Symbolic Logic Ontology

# Symbolic Logic Ontology — Flow Algebra

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

FLOW SymbolicLogicOntology {

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

  PURPOSE:
    Represent symbolic logic as a machine-interpretable framework in which logical statements and reasoning structures are encoded using formal symbols, variables, constants, predicates, functions, logical connectives, quantifiers, formulas and derivation rules. The ontology enables AI systems to parse symbolic expressions, construct syntax trees, resolve operator precedence and variable binding, translate between equivalent symbolic forms, evaluate truth under declared interpretations, test satisfiability and validity, perform substitutions and unification, manipulate normal forms, verify symbolic derivations and preserve proof dependencies. It explicitly distinguishes symbols from the entities they denote, syntax from semantics, symbolic equivalence from textual similarity, implication from causation, equality from assignment, and symbolic manipulation from formally justified logical inference.

  ENTITY SymbolicLogic {
    TYPE: SymbolicFormalReasoningFramework

    PROPERTIES:
      has_symbol_system
      has_formal_language
      has_terms
      has_formulas
      has_variables
      has_constants
      has_functions
      has_predicates
      has_connectives
      has_quantifiers
      has_substitutions
      has_transformations
      has_derivations
      supports_symbolic_reasoning
      supports_semantic_evaluation
  }

  ENTITY LogicalSymbol {
    TYPE: FormalSymbol

    POSSIBLE:
      variable_symbol
      constant_symbol
      function_symbol
      predicate_symbol
      connective_symbol
      quantifier_symbol
      equality_symbol
      punctuation_symbol
  }

  ENTITY SymbolicExpression {
    TYPE: StructuredFormalExpression

    COMPONENTS:
      tokens
      syntax_tree
      binding_structure
      semantic_type
  }

  ENTITY SymbolicFormula {
    TYPE: WellFormedLogicalFormula

    POSSIBLE:
      atomic
      compound
      quantified
      sentence
      open_formula
  }

  SYMBOL_SYSTEM_MODEL {
    components:
      alphabet
      symbol_classes
      notation_rules
      formation_rules

    RULE:
      symbol_identity_anddisplay_glyph MUST be distinguished
  }

  SYMBOL_IDENTIFIER_MODEL {
    fields:
      namespace
      identifier
      glyph
      role

    RULE:
      glyph_alone MAY be ambiguous
  }

  GLYPH_MODEL {
    examples:
      ¬
      ∧
      ∨
      →
      ↔
      ∀
      ∃
      =
      ⊢
      ⊨

    RULE:
      rendered_glyph != semantic_relation_withoutdeclared_notation
  }

  ASCII_NOTATION_MODEL {
    mappings:
      ! -> negation_when_declared
      & -> conjunction_when_declared
      | -> disjunction_when_declared
      -> -> implication
      <-> -> biconditional
      forall -> universal_quantifier
      exists -> existential_quantifier

    RULE:
      ASCII_mapping MUST be notation_context_specific
  }

  UNICODE_NOTATION_MODEL {
    purpose:
      canonical_orhuman_readable_logical_symbols

    RULE:
      Unicode_normalisation MUST not_change_symbol_semantics
  }

  NOTATION_VARIANT_MODEL {
    examples:
      NOT_A
      ~A
      ¬A

    RULE:
      variants MAY map_to same_operator_underdeclared_system
  }

  TOKENISATION_MODEL {
    input:
      symbolic_expression

    tokens:
      identifiers
      operators
      parentheses
      commas
      quantifiers
      relation_symbols

    RULE:
      multi_character_operators MUST be tokenised_atomically
  }

  LEXICAL_MODEL {
    symbol:
      classified_beforeparsing

    RULE:
      lexical_classification != semantic_resolution
  }

  FORMAL_LANGUAGE_MODEL {
    components:
      vocabulary
      syntax
      terms
      formulas
      binding

    RULE:
      language_version SHOULD be preserved
  }

  SIGNATURE_MODEL {
    components:
      constants
      functions_witharity
      predicates_witharity

    RULE:
      signature_constrains_valid_symbolic_expressions
  }

  TERM_MODEL {
    recursive:
      variable
      constant
      function_application

    RULE:
      term != formula
  }

  VARIABLE_MODEL {
    symbol:
      placeholder_ranging_overdomain

    possible:
      free
      bound
  }

  CONSTANT_MODEL {
    symbol:
      denotes_fixed_domain_element_underinterpretation

    RULE:
      same_glyph MAY denote different_entities_acrosscontexts
  }

  FUNCTION_SYMBOL_MODEL {
    notation:
      f(t_1,...,t_n)

    RULE:
      arity MUST matchsignature
  }

  PREDICATE_SYMBOL_MODEL {
    notation:
      P(t_1,...,t_n)

