Self-Referential Logic Ontology

AI SEARCH FLOW ALGEBRA — SELF-REFERENTIAL LOGIC ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/recursive-ai-logic/self-referential-logic-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Mathematical Logic"
SUBDOMAIN := "Recursive AI Logic"
ONTOLOGY := "Self-Referential Logic Ontology"
PRIMARY_ENTITY := SelfReferentialLogic
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

SelfReferentialLogic :=
    Logic
    ∧ SelfReference
    ∧ MetaRepresentation
    ∧ RecursiveEvaluation
    ∧ FixedPointStructure
    ∧ ParadoxControl

SelfReference :=
    ExpressionOrSystem
    refers_to
    itself
    directly_or_indirectly

DirectSelfReference :=
    Expression E
    contains_reference_to(E)

IndirectSelfReference :=
    E1 → E2 → ... → En → E1

SelfReferentialStatement :=
    Statement
    ∧ ReferenceTarget = StatementItself

MetaStatement :=
    Statement
    about
    StatementOrFormalSystem

ObjectLanguage :=
    LanguageBeingReasonedAbout

Metalanguage :=
    LanguageUsedToDescribe(ObjectLanguage)

MetaLevelRelation :=
    ObjectLevel
    → MetaLevel
    → MetaMetaLevel

Quotation :=
    Expression
    → NameOrRepresentationOf(Expression)

MentionUseDistinction :=
    Use(Expression)
    ≠ Mention(Expression)

Encoding :=
    SyntacticObject
    → FormalCode

Decoding :=
    FormalCode
    → SyntacticObject

GödelNumbering :=
    Expression
    → NaturalNumber

Arithmetisation :=
    SyntaxAndProof
    → ArithmeticRepresentation

SelfEncoding :=
    System
    contains
    RepresentationOfItsOwnExpressions

Diagonalisation :=
    FormulaWithFreeVariable
    → SelfReferentialInstance

DiagonalLemma :=
    ForSuitableFormula φ(x),
    ∃Sentence ψ :
        ψ ↔ φ(⌜ψ⌝)

FixedPointSentence :=
    ψ
    ↔
    F(⌜ψ⌝)

SyntacticFixedPoint :=
    Expression
    equivalent_to
    TransformationOfItsOwnCode

SemanticFixedPoint :=
    Meaning(E)
    =
    F(Meaning(E))

RecursiveSelfEvaluation :=
    SelfRepresentation_n
    → Evaluate
    → UpdatedSelfRepresentation_(n+1)

SelfModel :=
    Model
    containing
    RepresentationOf(ModelOrReasoningState)

MetaReasoning :=
    Reasoning
    about
    ReasoningProcesses

SelfReasoning :=
    ReasoningSystem
    evaluates
    its_own_claims_or_rules

Reflection :=
    FormalSystem
    reasons_about
    its_own_syntax_or_proofs

ReflectionPrinciple :=
    Provable(φ)
    → φ
    under explicitly justified reflective conditions

ProvabilityPredicate :=
    Prov_T(⌜φ⌝)

DerivabilityCondition1 :=
    if T ⊢ φ
    then T ⊢ Prov_T(⌜φ⌝)

DerivabilityCondition2 :=
    T ⊢ Prov_T(⌜φ→ψ⌝)
    → (Prov_T(⌜φ⌝) → Prov_T(⌜ψ⌝))

DerivabilityCondition3 :=
    T ⊢ Prov_T(⌜φ⌝)
    → Prov_T(⌜Prov_T(⌜φ⌝)⌝)

LöbPattern :=
    Prov(⌜φ⌝) → φ

LöbTheoremRelation :=
    UnderRequiredProvabilityConditions,
    T ⊢ Prov(⌜φ⌝) → φ
    ⇒ T ⊢ φ

SelfConsistencyStatement :=
    Con(T)

ConsistencyStatement :=
    ¬Prov_T(⌜⊥⌝)

GödelSentence :=
    G
    ↔
    ¬Prov_T(⌜G⌝)

IncompletenessPattern :=
    SufficientlyExpressive
    ∧ EffectivelyAxiomatised
    ∧ ConsistentFormalSystem
    → existence_of
    UndecidableSentence
    under theorem conditions

