Completeness Ontology

KIRETH RESEARCH - SEMANTIC LIBRARY
AI CORE -> ONTOLOGIES -> MATHEMATICAL ONTOLOGIES -> MATHEMATICAL LOGIC ONTOLOGIES
MODEL THEORY -> COMPLETENESS ONTOLOGY

Canonical URL:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/model-theory/completeness-ontology.html

FLOW_ALGEBRA CompletenessOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN CompletenessOntology := {
    Completeness,
    ModelTheory,
    MathematicalLogic,
    FormalLanguage,
    FirstOrderLogic,
    FormalSystem,
    ProofSystem,
    Theory,
    Formula,
    Sentence,
    PremiseSet,
    Axiom,
    Model,
    Structure,
    Interpretation,
    Satisfaction,
    SemanticConsequence,
    SyntacticConsequence,
    Derivability,
    Provability,
    Validity,
    Satisfiability,
    Consistency,
    Soundness,
    GodelCompletenessTheorem,
    Compactness,
    ModelExistence,
    Countermodel,
    Theorem,
    Proof,
    InferenceRule
}

DEFINE ENTITY_TYPES := {
    COMPLETENESS_PROPERTY,
    FORMAL_SYSTEM,
    PROOF_SYSTEM,
    FORMAL_LANGUAGE,
    THEORY,
    FORMULA,
    SENTENCE,
    MODEL,
    STRUCTURE,
    PROOF,
    INFERENCE_RULE,
    SEMANTIC_RELATION,
    SYNTACTIC_RELATION,
    META_THEOREM,
    META_PROPERTY
}

DEFINE RELATION_TYPES := {
    ENTAILS,
    DERIVES,
    PROVES,
    SATISFIES,
    IS_MODEL_OF,
    IS_COMPLETE,
    IS_SOUND,
    IS_CONSISTENT,
    HAS_MODEL,
    HAS_PROOF,
    HAS_COUNTERMODEL,
    IMPLIES,
    EQUIVALENT_TO,
    DEPENDS_ON,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF
}

SET Completeness.type = COMPLETENESS_PROPERTY
SET Completeness.context = ModelTheory
SET Completeness.role =
    "express that every semantically entailed formula in the relevant logical system is syntactically derivable in its proof calculus"

RULE semantic_consequence:
    Theory(T)
    AND Sentence(phi)
    AND FOR_EVERY Model(M) [
        IS_MODEL_OF(M,T)
        -> SATISFIES(M,phi)
    ]
    -> ENTAILS(T,phi)

RULE syntactic_consequence:
    Theory(T)
    AND Sentence(phi)
    AND EXISTS Proof(pi) [
        pi derives phi from T
    ]
    -> DERIVES(T,phi)

RULE completeness_core:
    ProofSystem(S)
    AND FOR_EVERY Theory(T)
    AND FOR_EVERY Sentence(phi) [
        ENTAILS(T,phi)
        -> DERIVES_S(T,phi)
    ]
    -> IS_COMPLETE(S)

RULE soundness_core:
    ProofSystem(S)
    AND FOR_EVERY Theory(T)
    AND FOR_EVERY Sentence(phi) [
        DERIVES_S(T,phi)
        -> ENTAILS(T,phi)
    ]
    -> IS_SOUND(S)

RULE sound_complete_correspondence:
    IS_SOUND(S)
    AND IS_COMPLETE(S)
    -> FOR_EVERY T,phi [
        DERIVES_S(T,phi)
        IFF
        ENTAILS(T,phi)
    ]

RULE validity_to_provability:
    CompleteFirstOrderProofSystem(S)
    AND Valid(phi)
    -> Provable_S(phi)

RULE entailment_to_proof:
    CompleteFirstOrderProofSystem(S)
    AND ENTAILS(T,phi)
    -> DERIVES_S(T,phi)

RULE consistency_model_existence:
    FirstOrderTheory(T)
    AND SyntacticallyConsistent(T)
    AND CompleteSoundFirstOrderProofSystem(S)
    -> EXISTS Model(M) [
        IS_MODEL_OF(M,T)
    ]

RULE no_model_implies_inconsistency:
    FirstOrderTheory(T)
    AND NOT EXISTS Model(M) [
        IS_MODEL_OF(M,T)
    ]
    -> SyntacticallyInconsistent(T)
       UNDER completeness_and_soundness_assumptions