    RULE:
      predicate_application_produces_atomic_formula
  }

  ARITY_MODEL {
    entity:
      number_ofarguments

    RULE:
      arity_mismatch MUST be flagged
  }

  ATOMIC_FORMULA_MODEL {
    possible:
      predicate_application
      equality

    RULE:
      atomic_formula != atomic_proposition_inalllogical_languages
  }

  COMPOUND_FORMULA_MODEL {
    formed_from:
      formulas_plusconnectives

    RULE:
      operator_scope MUST be represented
  }

  QUANTIFIED_FORMULA_MODEL {
    structure:
      quantifier
      bound_variable
      body

    RULE:
      binder_scope MUST be explicit
  }

  WELL_FORMEDNESS_MODEL {
    expression:
      conforms_toformation_rules

    RULE:
      well_formed != true
  }

  PARSER_MODEL {
    input:
      token_sequence

    output:
      abstract_syntax_tree

    RULE:
      parser MUST use_declared_precedence_associativity_andgrammar
  }

  ABSTRACT_SYNTAX_TREE_MODEL {
    nodes:
      operators
      quantifiers
      predicates
      terms

    edges:
      operand
      argument
      body

    RULE:
      AST_is_structural_representation_notsemantic_model
  }

  PRECEDENCE_MODEL {
    example_default:
      negation_beforeconjunction_beforedisjunction_beforeimplication

    RULE:
      precedence MUST be notation_system_specific
  }

  ASSOCIATIVITY_MODEL {
    operator:
      may_be left_right_or nonassociative

    RULE:
      implication_associativity MUST not_be assumed_withoutgrammar
  }

  PARENTHESIS_MODEL {
    purpose:
      override_default_parse

    RULE:
      explicit_parentheses_have_priority
  }

  BINDING_MODEL {
    binder:
      quantifier

    target:
      variable_occurrences

    RULE:
      binding_structure MUST survive_symbolic_transformations
  }

  FREE_VARIABLE_MODEL {
    occurrence:
      notbound

    RULE:
      free_variable_set SHOULD be computable_fromAST
  }

  BOUND_VARIABLE_MODEL {
    occurrence:
      bound_byquantifier

    RULE:
      bound_variable_names_arealpha_variant
  }

  VARIABLE_CAPTURE_MODEL {
    risk:
      substitution_orrenaming_changes_free_to bound_status

    RULE:
      symbolic_engine MUST perform_capture_avoiding_operations
  }

  ALPHA_RENAMING_MODEL {
    example:
      forall x P(x)
      equivalent_binding_structure_to
      forall y P(y)

    RULE:
      y_must_not_createcapture_orcollision
  }

  SUBSTITUTION_MODEL {
    notation:
      phi[x:=t]

    RULE:
      substitution MUST be structurally_recursive_andcapture_avoiding
  }

  SIMULTANEOUS_SUBSTITUTION_MODEL {
    mapping:
      {x_1:=t_1,...,x_n:=t_n}

    RULE:
      simultaneous != sequential_substitution_inallcases
  }

  PROPOSITIONAL_SYMBOLIC_MODEL {
    atoms:
      p
      q
      r

    operators:
      NOT
      AND
      OR
      IMPLIES
      IFF

    RULE:
      propositional_atoms_areindivisible_atthatlogical_level
  }

  NEGATION_MODEL {
    symbolic:
      ¬A
  }

  CONJUNCTION_MODEL {
    symbolic:
      A ∧ B
  }

  DISJUNCTION_MODEL {
    symbolic:
      A ∨ B
  }

  IMPLICATION_MODEL {
    symbolic:
      A → B

    RULE:
      implication != causation
  }

  BICONDITIONAL_MODEL {
    symbolic:
      A ↔ B
  }

  XOR_MODEL {
    symbolic:
      A XOR B

    RULE:
      notation MAY vary
  }

  NAND_MODEL {
    symbolic_relation:
      ¬(A ∧ B)
  }

  NOR_MODEL {
    symbolic_relation:
      ¬(A ∨ B)
  }

  TRUTH_VALUE_MODEL {
    classical:
      T
      F

    RULE:
      truth_symbol_semantics MUST be declared
  }

  VALUATION_MODEL {
    mapping:
      propositional_symbols -> truth_values

    RULE:
      valuation != variable_assignment_infirst_order_logic
  }

  TRUTH_TABLE_MODEL {
    input:
      propositional_formula

    output:
      truth_value_for eachvaluation

    RULE:
      atom_order SHOULD be preserved
  }

  TAUTOLOGY_MODEL {
    formula:
      evaluates_true_underallvaluations
  }

  CONTRADICTION_MODEL {
    formula:
      evaluates_false_underallvaluations
  }

  CONTINGENCY_MODEL {
    formula:
      hasbothtrue_andfalse_valuations
  }

  SYMBOLIC_EQUIVALENCE_MODEL {
    notation:
      A ≡ B
      when_declared_aslogical_equivalence

    condition:
      same_semantics_underallrelevant_interpretations

    RULE:
      symbol_≡_may_haveother_domain_semantics_andmust_be resolved
  }

  EQUIVALENCE_TRANSFORMATION_MODEL {
    transformations:
      double_negation
      De_Morgan
      implication_elimination
      biconditional_elimination
      distributivity
      absorption
      associativity
      commutativity