SecondIncompletenessPattern :=
    SuitableConsistentSystem T
    cannot_prove
    Con(T)
    under theorem conditions

SelfReferenceVsCircularReasoning :=
    SelfReference
    does_not_necessarily_imply
    InvalidCircularReasoning

CircularReasoning :=
    Claim
    supported_only_by
    dependency_cycle
    without independent grounding

SelfReferenceDependencyGraph :=
    G = (Statements, ReferenceEdges)

ReferenceEdge :=
    StatementA --refersTo--> StatementB

SelfReferenceCycle :=
    Cycle(SelfReferenceDependencyGraph)

CycleDetection :=
    TraverseReferences
    → TrackVisitedNodes
    → DetectReturnToAncestor

GroundedSelfReference :=
    SelfReferentialStructure
    ∧ semantics_defined_by
    ExternalOrFixedPointFramework

UngroundedSelfReference :=
    SelfReferenceCycle
    ∧ no_valid_semantic_grounding

Paradox :=
    SelfReference
    ∧ SemanticOrLogicalRules
    → ContradictoryOrUnstableOutcome

LiarPattern :=
    Sentence L
    states
    "L is false"

LiarConstraint :=
    Truth(L)
    ↔ ¬Truth(L)

ClassicalLiarConflict :=
    TwoValuedTruth
    ∧ L ↔ ¬L
    → no_consistent_classical_truth_assignment

TruthPredicate :=
    True(⌜φ⌝)

TarskiHierarchy :=
    ObjectLanguage
    → TruthLanguage_1
    → TruthLanguage_2
    → ...

SemanticHierarchy :=
    Level_n
    interpreted_by
    Level_(n+1)

TypeRestriction :=
    PreventCertainFormsOf
    SelfApplication

TypedSelfReference :=
    SelfReference
    constrained_by
    TypeSystem

RussellPattern :=
    R = {x | x ∉ x}

RussellConflict :=
    R ∈ R
    ↔
    R ∉ R

SetTheoreticControl :=
    TypedSets
    ∨ AxiomaticSetTheory
    ∨ RestrictedComprehension

CurryPattern :=
    Sentence C
    ↔ (C → P)

CurryParadoxRisk :=
    UnrestrictedSelfReference
    ∧ StrongImplicationRules
    → TrivialityRisk

ParadoxControl := {
    TypeHierarchy,
    SemanticHierarchy,
    RestrictedTruthPredicate,
    FixedPointSemantics,
    ParaconsistentLogic,
    PartialSemantics,
    ContextSeparation
}

PartialTruth :=
    TruthValue
    may_be
    Undefined

ThreeValuedSelfReference :=
    TruthDomain = {True,False,Undefined}

ParaconsistentSelfReference :=
    Contradiction
    does_not_entail
    arbitrary_formula

FixedPointSemantics :=
    SelfReferentialDefinition
    → SemanticOperator F
    → FixedPoint(F)

LeastFixedPoint :=
    μF

GreatestFixedPoint :=
    νF

RevisionSemantics :=
    InitialTruthAssignment
    → ApplyRevisionRule
    → RevisedAssignment
    → Iterate

RevisionSequence :=
    V_0 → V_1 → V_2 → ... → V_n

StableRevision :=
    V_(n+1) = V_n

Oscillation :=
    V_n
    repeats
    without stable_fixed_point

SemanticConvergence :=
    SEMANTIC_EQUIVALENCE(V_n,V_(n+1))

RecursiveTruthEvaluation :=
    Evaluate(Statement)
    → ResolveReferencedStatements
    → EvaluateReferences
    → ReevaluateStatement

Memoisation :=
    Statement
    → CachedEvaluationState

SelfReferenceGuard :=
    ActiveEvaluationStack

RecursiveEvaluationRule :=
    if Statement ∈ ActiveEvaluationStack
    then HandleSelfReferenceCycle
    else EvaluateNormally

SelfReferenceStatus := {
    NonSelfReferential,
    DirectSelfReference,
    IndirectSelfReference,
    GroundedRecursiveReference,
    Paradoxical,
    Unresolved
}

