules
      proof_steps
      semantic_model
      version

    RULE:
      validation MUST matchformal_system
  }

  UPDATE_MODEL {
    ON new_formula:
      parse
      resolve_symbols
      identify_logic
      validate_well_formedness
      attach_semantics
      preserve_provenance

    ON new_proof:
      preserve_premises
      validate_each_step
      attach_dependencies
      register_conclusion

    ON new_model:
      define_domain_andinterpretation
      evaluate_satisfaction
      attach_countermodel_orwitness_whenrelevant

    ON axiom_orlogic_revision:
      preserve_previous_system
      register_new_version
      identify_dependent_theorems_andproofs
      revalidate_undernew_system_withoutoverwritinghistory

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  INTEROPERABILITY_MODEL {
    exchange:
      formulas
      symbols
      variable_binding
      proof_objects
      models
      theory_identifiers
      versions

    RULE:
      text_exchange_withoutformal_semantics_is_incomplete
  }

  METADATA_MODEL {
    fields:
      logic_entity_identifier
      logic_system
      language_identifier
      formula
      syntax_tree
      free_variables
      bound_variables
      domain
      interpretation
      axioms
      inference_rules
      proof_identifier
      model_identifier
      satisfiability_status
      validity_status
      consistency_status
      decidability_status
      solver_identifier
      proof_certificate
      countermodel
      semantic_confidence
      version
      provenance

    RULE:
      logic_metadata MUST support_reconstruction_offormal_reasoning
  }

  RELATIONSHIPS {
    MathematicalLogicOntologies -> CONTAINS -> LogicalSystem
    LogicalSystem -> HAS -> FormalLanguage
    FormalLanguage -> CONTAINS -> Formula
    Theory -> HAS -> Axiom
    Theory -> MAY_DERIVE -> Theorem
    Proof -> DERIVES -> Conclusion
    Proof -> USES -> InferenceRule
    Structure -> SATISFIES -> Formula
    Countermodel -> REFUTES -> ValidityClaim
    Solver -> PRODUCES -> Result
    ProofCertificate -> SUPPORTS -> SolverResult
  }

  SEMANTIC_EQUIVALENCE {
    "mathematical logic" ~= formal_study_ofreasoning_languages_proofs_andmodels
    "logical validity" ~= truth_underallinterpretations_in declared_logic
    "provability" ~= existence_offormal_derivation_in declared_proof_system

    syntax != semantics
    truth != provability
    validity != truth_inparticular_model
    satisfiable != valid
    consistency != completeness
    soundness != completeness
    implication != causation
    theorem != model
    proof != heuristic_argument
    fuzzy_truth_degree != probability
    computable != computationally_feasible
  }

  CONSTRAINTS {
    logic_system_must_be identified = true
    syntax_andsemantics_must_be distinguished = true
    variable_binding_andquantifier_scope_must_be preserved = true
    theoremhood_andsemantic_validity_must_be distinguished = true
    satisfiability_andvalidity_must_be distinguished = true
    consistency_soundness_andcompleteness_must_be distinguished = true
    inference_rules_must_preserve_side_conditions = true
    proof_steps_must_be justified = true
    object_language_andmetalanguage_must_be distinguished = true
    nonclassical_logics_must_not_inherit_classical_rules_withoutbasis = true
    automated_solver_timeout_must_not_be interpreted_asfalse = true
    heuristic_AI_reasoning_must_not_be represented_asformal_proof_unlessverified = true
    versions_andprovenance_must_be preserved = true
  }

  LOGIC_MODEL {
    RECORD:
      logic_entity_identifier
      logic_domain
      formal_language
      formula
      syntax_tree
      term_structure
      free_variables
      bound_variables
      quantifier_scope
      interpretation_domain
      valuation
      model
      theory
      axioms
      inference_rules
      premises
      conclusion
      proof_identifier
      proof_steps
      semantic_consequence
      syntactic_derivability
      satisfiability_status
      validity_status
      consistency_status
      soundness_context
      completeness_context
      decidability_status
      computability_context
      solver_identifier
      solver_result
      proof_certificate
      countermodel
      ambiguity_status
      semantic_confidence
      version
      provenance

    RULE:
      logical_record MUST remain_linked_toformal_system_language_semantics_proof_or model_andprovenance
  }

  REASONING_RULES {
    IF logical_expression_received
      THEN identify_logic_system
      parse_binding_scope_andconnectives
      validate_well_formedness