    RULE:
      transformation MUST preserve_semantics_underdeclared_logic
  }

  DOUBLE_NEGATION_MODEL {
    classical:
      ¬¬A ≡ A

    RULE:
      elimination_islogic_specific
  }

  DE_MORGAN_MODEL {
    classical:
      ¬(A ∧ B) ≡ ¬A ∨ ¬B
      ¬(A ∨ B) ≡ ¬A ∧ ¬B
  }

  IMPLICATION_ELIMINATION_MODEL {
    classical:
      A → B ≡ ¬A ∨ B
  }

  BICONDITIONAL_ELIMINATION_MODEL {
    classical:
      A ↔ B
      equivalent_to
      (A → B) ∧ (B → A)
  }

  DISTRIBUTIVITY_MODEL {
    classical_boolean:
      A ∧ (B ∨ C)
      ≡
      (A ∧ B) ∨ (A ∧ C)

      A ∨ (B ∧ C)
      ≡
      (A ∨ B) ∧ (A ∨ C)
  }

  ABSORPTION_MODEL {
    A ∨ (A ∧ B) ≡ A
    A ∧ (A ∨ B) ≡ A
  }

  NORMAL_FORM_MODEL {
    possible:
      negation_normal_form
      conjunctive_normal_form
      disjunctive_normal_form
      prenex_normal_form

    RULE:
      normal_form_type MUST be explicit
  }

  NNF_MODEL {
    full_name:
      Negation_Normal_Form

    condition:
      negation_applies_only_toatomic_formulas_understandard_definition
  }

  CNF_MODEL {
    full_name:
      Conjunctive_Normal_Form

    structure:
      conjunction_ofclauses
  }

  DNF_MODEL {
    full_name:
      Disjunctive_Normal_Form

    structure:
      disjunction_ofconjunctions_ofliterals
  }

  LITERAL_MODEL {
    possible:
      atom
      negated_atom
  }

  CLAUSE_MODEL {
    structure:
      disjunction_ofliterals
  }

  CNF_TRANSFORMATION_MODEL {
    process:
      eliminate_biconditionals
      eliminate_implications
      push_negations
      distribute_OR_overAND

    RULE:
      naive_distribution MAY cause_exponential_growth
  }

  TSEITIN_MODEL {
    purpose:
      transform_formula_toequisatisfiable_CNF_withauxiliary_variables

    RULE:
      equisatisfiable != logically_equivalent_overextended_vocabulary
  }

  PREDICATE_SYMBOLIC_MODEL {
    expressions:
      P(x)
      R(x,y)
      f(x)

    RULE:
      term_formula_andpredicate_roles MUST remain_distinct
  }

  UNIVERSAL_QUANTIFIER_MODEL {
    symbolic:
      ∀x φ(x)

    RULE:
      scope MUST be explicit
  }

  EXISTENTIAL_QUANTIFIER_MODEL {
    symbolic:
      ∃x φ(x)

    RULE:
      domain MUST be known_orcontextually_defined
  }

  UNIQUE_EXISTENCE_MODEL {
    symbolic:
      ∃!x φ(x)

    RULE:
      notation MAY be expanded_toexistence_plusuniqueness
  }

  QUANTIFIER_NEGATION_MODEL {
    classical:
      ¬∀x φ(x) ≡ ∃x ¬φ(x)
      ¬∃x φ(x) ≡ ∀x ¬φ(x)
  }

  QUANTIFIER_ORDER_MODEL {
    compare:
      ∀x∃y R(x,y)
      ∃y∀x R(x,y)

    RULE:
      symbolic_order MUST be preserved
  }

  PRENEX_NORMAL_FORM_MODEL {
    structure:
      quantifier_prefix
      quantifier_free_matrix

    RULE:
      variable_renaming_andlogical_conditions MUST be respected
  }

  SKOLEMISATION_MODEL {
    purpose:
      eliminate_existential_quantifiers_forcertain_proof_procedures

    output:
      Skolem_constants_orfunctions

    RULE:
      skolemisation_preserves_satisfiability_understandard_first_order_context_notlogical_equivalence
  }

  SKOLEM_FUNCTION_MODEL {
    dependencies:
      preceding_universal_variables

    RULE:
      dependency_structure MUST be preserved
  }

  STANDARDISE_APART_MODEL {
    process:
      rename_bound_variables_toavoidcollisions