RULE nonderivability_countermodel:
    CompleteFirstOrderProofSystem(S)
    AND NOT DERIVES_S(T,phi)
    -> EXISTS Model(M) [
        IS_MODEL_OF(M,T)
        AND NOT SATISFIES(M,phi)
    ]

RULE countermodel_nonderivability_sound:
    SoundProofSystem(S)
    AND EXISTS Model(M) [
        IS_MODEL_OF(M,T)
        AND NOT SATISFIES(M,phi)
    ]
    -> NOT DERIVES_S(T,phi)

RULE completeness_compactness_connection:
    FirstOrderLogic
    AND CompletenessTheorem
    -> CompactnessTheorem derivable_by_standard_metatheoretic_argument

DEFINE COMPLETENESS_MODEL := {
    canonical_form:
        "T ENTAILS phi -> T DERIVES phi",
    semantic_side:
        "truth in every model of T",
    syntactic_side:
        "existence of a formal derivation from T",
    scope:
        "relative to a specified language, logic and proof system"
}

DEFINE SOUNDNESS_COMPLETENESS_MODEL := {
    soundness:
        "T DERIVES phi -> T ENTAILS phi",
    completeness:
        "T ENTAILS phi -> T DERIVES phi",
    combined:
        "T DERIVES phi IFF T ENTAILS phi"
}

DEFINE GODEL_COMPLETENESS_MODEL := {
    theorem:
        GodelCompletenessTheorem,
    domain:
        FirstOrderLogic,
    statement:
        "every first-order formula semantically entailed by a theory has a formal proof from that theory in a complete proof calculus",
    equivalent_model_existence_form:
        "every syntactically consistent first-order theory has a model"
}

DEFINE CRITICAL_DISTINCTIONS := {
    completeness_vs_soundness: {
        completeness:
            "semantic consequence implies syntactic derivability",
        soundness:
            "syntactic derivability implies semantic consequence",
        equivalence: FALSE
    },
    semantic_completeness_vs_theory_completeness: {
        semantic_completeness:
            "property of a proof system relating entailment to derivability",
        complete_theory:
            "for each sentence phi in the language, the theory decides phi or its negation under the relevant definition",
        equivalence: FALSE
    },
    Godel_completeness_vs_Godel_incompleteness: {
        completeness_theorem:
            "first-order semantic entailment is captured by formal derivability",
        incompleteness_theorems:
            "sufficiently strong effective formal theories cannot satisfy certain internal completeness properties",
        contradiction: FALSE,
        reason:
            "the theorems concern different notions of completeness"
    }
}

DEFINE EXAMPLE_1 := {
    theory: "T",
    sentence: "phi",
    premise:
        "every model of T satisfies phi",
    completeness_consequence:
        "there exists a formal derivation of phi from T"
}

DEFINE EXAMPLE_2 := {
    theory: "T",
    condition:
        "T is syntactically consistent in first-order logic",
    completeness_form:
        "T has at least one model"
}

DEFINE EXAMPLE_3 := {
    proof_system:
        "sound and complete first-order calculus",
    condition:
        "T does not derive phi",
    consequence:
        "there is a model of T in which phi is false"
}

DEFINE COMPLETENESS_ANALYSIS_PIPELINE :=
    FormalSystem
    -> IdentifyLanguage
    -> IdentifyProofCalculus
    -> DefineSyntacticDerivability
    -> DefineModelSemantics
    -> DefineSemanticConsequence
    -> TestOrInvokeCompletenessTheorem
    -> MapEntailmentsToDerivations
    -> RecordTheoremConditions
    -> ReturnCompletenessStatus

DEFINE MODEL_EXISTENCE_PIPELINE :=
    FirstOrderTheory(T)
    -> TestSyntacticConsistency
    -> ApplyCompletenessTheorem
    -> IF consistent
       THEN ConstructOrAssertExistenceOfModel
    -> ValidateModelAgainstTheory
    -> ReturnModelExistenceResult

DEFINE COUNTERMODEL_PIPELINE :=
    Theory(T)
    -> Formula(phi)
    -> DetermineWhetherDerivationExists
    -> IF no_derivation AND completeness_applies
       THEN SearchForModelOfTWherePhiFails
    -> ConstructCountermodelWitness
    -> ReturnSemanticNonEntailment