    IF formula_is_propositional
      THEN MAY evaluate_truth_table_SAT
      ornormal_form_underdeclared_semantics

    IF formula_is_first_order
      THEN resolve_domain_symbols_assignments_andquantifiers
      before_satisfaction_reasoning

    IF proof_claimed
      THEN verify_each_step_againstaxioms_premises_rules_andside_conditions

    IF validity_claimed
      THEN require_semantic_argument_proof_undercomplete_system_orabsence_ofcountermodel_undercomplete_decision_context

    IF countermodel_found
      THEN reject_validity_claim_withinscope

    IF theory_inconsistent_underclassical_logic
      THEN flag_explosion_risk
      rather_thanderive_arbitrary_claim_withoutcontext

    IF nonclassical_logic_selected
      THEN load_logic_specific_truth_andinference_rules

    IF automated_solver_returns_unknown_or timeout
      THEN preserve_unknown_status
      anddo_not infer_falsehood

    IF heuristic_AI_generates_candidate_proof
      THEN validate_withformal_rules_orproof_checker_beforemarking_proven

    IF axiom_orproof_system_changes
      THEN recursively_revalidate_dependent_theorems_andcertificates
  }

  VALIDATION {
    CHECK logic_entity_identifier
    CHECK logic_system
    CHECK formal_language
    CHECK syntax_tree
    CHECK term_formula_distinction
    CHECK free_andbound_variables
    CHECK variable_capture
    CHECK quantifier_scope
    CHECK domain_andinterpretation
    CHECK valuation
    CHECK satisfiability
    CHECK validity
    CHECK semantic_consequence
    CHECK syntactic_derivability
    CHECK axioms
    CHECK inference_rules
    CHECK proof_steps
    CHECK soundness_andcompleteness_context
    CHECK consistency
    CHECK model_orcountermodel
    CHECK decidability_andcomputability_context
    CHECK solver_result
    CHECK proof_certificate
    CHECK nonclassical_logic_semantics
    CHECK ambiguity
    CHECK semantic_confidence
    CHECK version
    CHECK provenance
  }

  CONFIDENCE {
    syntax_parsing_confidence: 1.00
    propositional_logic_confidence: 1.00
    first_order_logic_confidence: 1.00
    proof_theory_confidence: 1.00
    model_theory_confidence: 1.00
    computability_confidence: 1.00
    nonclassical_logic_confidence: 0.99
    automated_reasoning_confidence: 0.99

    RULE:
      confidence decreases when logic system, scope, domain,
      axioms, proof rules, semantic model, solver status,
      version orprovenance is missing or ambiguous
  }

  AI_CONSUMPTION {
    canonical_entity: MathematicalLogicOntologies
    canonical_role: FormalReasoningKnowledgeFramework

    interpretation_order:
      identify_logic_system
      identify_formal_language
      parse_terms_formulas_andbinding
      resolve_variables_quantifiers_andconnectives
      distinguish_syntax_fromsemantics
      resolve_domain_interpretation_andvaluation
      determine_satisfiability_validity_orconsequence_question
      identify_theory_axioms_andinference_rules
      verify_orconstruct_formal_derivation
      search_forcountermodel_whenappropriate
      distinguish_truth_provability_validity_consistency_soundness_andcompleteness
      resolve_computability_or decidability_context
      identify_automated_solver_orproof_checker
      preserve_unknown_timeout_orpartial_results
      detect_ambiguity_orcontradiction
      assign_semantic_confidence
      preserve_versions_andprovenance
      emit_machine_resolvable_logic_structure

    ambiguity_controls:
      distinguish_syntax_fromsemantics
      distinguish_truth_fromprovability
      distinguish_validity_fromtruth_inparticular_model
      distinguish_satisfiable_fromvalid
      distinguish_consistency_fromcompleteness
      distinguish_soundness_fromcompleteness
      distinguish_implication_fromcausation
      distinguish_theorem_frommodel
      distinguish_proof_fromheuristic_argument
      distinguish_fuzzy_truth_degree_fromprobability
      distinguish_computable_fromcomputationally_feasible
      preserve_logic_language_scope_semantics_proof_model_version_andprovenance
  }

  PROVENANCE {
    ontology_scope:
      mathematical_logic_formal_languages_proofs_models_computability_andAI_reasoning

    parent_path:
      AI Core
      -> Ontologies
      -> Mathematical Ontologies
      -> Mathematical Logic Ontologies