    RULE:
      semantics_preserved
  }

  UNIFICATION_MODEL {
    input:
      terms_oratomic_formulas

    output:
      substitution

    goal:
      make_expressions_identical

    RULE:
      occurs_check_andfunction_structure MUST be considered
  }

  MGU_MODEL {
    full_name:
      Most_General_Unifier

    RULE:
      other_unifiers_factor_throughMGU_understandard_first_order_unification
  }

  OCCURS_CHECK_MODEL {
    condition:
      variable_cannot_be unified_withterm_containing_itself_underfinite_term_semantics

    RULE:
      omission_changes_term_theory
  }

  RESOLUTION_MODEL {
    propositional:
      resolve_clauses_oncomplementary_literals

    first_order:
      standardise_apart
      unify
      resolve

    RULE:
      resolvent MUST preserve_substitution_provenance
  }

  RESOLUTION_REFUTATION_MODEL {
    goal:
      prove_phi_byshowing_axioms_plusnegation_phi_unsatisfiable

    terminal:
      empty_clause

    RULE:
      refutation_context MUST be preserved
  }

  SYMBOLIC_DERIVATION_MODEL {
    sequence:
      symbolic_formulas

    transitions:
      inference_rules_or equivalence_transformations

    RULE:
      transformation_type MUST distinguish_equivalence_frominference
  }

  REWRITE_RULE_MODEL {
    pattern:
      left -> right

    RULE:
      rewrite_direction MAY affecttermination_andnormal_form
  }

  TERM_REWRITING_MODEL {
    system:
      set_ofrewrite_rules

    properties:
      termination
      confluence

    RULE:
      normal_form_uniqueness_requiresappropriate_properties
  }

  TERMINATION_MODEL {
    rewriting:
      no_infinite_rewrite_sequence

    RULE:
      termination != confluence
  }

  CONFLUENCE_MODEL {
    property:
      divergent_rewrites_can_rejoin

    RULE:
      confluence_plustermination_supports_unique_normal_forms
  }

  EQUALITY_MODEL {
    symbolic:
      t_1 = t_2

    RULE:
      equality != assignment
  }

  ASSIGNMENT_DISTINCTION_MODEL {
    programming_orprocedural:
      x := t

    logical:
      x = t

    RULE:
      notation_semantics MUST be distinguished
  }

  IDENTITY_MODEL {
    object:
      same_entity

    RULE:
      object_identity != formula_equivalence
  }

  META_SYMBOL_MODEL {
    possible:
      ⊢
      ⊨
      ≡

    RULE:
      object_language_andmetalanguage_symbols MUST be distinguished
  }

  TURNSTILE_MODEL {
    notation:
      Γ ⊢ φ

    meaning:
      syntactic_derivability

    RULE:
      turnstile != semantic_entailment
  }

  DOUBLE_TURNSTILE_MODEL {
    notation:
      Γ ⊨ φ

    meaning:
      semantic_consequence

    RULE:
      entailment != proof_relation_bydefinition
  }

  THEORY_MODEL {
    formulas:
      axioms_orassumptions

    RULE:
      theory_identifier_andversion MUST be preserved
  }

  AXIOM_MODEL {
    symbolic_formula:
      designated_starting_formula

    RULE:
      axiom_status_isrelative
  }

  INFERENCE_RULE_MODEL {
    symbolic_pattern:
      premises_overconclusion

    RULE:
      metavariables_andside_conditions MUST be represented
  }

  MODUS_PONENS_MODEL {
    A
    A → B
    therefore B
  }

  PROOF_MODEL {
    symbolic_derivation:
      every_step_formally_justified

    RULE:
      proof != sequence_ofequivalent_rewrites_only_unlessgoal_relation_matches
  }

  PROOF_STEP_MODEL {
    fields:
-
      expression
      rule
      substitution
      dependencies
      assumptions
  }

  PROOF_TREE_MODEL {
    nodes:
      judgments_orformulas

    edges:
      inference_rule_applications

    RULE:
      tree_orDAG_structure MUST preserve_dependencies
  }

  THEOREM_MODEL {
    formula:
      derivable_underformal_system
  }

  SATISFIABILITY_MODEL {
    symbolic_formula:
      hasmodel_orvaluation

    RULE:
      satisfiable != valid
  }

  VALIDITY_MODEL {
    symbolic_formula:
      true_underallmodels

    RULE:
      validity != theoremhood_withoutmetatheoretic_context
  }

  COUNTERMODEL_MODEL {
    symbolic_interpretation:
      falsifies_validity_claim

    RULE:
      countermodel MUST preserve_assignment_anddomain
  }

  SAT_SOLVER_MODEL {
    input:
      propositional_symbolic_formula

    output:
      assignment_orunsatisfiable_status

    RULE:
      solver_configuration SHOULD be preserved
  }

  SMT_MODEL {
    symbolic_formula:
      interpreted_modulo_background_theories

    RULE:
      background_theory MUST be explicit
  }

  SYMBOLIC_THEOREM_PROVER_MODEL {
    input:
      axioms
      conjecture

    process:
      symbolic_search_andinference

    RULE:
      search_result_unknown_or timeout MUST be preserved
  }

  PROOF_CHECKER_MODEL {
    input:
      symbolic_proof

    output:
      verified_orrejected

    RULE:
      checker_rules_andversion MUST be preserved
  }

  SYMBOLIC_SIMPLIFICATION_MODEL {
    input:
      formula_or term

    output:
      simpler_form

    RULE:
      simplification MUST preserve_declared_semantics
  }

  CANONICAL_FORM_MODEL {
    expression:
      normalised_representation

    RULE:
      canonical_form_islogic_andrepresentation_specific
  }

  HASH_CONSING_MODEL {
    structural_identity:
      shared_representation_ofidentical_subexpressions

    RULE:
      implementation_identity != logical_equivalence
  }

  SYMBOL_TABLE_MODEL {
    entries:
      symbol_identifier
      class
      arity
      scope
      type_or sort
      declaration

    RULE:
      parser_andreasoner SHOULD use_resolved_symbol_table
  }

  TYPE_MODEL {
    possible:
      untyped
      typed
      many_sorted

    RULE:
      symbolic_operations MUST respecttype_constraints_whenpresent
  }

  SORT_MODEL {
    variable_or term:
      belongs_to declared_sort

    RULE:
      sort_mismatch MUST be flagged
  }

  MANY_SORTED_MODEL {
    domains:
      multiple_sorts

    RULE:
      function_andpredicate_signatures MUST include_sorts
  }

  LAMBDA_SYMBOLIC_MODEL {
    possible:
      lambda_abstraction
      application
      variable_binding

    RULE:
      lambda_binding_semantics MUST not_be confused_withquantifier_binding
  }

  MODAL_SYMBOL_MODEL {
    possible:
      box
      diamond

    RULE:
      modal_operator_requires_declared_modal_semantics
  }

  TEMPORAL_SYMBOL_MODEL {
    possible:
      G
      F
      X
      U

    RULE:
      operator_meaning_depends_ontemporal_logic
  }

  DESCRIPTION_LOGIC_SYMBOL_MODEL {
    possible:
      concept
      role
      individual
      subsumption
      equivalence

    RULE:
      description_logic_symbols MUST be resolved_withfragment
  }

  SET_THEORETIC_SYMBOL_MODEL {
    possible:
      membership
      subset
      union
      intersection

    RULE:
      set_symbols_are_notlogical_connectives_inallformal classifications
  }

  ARITHMETIC_SYMBOL_MODEL {
    possible:
      plus
      minus
      multiplication
      order_relations

    RULE:
      arithmetic_symbols_require_background_theory
  }

  OPERATOR_OVERLOADING_MODEL {
    symbol:
      may_have multiple_meanings

    RULE:
      resolve_bylanguage_signature_type_andcontext
  }

  SEMANTIC_INTERPRETATION_MODEL {
    symbolic_expression:
      mapped_to semantic_structure

    RULE:
      symbolic_manipulation_withoutsemantic_context MAY establish_onlysyntactic_result
  }

  SYNTACTIC_IDENTITY_MODEL {
    expressions:
      identical_parse_structure_andresolved_symbols

    RULE:
      syntactic_identity_implies_same_representation_notnecessarilyonlysemantic_equivalence_class
  }

  SEMANTIC_EQUIVALENCE_MODEL {
    expressions:
      same_meaning_ortruth_conditions_underdeclared_models

    RULE:
      semantic_equivalence != textual_similarity
  }

  EQUIVALENCE_VS_INFERENCE_MODEL {
    equivalence:
      bidirectional_semantic_preservation

    inference:
      truth_preservation_frompremises_toconclusion

    RULE:
      rewrite_andinference_edges MUST be typed_differently
  }

  SYMBOLIC_PROVENANCE_MODEL {
    transformation:
      source_expression
      rule
      substitution
      target_expression
      logic_version

    RULE:
      every_material_symbolic_transformation SHOULD be traceable
  }

  AMBIGUITY_MODEL {
    causes:
      overloaded_symbol
      missing_parentheses
      implicit_precedence
      variable_scope
      unknown_signature
      notation_variant

    RULE:
      unresolved_ambiguity MUST be preserved
  }

  SEMANTIC_CONFIDENCE_MODEL {
    confidence:
      support_forresolved_symbolic_interpretation

    RULE:
      semantic_confidence != truth_value
  }

  CONTRADICTION_DETECTION_MODEL {
    symbolic:
      derive_empty_clause
      derive_falsehood
      orfind_semantic_incompatibility