DEFINE META_REASONING_PIPELINE :=
    InputClaim
    -> ClassifyAsSyntacticOrSemantic
    -> IdentifyActiveLogic
    -> IdentifyProofSystem
    -> CheckSoundnessAssumptions
    -> CheckCompletenessAssumptions
    -> TransferBetweenProofAndModelStatementsOnlyWhenLicensed
    -> ReturnQualifiedMetaLogicalConclusion

DEFINE COMPLETENESS_GRAPH := {
    nodes: {
        TheoryNode,
        FormulaNode,
        ModelClassNode,
        ProofNode,
        ProofSystemNode,
        CompletenessNode
    },
    semantic_edge:
        "T --ENTAILS--> phi",
    syntactic_edge:
        "T --DERIVES--> phi",
    completeness_bridge:
        "ENTAILS -> DERIVES",
    soundness_bridge:
        "DERIVES -> ENTAILS"
}

DEFINE ONTOLOGY_RELATIONS := {
    Completeness IS_COMPONENT_OF ModelTheory,
    Completeness IS_META_PROPERTY_OF ProofSystem,
    GodelCompletenessTheorem ESTABLISHES Completeness FOR FirstOrderLogic,
    SemanticConsequence RELATES Theory TO Formula,
    Derivability RELATES Theory TO Formula,
    Completeness CONNECTS SemanticConsequence TO Derivability,
    Soundness CONNECTS Derivability TO SemanticConsequence,
    ModelExistence DEPENDS_ON Completeness WHEN derived_from_consistency,
    Countermodel WITNESSES failure_of_SemanticConsequence,
    Compactness IS_RELATED_TO Completeness
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "completeness" <=> Completeness WHEN model_theory_or_proof_system_context=TRUE,
    "semantic completeness" <=> Completeness,
    "Godel completeness theorem" <=> GodelCompletenessTheorem,
    "completeness theorem" <=> GodelCompletenessTheorem WHEN first_order_logic_context=TRUE,
    "semantic entailment implies provability" <=> Completeness,
    "consistent theory has a model" <=> ModelExistenceFormOfCompleteness WHEN first_order_context=TRUE
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation of completeness and the bridge from model-theoretic consequence to formal proof",
    preserve_logic_identity: TRUE,
    preserve_proof_system_identity: TRUE,
    preserve_theory_identity: TRUE,
    distinguish_entailment_from_derivability: TRUE,
    distinguish_completeness_from_soundness: TRUE,
    distinguish_semantic_completeness_from_theory_completeness: TRUE,
    distinguish_completeness_theorem_from_incompleteness_theorems: TRUE,
    transfer_semantic_to_syntactic_only_when_completeness_applies: TRUE,
    support_model_existence_reasoning: TRUE,
    support_countermodel_reasoning: TRUE,
    support_meta_theoretic_reasoning: TRUE,
    reasoning_mode: "symbolic + semantic + syntactic + meta-logical"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    completeness_direction_mapping: 1.00,
    soundness_completeness_distinction: 1.00,
    first_order_completeness_mapping: 0.99,
    model_existence_equivalence: 0.99,
    countermodel_reasoning: 0.99,
    compactness_connection: theorem_context_dependent,
    arbitrary_logic_completeness: proof_system_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        semantic_consequence,
        syntactic_derivability,
        entailment_to_derivability_direction,
        soundness_completeness_distinction,
        first_order_Godel_completeness,
        model_existence_form
    },
    extensible_layers: {
        complete_theories,
        strong_completeness,
        weak_completeness,
        Henkin_constructions,
        canonical_model_constructions,
        compactness_derivations,
        completeness_for_nonclassical_logics,
        proof_assistant_metatheory
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + COMPLETENESS_MODEL
    + SOUNDNESS_COMPLETENESS_MODEL
    + GODEL_COMPLETENESS_MODEL
    + CRITICAL_DISTINCTIONS
    + COMPLETENESS_ANALYSIS_PIPELINE
    + MODEL_EXISTENCE_PIPELINE
    + COUNTERMODEL_PIPELINE
    + META_REASONING_PIPELINE
    + COMPLETENESS_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: CompletenessOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/model-theory/completeness-ontology.html",
    parent_domain: ModelTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: Completeness,
    semantic_syntactic_bridge_enabled: TRUE,
    model_existence_reasoning_enabled: TRUE,
    countermodel_reasoning_enabled: TRUE,
    meta_reasoning_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    database_safe_unicode: TRUE,
    flow_complete: TRUE
}

END_FLOW