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_entity
      ontology_path
      domain_hierarchy
      syntax_model
      alphabet_model
      term_model
      atomic_formula_model
      well_formed_formula_model
      parsing_model
      variable_model
      free_variable_model
      bound_variable_model
      variable_capture_model
      substitution_model
      propositional_logic_model
      negation_model
      conjunction_model
      disjunction_model
      implication_model
      biconditional_model
      XOR_model
      NAND_model
      NOR_model
      truth_table_model
      tautology_model
      contradiction_formula_model
      contingency_model
      logical_equivalence_model
      propositional_satisfiability_model
      SAT_model
      CNF_model
      DNF_model
      clause_model
      literal_model
      normal_form_transformation_model
      predicate_logic_model
      universal_quantifier_model
      existential_quantifier_model
      unique_existence_model
      quantifier_scope_model
      quantifier_negation_model
      first_order_logic_model
      first_order_structure_model
      assignment_model
      satisfaction_model
      sentence_model
      validity_model
      model_model
      countermodel_model
      theory_model
      consequence_model
      syntactic_derivability_model
      soundness_model
      completeness_model
      consistency_model
      inconsistency_model
      explosion_model
      independence_model
      axiom_model
      axiom_schema_model
      inference_rule_model
      modus_ponens_model
      modus_tollens_model
      universal_instantiation_model
      existential_generalisation_model
      proof_model
      formal_proof_model
      direct_proof_model
      proof_by_contradiction_model
      proof_by_contraposition_model
      mathematical_induction_model
      structural_induction_model
      proof_theory_model
      natural_deduction_model
      sequent_calculus_model
      cut_rule_model
      cut_elimination_model
      model_theory_model
      elementary_equivalence_model
      isomorphism_model
      definability_model
      compactness_model
      Lowenheim_Skolem_model
      higher_order_logic_model
      type_theory_model
      typing_judgment_model
      simple_type_model
      dependent_type_model
      lambda_calculus_model
      alpha_equivalence_model
      beta_reduction_model
      eta_model
      set_theory_model
      membership_model
      subset_model
      extensionality_model
      empty_set_model
      power_set_model
      cardinality_model
      ordinal_model
      modal_logic_model
      Kripke_model
      necessity_model
      possibility_model
      temporal_logic_model
      linear_temporal_logic_model
      computation_tree_logic_model
      computability_model
      Turing_machine_model
      recursive_function_model
      decidability_model
      semidecidability_model
      undecidability_model
      halting_problem_model
      reduction_model
      Godel_incompleteness_model
      first_incompleteness_model
      second_incompleteness_model
      Church_Turing_model
      decision_procedure_model
      automated_reasoning_model
      resolution_model
      unification_model
      most_general_unifier_model
      forward_chaining_model
      backward_chaining_model
      theorem_prover_model
      proof_checker_model
      SAT_solver_model
      SMT_model
      model_checking_model
      constraint_logic_model
      nonclassical_logic_model
      intuitionistic_logic_model
      paraconsistent_logic_model
      many_valued_logic_model
      fuzzy_logic_model
      description_logic_model
      ontology_logic_model
      open_world_model
      closed_world_model
      monotonic_reasoning_model
      nonmonotonic_reasoning_model
      default_logic_model
      belief_revision_model
      meta_logic_model
      object_language_model
      metalanguage_model
      truth_model
      provability_model
      validity_truth_distinction_model
      satisfiability_consistency_model
      formal_semantics_model
      denotation_model
      interpretation_model
      semantic_equivalence_model
      syntactic_equivalence_model
      proof_equivalence_model
      dependency_graph_model
      contradiction_detection_model
      ambiguity_model
      confidence_model
      AI_logic_model
      heuristic_reasoning_model
      formal_verification_model
      proof_certificate_model
      counterexample_model
      explanation_model
      version_model
      provenance_model
      validation_model
      update_model
      interoperability_model
      metadata_model
      relationships
      semantic_equivalence
      constraints
      logic_model
      reasoning_rules
      validation
      confidence_model
      AI_consumption_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      formal_language_or syntax_semantics_change
      propositional_predicate_orfirst_order_logic_changes
      proof_theory_model_theory_ortype_theory_changes
      set_theory_modal_ortemporal_logic_changes
      computability_decidability_orincompleteness_semantics_change
      automated_reasoning_solver_orproof_certificate_changes
      nonclassical_description_orontology_logic_changes
      truth_validity_provability_orconsistency_semantics_change
      AI_logic_heuristic_orformal_verification_changes
      logic_record_changes
      constraint_model_changes
      reasoning_rule_changes
  }

}

END FLOW