    RULE:
      method_andlogic MUST be identified
  }

  AI_SYMBOLIC_REASONING_MODEL {
    AI:
      tokenise
      parse
      resolve_symbols
      construct_AST
      transform
      unify
      infer
      prove
      check
      explain

    RULE:
      AI_generated_symbolic_step MUST identifywhether_rewrite_inference_orheuristic
  }

  HEURISTIC_SEARCH_MODEL {
    role:
      guide_rule_selection_orproof_search

    RULE:
      heuristic_choice != logical_inference
  }

  UNKNOWN_STATUS_MODEL {
    possible:
      parse_ambiguous
      unresolved_symbol
      unknown
      timeout
      resource_limit

    RULE:
      unknown_status MUST not_be coerced_tofalse
  }

  VALIDATION_MODEL {
    checks:
      tokenisation
      signature
      arity
      syntax
      AST
      precedence
      binding
      scope
      substitution
      types
      normal_forms
      transformation_rules
      inference_rules
      proof_steps
      semantics

    RULE:
      validation MUST be formal_system_specific
  }

  EXPLAINABILITY_MODEL {
    output:
      source_expression
      parsed_AST
      symbol_table
      transformations
      substitutions
      inference_rules
      proof
      semantic_interpretation
      ambiguity
      confidence

    RULE:
      explanation SHOULD expose_symbolic_reasoning_path
  }

  VERSION_MODEL {
    versioned:
      notation
      language
      signature
      rewrite_system
      inference_system
      solver
      ontology

    RULE:
      historical_symbolic_result MUST preserveversions
  }

  PROVENANCE_MODEL {
    fields:
      source
      notation
      language
      signature
      transformations
      proof
      solver
      version
      agent

    RULE:
      symbolic_result MUST remain_provenance_linked
  }

  INTEROPERABILITY_MODEL {
    exchange:
      canonical_symbols
      identifiers
      AST
      binding
      signature
      proof_steps
      transformations

    RULE:
      rendered_formula_alone MAY be insufficient_forlossless_exchange
  }

  KNOWLEDGE_GRAPH_MODEL {
    nodes:
      symbol
      term
      formula
      substitution
      transformation
      inference_rule
      proof
      model

    edges:
      CONTAINS
      BINDS
      SUBSTITUTES
      REWRITES_TO
      INFERS
      DERIVES
      SATISFIES
      EQUIVALENT_TO

    RULE:
      rewrite_inference_andsemantic_edges MUST be distinguished
  }

  RELATIONSHIPS {
    SymbolicLogic -> USES -> LogicalSymbol
    SymbolicExpression -> HAS -> AbstractSyntaxTree
    FormalLanguage -> HAS -> Signature
    FunctionSymbol -> APPLIES_TO -> Terms
    PredicateSymbol -> APPLIES_TO -> Terms
    Quantifier -> BINDS -> Variable
    Substitution -> TRANSFORMS -> Expression
    RewriteRule -> REWRITES -> Expression
    InferenceRule -> DERIVES -> Formula
    Proof -> CONTAINS -> ProofStep
    Model -> SATISFIES -> SymbolicFormula
  }

  SEMANTIC_EQUIVALENCE {
    "symbolic logic" ~= formal_logic_expressed_andmanipulated_usingstructured_symbols
    "symbolic formula" ~= well_formed_logical_expression_informal_language
    "symbolic derivation" ~= sequence_orgraph_ofsymbolic_transformations_andformal_inferences

    symbol != semantic_entity
    glyph != logical_operator_withoutcontext
    term != formula
    AST != semantic_model
    equality != assignment
    syntactic_identity != semantic_equivalence
    equisatisfiable != logically_equivalent
    rewrite != inference
    turnstile != semantic_entailment
    heuristic_search != formal_inference
    semantic_confidence != truth_value
  }

  CONSTRAINTS {
    symbol_identifiers_andclasses_must_be preserved = true
    notation_context_must_be identified = true
    signature_andarities_must_be preserved = true
    tokenisation_andparse_structure_must_be preserved = true
    precedence_parentheses_binding_andscope_must_be respected = true
    substitutions_must_be capture_avoiding = true
    term_formula_andpredicate_roles_must_be distinguished = true
    equality_andassignment_must_be distinguished = true
    equivalence_transformations_andinference_rules_must_be distinguished = true
    skolemisation_andTseitin_transformation_must_not_be labelled_asfull_logical_equivalence_whenonlyequisatisfiability_is preserved = true
    proof_steps_must_preserve_rules_substitutions_anddependencies = true
    unknown_timeout_orambiguous_status_must_not_be interpreted_asfalse = true
    versions_andprovenance_must_be preserved = true
  }