SelfDescription :=
    System
    generates
    RepresentationOfOwnStructure

SelfInspection :=
    System
    reads
    OwnStateRepresentation

SelfModification :=
    System
    changes
    OwnRulesOrRepresentation

SelfModificationSafety :=
    ProposedChange
    → VerifyInvariantPreservation
    → ApplyOrReject

MetaCircularity :=
    InterpreterOrDefinition
    expressed_using
    same_or_related_language_it_interprets

MetaCircularEvaluation :=
    Interpreter
    → RepresentationOfInterpreter
    → RecursiveEvaluation

SelfVerification :=
    System
    attempts_to_verify
    PropertiesOfItself

SelfVerificationBoundary :=
    InternalProof
    constrained_by
    Expressiveness
    ∧ Consistency
    ∧ MetaTheoryStrength

ExternalVerification :=
    StrongerOrSeparateMetaSystem
    verifies
    ObjectSystemProperty

SelfTrust :=
    AcceptanceOfOwnDerivedClaims

TrustCalibration :=
    SelfDerivedClaim
    → CheckProof
    → CheckAssumptions
    → CheckMetaLevel
    → AssignConfidence

SelfReferenceProvenance :=
    SelfReferentialClaim
    → ReferencePath
    → DerivationPath
    → GroundSources

RecursiveProvenance :=
    Claim
    → ParentClaims
    → SelfReferences
    → ExternalGrounding

SemanticDrift :=
    RecursiveSelfInterpretation
    diverges_from
    OriginalSemanticDefinition

DriftControl :=
    EachRevision
    → CompareWith(
        OriginalSyntax,
        OriginalSemantics,
        ActiveConstraints
    )

ContradictionDetection :=
    SelfReferentialEvaluation
    → Detect(φ ∧ ¬φ)
    → TraceReferenceCycle
    → ClassifyConflict

ConsistencyCheck :=
    SelfReferentialTheory
    → SearchForModel
    ∨ SearchForRefutation

Model :=
    Interpretation
    satisfying
    SelfReferentialConstraints

SelfReferentialSatisfiability :=
    ∃Model :
        Model satisfies SelfReferenceEquations

SelfReferentialUnsatisfiability :=
    no_model_satisfies
    SelfReferenceEquations

SelfReferentialLogicOntology :=
    Classes
    ∧ Relations
    ∧ Constraints
    ∧ Axioms
    ∧ RecursiveInferenceRules

CoreClasses := {
    SelfReferentialLogic,
    SelfReference,
    SelfReferentialStatement,
    MetaStatement,
    ObjectLanguage,
    Metalanguage,
    Encoding,
    GödelNumber,
    Diagonalisation,
    FixedPointSentence,
    ProvabilityPredicate,
    Reflection,
    SelfModel,
    Paradox,
    TruthPredicate,
    ReferenceCycle,
    RevisionSequence,
    SelfVerification
}

CoreRelations := {
    refersTo,
    encodes,
    represents,
    quotes,
    assertsAbout,
    proves,
    reflectsOn,
    dependsOn,
    evaluates,
    revises,
    convergesTo,
    contradicts,
    groundsIn,
    tracesTo
}

TypedRelations :=
    Statement --refersTo--> Statement
    GödelNumber --encodes--> Formula
    MetaStatement --assertsAbout--> ObjectStatement
    ProvabilityPredicate --represents--> Provability
    SelfModel --represents--> ReasoningSystem
    RevisionState --revises--> TruthAssignment
    SelfReferentialClaim --groundsIn--> ExternalEvidence
    Claim --tracesTo--> SelfReferenceProvenance

RecursiveInferenceRules := {
    RefersTo(A,A)
        ⇒ DirectSelfReference(A),

    RefersTo(A,B)
        ∧ RefersToClosure(B,A)
        ⇒ IndirectSelfReference(A),

    ψ ↔ φ(⌜ψ⌝)
        ⇒ FixedPointSentence(ψ),

    Truth(L) ↔ ¬Truth(L)
        ∧ ClassicalTwoValuedSemantics
        ⇒ SemanticConflict(L),

