Soundness Ontology

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

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

FLOW_ALGEBRA SoundnessOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN SoundnessOntology := {
    Soundness,
    ModelTheory,
    MathematicalLogic,
    FormalLanguage,
    FormalSystem,
    ProofSystem,
    ProofCalculus,
    Theory,
    Formula,
    Sentence,
    PremiseSet,
    Axiom,
    InferenceRule,
    Proof,
    Derivation,
    Provability,
    Derivability,
    Model,
    Structure,
    Interpretation,
    Satisfaction,
    SemanticConsequence,
    SyntacticConsequence,
    Validity,
    Satisfiability,
    Consistency,
    Completeness,
    Countermodel,
    SoundInferenceRule,
    SoundProofSystem,
    TruthPreservation
}

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

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

SET Soundness.type = SOUNDNESS_PROPERTY
SET Soundness.context = ModelTheory
SET Soundness.role =
    "express that every formula derivable in a proof system from given premises is semantically entailed by those premises"

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

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

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

RULE rule_soundness:
    InferenceRule(R)
    AND FOR_EVERY Structure(M)
    AND FOR_EVERY Assignment(s) [
        all_instantiated_premises_of_R_satisfied(M,s)
        -> instantiated_conclusion_of_R_satisfied(M,s)
    ]
    -> SoundInferenceRule(R)

RULE system_soundness_from_components:
    ProofSystem(S)
    AND all_axioms_of_S_are_valid_or_semantically_licensed
    AND all_inference_rules_of_S_are_sound
    -> IS_SOUND(S)

RULE theorem_validity:
    SoundProofSystem(S)
    AND DERIVES_S(EMPTY_THEORY,phi)
    -> Valid(phi)

RULE derivation_truth_preservation:
    SoundProofSystem(S)
    AND IS_MODEL_OF(M,T)
    AND DERIVES_S(T,phi)
    -> SATISFIES(M,phi)

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

RULE unsoundness_witness:
    ProofSystem(S)
    AND DERIVES_S(T,phi)
    AND EXISTS Model(M) [
        IS_MODEL_OF(M,T)
        AND NOT SATISFIES(M,phi)
    ]
    -> NOT IS_SOUND(S)
    AND CounterexampleToSoundness(M,T,phi)

RULE soundness_consistency_connection:
    SoundProofSystem(S)
    AND EXISTS Model(M) [
        IS_MODEL_OF(M,T)
    ]
    -> T cannot_derive_semantic_falsehood
       UNDER S

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

RULE local_to_global_proof_soundness:
    all_initial_steps_semantically_valid_relative_to_premises
    AND every_inference_step_preserves_truth
    -> final_conclusion_semantically_follows_from_premises

DEFINE SOUNDNESS_MODEL := {
    canonical_form:
        "T DERIVES phi -> T ENTAILS phi",
    syntactic_side:
        "formal derivability in a specified proof system",
    semantic_side:
        "truth in every model of the premises",
    interpretation:
        "formal proofs do not derive semantically invalid consequences from semantically satisfied premises"
}

DEFINE RULE_SOUNDNESS_MODEL := {
    input:
        "truth of every premise under an interpretation",
    rule:
        InferenceRule,
    output_requirement:
        "truth of the rule conclusion under the same interpretation",
    property:
        TruthPreservation
}

DEFINE PROOF_SYSTEM_SOUNDNESS_MODEL := {
    sufficient_pattern: {
        "axioms are semantically valid or appropriately satisfied",
        "each primitive inference rule preserves truth",
        "proofs are finite or otherwise admitted rule-governed derivations"
    },
    consequence:
        "every theorem is valid and every derivable consequence is semantically entailed"
}

DEFINE SOUNDNESS_COMPLETENESS_DISTINCTION := {
    Soundness:
        "DERIVES -> ENTAILS",
    Completeness:
        "ENTAILS -> DERIVES",
    combined:
        "DERIVES IFF ENTAILS",
    equivalence_of_properties: FALSE
}

DEFINE CRITICAL_DISTINCTIONS := {
    soundness_vs_consistency: {
        soundness:
            "proof-theoretic derivations preserve semantic consequence",
        consistency:
            "the system does not derive a contradiction in the relevant sense",
        implication:
            "soundness plus a suitable model/semantics can establish consistency, but the notions are distinct"
    },
    soundness_vs_validity: {
        soundness:
            "property of a proof system or inference rule",
        validity:
            "semantic property of a formula or argument"
    },
    soundness_vs_completeness: {
        soundness:
            "no derivable semantic false positives",
        completeness:
            "no semantically entailed consequences are missed by derivability"
    }
}

DEFINE EXAMPLE_1 := {
    rule:
        ModusPonens,
    premises: {
        "P",
        "P IMPLIES Q"
    },
    conclusion:
        "Q",
    soundness_condition:
        "every interpretation satisfying both premises also satisfies Q"
}

DEFINE EXAMPLE_2 := {
    proof_system:
        "sound first-order calculus",
    theory:
        "T",
    derivation:
        "T DERIVES phi",
    consequence:
        "T ENTAILS phi"
}

DEFINE EXAMPLE_3 := {
    candidate_unsoundness_witness: {
        "T DERIVES phi",
        "M is a model of T",
        "M does not satisfy phi"
    },
    consequence:
        "the proof system is not sound"
}

DEFINE RULE_VERIFICATION_PIPELINE :=
    InferenceRule(R)
    -> EnumerateOrCharacteriseInterpretations
    -> AssumeAllPremisesSatisfied
    -> EvaluateConclusion
    -> SearchForCounterinterpretation
    -> IF counterinterpretation_exists
       THEN RuleUnsound
       ELSE RuleSoundUnderAnalysedSemantics

DEFINE PROOF_SOUNDNESS_PIPELINE :=
    ProofSystem(S)
    -> IdentifyAxioms
    -> IdentifyPrimitiveRules
    -> EstablishSemanticValidityOfAxioms
    -> EstablishTruthPreservationOfRules
    -> InductOverProofLengthOrStructure
    -> ShowEveryDerivedStepSemanticallyFollows
    -> ConcludeSystemSoundness

DEFINE COUNTERMODEL_PIPELINE :=
    Theory(T)
    -> Formula(phi)
    -> SearchForModel(M)OfT
    -> Evaluate(phi,M)
    -> IF phi_false_in_M
       THEN Countermodel(M,T,phi)
    -> IF S_is_sound
       THEN ConcludeNotDerivable(T,phi)

DEFINE META_REASONING_PIPELINE :=
    InputClaim
    -> IdentifyProofSystem
    -> IdentifySemanticFramework
    -> DistinguishDerivabilityFromEntailment
    -> CheckSoundnessStatus
    -> TransferSyntacticResultToSemanticResultOnlyIfSoundnessApplies
    -> CheckForCountermodels
    -> ReturnQualifiedConclusion

DEFINE SOUNDNESS_GRAPH := {
    nodes: {
        TheoryNode,
        FormulaNode,
        ProofNode,
        ProofSystemNode,
        ModelClassNode,
        SoundnessNode
    },
    syntactic_edge:
        "T --DERIVES--> phi",
    semantic_edge:
        "T --ENTAILS--> phi",
    soundness_bridge:
        "DERIVES -> ENTAILS",
    countermodel_edge:
        "M --HAS_COUNTERMODEL_FOR--> phi relative to T"
}

DEFINE ONTOLOGY_RELATIONS := {
    Soundness IS_COMPONENT_OF ModelTheory,
    Soundness IS_META_PROPERTY_OF ProofSystem,
    SoundInferenceRule PRESERVES_TRUTH,
    ProofSystem IS_SOUND WHEN derivability_implies_entailment,
    Derivability RELATES Theory TO Formula,
    SemanticConsequence RELATES Theory TO Formula,
    Soundness CONNECTS Derivability TO SemanticConsequence,
    Countermodel WITNESSES failure_of_SemanticConsequence,
    Countermodel BLOCKS Derivability WHEN ProofSystemIsSound,
    Completeness IS_DISTINCT_FROM Soundness,
    Consistency IS_RELATED_TO Soundness
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "soundness" <=> Soundness WHEN logic_or_proof_system_context=TRUE,
    "semantic soundness" <=> Soundness,
    "sound proof system" <=> SoundProofSystem,
    "sound inference rule" <=> SoundInferenceRule,
    "truth preserving" <=> TruthPreservation,
    "derivability implies entailment" <=> Soundness
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation of proof-system soundness and truth-preserving inference",
    preserve_proof_system_identity: TRUE,
    preserve_semantic_framework: TRUE,
    preserve_theory_identity: TRUE,
    distinguish_derivability_from_entailment: TRUE,
    distinguish_soundness_from_completeness: TRUE,
    distinguish_soundness_from_consistency: TRUE,
    validate_inference_rule_truth_preservation: TRUE,
    support_inductive_proof_soundness_analysis: TRUE,
    support_countermodel_reasoning: TRUE,
    transfer_syntactic_to_semantic_only_when_soundness_applies: TRUE,
    reasoning_mode: "symbolic + semantic + proof-theoretic + meta-logical"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    soundness_direction_mapping: 1.00,
    derivability_entailment_distinction: 1.00,
    rule_soundness_mapping: 0.99,
    proof_system_soundness_mapping: 0.99,
    countermodel_reasoning: 0.99,
    soundness_consistency_relation: context_dependent,
    arbitrary_system_soundness: proof_system_and_semantics_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        derivability_to_entailment_direction,
        truth_preservation,
        syntactic_semantic_distinction,
        rule_soundness,
        proof_system_soundness,
        countermodel_blocking_under_soundness
    },
    extensible_layers: {
        local_soundness,
        global_soundness,
        soundness_of_natural_deduction,
        soundness_of_sequent_calculus,
        soundness_of_resolution,
        modal_logic_soundness,
        intuitionistic_logic_soundness,
        type_theoretic_soundness,
        proof_assistant_metatheory
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + SOUNDNESS_MODEL
    + RULE_SOUNDNESS_MODEL
    + PROOF_SYSTEM_SOUNDNESS_MODEL
    + SOUNDNESS_COMPLETENESS_DISTINCTION
    + CRITICAL_DISTINCTIONS
    + RULE_VERIFICATION_PIPELINE
    + PROOF_SOUNDNESS_PIPELINE
    + COUNTERMODEL_PIPELINE
    + META_REASONING_PIPELINE
    + SOUNDNESS_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: SoundnessOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/model-theory/soundness-ontology.html",
    parent_domain: ModelTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: Soundness,
    truth_preservation_enabled: TRUE,
    rule_soundness_reasoning_enabled: TRUE,
    proof_system_soundness_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