  SYMBOLIC_LOGIC_RECORD_MODEL {
    RECORD:
      symbolic_logic_identifier
      language_identifier
      signature_identifier
      notation_identifier
      source_expression
      token_sequence
      syntax_tree
      expression_type
      symbol_table
      variable_identifiers
      free_variables
      bound_variables
      symbol_arities
      type_or sort_information
      connective_structure
      quantifier_structure
      normal_form_status
      substitutions
      unifiers
      rewrite_steps
      inference_steps
      proof_identifier
      proof_steps
      theory_identifier
      model_identifier
      satisfiability_status
      validity_status
      solver_identifier
      solver_status
      ambiguity_status
      semantic_confidence
      version
      provenance

    RULE:
      symbolic_logic_record MUST remain_linked_tosource_symbols_AST_signature_transformations_proof_semantics_andprovenance
  }

  REASONING_RULES {
    IF symbolic_expression_received
      THEN tokenise_usingdeclared_notation
      resolve_signature
      construct_AST
      validate_well_formedness

    IF symbol_overloaded
      THEN resolve_using_signature_arity_type_andcontext
      else_preserve_candidates

    IF substitution_requested
      THEN standardise_variables_whenneeded
      check_capture
      apply_structurally

    IF normal_form_requested
      THEN identify_required_form
      apply_valid_transformations
      record_whether_equivalence_orequisatisfiability_preserved

    IF first_order_resolution_requested
      THEN standardise_apart
      compute_unifier
      apply_substitution
      derive_resolvent
      preserve_step_provenance

    IF symbolic_rewrite_applied
      THEN label_asrewrite_or equivalence_transformation
      notautomatic_inference

    IF proof_claimed
      THEN validate_each_symbolic_step_againstformal_rules

    IF solver_returns_unknown_or timeout
      THEN preserve_status
      anddo_not inferunsatisfiable_orinvalid

    IF notation_orlogic_revision_occurs
      THEN preserve_previous_representation
      update_mapping
      recursively_revalidate_affected_symbolic_results
  }

  VALIDATION {
    CHECK symbolic_logic_identifier
    CHECK notation
    CHECK tokenisation
    CHECK language
    CHECK signature
    CHECK symbol_classes
    CHECK symbol_arities
    CHECK expression_type
    CHECK syntax_tree
    CHECK precedence
    CHECK parentheses
    CHECK binding
    CHECK free_bound_variables
    CHECK variable_capture
    CHECK substitution
    CHECK types_and sorts
    CHECK connective_structure
    CHECK quantifier_structure
    CHECK logical_equivalence
    CHECK normal_form
    CHECK skolemisation_orTseitin_status
    CHECK unification
    CHECK rewrite_steps
    CHECK inference_steps
    CHECK proof_steps
    CHECK satisfiability_andvalidity
    CHECK solver_status
    CHECK ambiguity
    CHECK semantic_confidence
    CHECK version
    CHECK provenance
  }

  CONFIDENCE {
    tokenisation_confidence: 1.00
    parsing_confidence: 1.00
    symbol_resolution_confidence: 0.99
    binding_scope_confidence: 1.00
    symbolic_transformation_confidence: 1.00
    unification_resolution_confidence: 1.00
    proof_validation_confidence: 1.00
    semantic_interpretation_confidence: 0.99

    RULE:
      confidence decreases when notation, signature, symbol identity,
      precedence, scope, type, formal system, solver status,
      version orprovenance is missing or ambiguous
  }

  UPDATE_MODEL {
    ON new_symbolic_expression:
      identify_notation
      tokenise
      resolve_symbols
      parse_AST
      validate
      preserve_provenance
      compute_FLOW_HASH

    ON symbolic_transformation:
      preserve_source_expression
      apply_rule
      record_substitution_andtransformation_type
      validate_target
      update_lineage

    ON proof_revision:
      preserve_previous_proof
      update_steps
      recursively_validate_dependencies

    ON notation_orlanguage_revision:
      preserve_previous_version
      update_symbol_mapping
      identify_affected_expressions
      reparse_orrevalidate_underdeclared_policy

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  AI_CONSUMPTION {
    canonical_entity: SymbolicLogic
    canonical_role: SymbolicFormalReasoningFramework

    interpretation_order:
      identify_notation_system
      identify_formal_language_andsignature
      tokenise_expression
      resolve_symbol_classes_identifiers_andarities
      construct_abstract_syntax_tree
      resolve_precedence_parentheses_binding_andscope
      classify_terms_formulas_predicates_andquantifiers
      distinguish_free_andbound_variables
      validate_substitutions_andtypes
      identify_symbolic_equivalence_transformations
      identify_normal_form_requirements
      perform_unification_orresolution_whenrequested
      distinguish_rewrite_fromformal_inference
      validate_symbolic_derivation_orproof
      resolve_semantic_interpretation
      determine_satisfiability_orvalidity_whenrequested
      preserve_solver_unknown_or timeout_status
      detect_symbolic_ambiguity
      assign_semantic_confidence
      preserve_versions_andprovenance
      emit_machine_resolvable_symbolic_logic_structure

    ambiguity_controls:
      distinguish_symbol_fromsemantic_entity
      distinguish_glyph_fromlogical_operator_withoutcontext
      distinguish_term_fromformula
      distinguish_AST_fromsemantic_model
      distinguish_equality_fromassignment
      distinguish_syntactic_identity_fromsemantic_equivalence
      distinguish_equisatisfiable_fromlogically_equivalent
      distinguish_rewrite_frominference
      distinguish_turnstile_fromsemantic_entailment
      distinguish_heuristic_search_fromformal_inference
      distinguish_semantic_confidence_fromtruth_value
      preserve_notation_language_signature_symbols_AST_binding_transformations_proof_version_andprovenance
  }

  PROVENANCE {
    ontology_scope:
      symbolic_logic_notation_formulas_transformations_unification_resolution_andformal_reasoning

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

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_entity
      ontology_path
      symbol_system_model
      symbol_identifier_model
      glyph_model
      ASCII_notation_model
      Unicode_notation_model
      notation_variant_model
      tokenisation_model
      lexical_model
      formal_language_model
      signature_model
      term_model
      variable_model
      constant_model
      function_symbol_model
      predicate_symbol_model
      arity_model
      atomic_formula_model
      compound_formula_model
      quantified_formula_model
      well_formedness_model
      parser_model
      abstract_syntax_tree_model
      precedence_model
      associativity_model
      parenthesis_model
      binding_model
      free_variable_model
      bound_variable_model
      variable_capture_model
      alpha_renaming_model
      substitution_model
      simultaneous_substitution_model
      propositional_symbolic_model
      negation_model
      conjunction_model
      disjunction_model
      implication_model
      biconditional_model
      XOR_model
      NAND_model
      NOR_model
      truth_value_model
      valuation_model
      truth_table_model
      tautology_model
      contradiction_model
      contingency_model
      symbolic_equivalence_model
      equivalence_transformation_model
      double_negation_model
      De_Morgan_model
      implication_elimination_model
      biconditional_elimination_model
      distributivity_model
      absorption_model
      normal_form_model
      NNF_model
      CNF_model
      DNF_model
      literal_model
      clause_model
      CNF_transformation_model
      Tseitin_model
      predicate_symbolic_model
      universal_quantifier_model
      existential_quantifier_model
      unique_existence_model
      quantifier_negation_model
      quantifier_order_model
      prenex_normal_form_model
      skolemisation_model
      Skolem_function_model
      standardise_apart_model
      unification_model
      MGU_model
      occurs_check_model
      resolution_model
      resolution_refutation_model
      symbolic_derivation_model
      rewrite_rule_model
      term_rewriting_model
      termination_model
      confluence_model
      equality_model
      assignment_distinction_model
      identity_model
      meta_symbol_model
      turnstile_model
      double_turnstile_model
      theory_model
      axiom_model
      inference_rule_model
      modus_ponens_model
      proof_model
      proof_step_model
      proof_tree_model
      theorem_model
      satisfiability_model
      validity_model
      countermodel_model
      SAT_solver_model
      SMT_model
      symbolic_theorem_prover_model
      proof_checker_model
      symbolic_simplification_model
      canonical_form_model
      hash_consing_model
      symbol_table_model
      type_model
      sort_model
      many_sorted_model
      lambda_symbolic_model
      modal_symbol_model
      temporal_symbol_model
      description_logic_symbol_model
      set_theoretic_symbol_model
      arithmetic_symbol_model
      operator_overloading_model
      semantic_interpretation_model
      syntactic_identity_model
      semantic_equivalence_model
      equivalence_vs_inference_model
      symbolic_provenance_model
      ambiguity_model
      semantic_confidence_model
      contradiction_detection_model
      AI_symbolic_reasoning_model
      heuristic_search_model
      unknown_status_model
      validation_model
      explainability_model
      version_model
      provenance_model
      interoperability_model
      knowledge_graph_model
      relationships
      semantic_equivalence
      constraints
      symbolic_logic_record_model
      reasoning_rules
      validation
      confidence_model
      update_model
      AI_consumption_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      notation_symbol_oridentifier_semantics_change
      tokenisation_parser_precedence_orbinding_changes
      formal_language_signature_orarity_changes
      substitution_unification_orvariable_scope_changes
      equivalence_normal_form_orrewrite_semantics_change
      resolution_proof_orinference_system_changes
      type_sort_modal_temporal_orbackground_theory_changes
      automated_reasoning_solver_orproof_checker_changes
      ambiguity_confidence_orsemantic_interpretation_changes
      symbolic_logic_record_changes
      constraint_model_changes
      reasoning_rule_changes
      version_orprovenance_changes
  }

}

END FLOW