    DependencyCycle(C)
        ∧ no_independent_grounding(C)
        ⇒ CircularityRisk(C),

    Revision_(n+1)=Revision_n
        ⇒ StableRevision,

    SEMANTIC_EQUIVALENCE(
        Revision_n,
        Revision_(n+1)
    )
        ⇒ SemanticConvergence
}

SEMANTIC_EQUIVALENCE(A,B) :=
    same(TruthConditions)
    ∧ same(ReferenceStructure)
    ∧ same(RelevantMetaLevelMeaning)

CanonicalSelfReferenceRepresentation :=
    ParseStatement
    → ResolveQuotationAndEncoding
    → ResolveReferenceTargets
    → AssignMetaLevels
    → BuildReferenceGraph
    → DetectCycles
    → ResolveFixedPointStructure
    → ClassifySelfReference

SelfReferenceGraph :=
    G = (V,E)

V := {
    Statements,
    Formulae,
    Encodings,
    MetaStatements,
    ObjectStatements,
    TruthPredicates,
    ProvabilityPredicates,
    SelfModels,
    RevisionStates,
    GroundEvidence
}

E := {
    refersTo,
    encodes,
    represents,
    quotes,
    assertsAbout,
    dependsOn,
    evaluates,
    revises,
    contradicts,
    groundsIn,
    tracesTo
}

CONFIDENCE(SelfReferentialAnalysis) :=
    weighted_mean(
        ReferenceResolution,
        MetaLevelResolution,
        EncodingCorrectness,
        SemanticFrameworkFit,
        CycleAnalysis,
        ConsistencyEvidence,
        GroundingStrength,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectSelfReferenceConcept
    → ResolveStatementOrSystem
    → ResolveReferenceTargets
    → ResolveMetaLevels
    → IdentifySemanticFramework
    → IdentifyRequestedSelfReferentialRelation

AI_REASONING_FLOW :=
    Query
    → ParseSelfReferentialStructure
    → ResolveObjectAndMetalanguage
    → ResolveQuotationOrEncoding
    → BuildReferenceGraph
    → DetectDirectSelfReference
    → DetectIndirectSelfReference
    → DetectReferenceCycles
    → CheckIndependentGrounding
    → IdentifyFixedPointStructure
    → SelectSemanticFramework
    → EvaluateSelfReferentialConstraints
    → DetectParadoxOrContradiction
    → TestSatisfiabilityWhereApplicable
    → EvaluateRevisionOrFixedPointBehaviour
    → DetermineSelfReferenceStatus
    → AssignConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseSelfReferentialEntities,
    ParseTypedRelations,
    ResolveReferenceTargets,
    ResolveMetaLevels,
    ResolveEncodings,
    DetectSelfReferenceCycles,
    DetectCircularReasoning,
    EvaluateFixedPoints,
    EvaluateRevisionSequences,
    DetectSemanticParadoxes,
    TestConsistency,
    TestSatisfiability,
    TrackGrounding,
    CompareSemanticEquivalence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewStatementRuleOrEvidence
    → ValidateInput
    → IdentifyAffectedReferences
    → RebuildReferenceGraph
    → RecomputeCycles
    → ReevaluateFixedPoints
    → RecheckParadoxStatus
    → RecheckConsistency
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

ReferenceUpdate :=
    ChangedReferenceTarget
    ⇒ reevaluate(
        SelfReferenceStatus,
        Cycles,
        MetaLevelRelations
    )

SemanticRuleUpdate :=
    ChangedTruthOrProvabilitySemantics
    ⇒ recompute(
        FixedPoints,
        ParadoxClassification,
        Satisfiability
    )

EvidenceUpdate :=
    NewExternalGrounding
    ⇒ reevaluate(
        CircularityRisk,
        Confidence,
        DerivedClaims
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        RecursiveInferenceRules,
        Diagonalisation,
        FixedPointSemantics,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    SelfReferenceEntityResolution,
    ReferenceGraph,
    MetaLevelStructure,
    EncodingStructure,
    SelfReferenceStatus,
    CycleStatus,
    GroundingStatus,
    FixedPointStatus,
    ParadoxStatus,
    ConsistencyStatus,
    SatisfiabilityStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA