Logical Consistency Ontology

AI SEARCH FLOW ALGEBRA — LOGICAL CONSISTENCY ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/logical-semantics/logical-consistency-ontology.html

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

BEGIN_FLOW

LogicalConsistency :=
    AbsenceOfContradiction
    ∧ JointSatisfiability
    ∧ CoherentInference
    ∧ NonTriviality

FormalSystem :=
    Language
    ∧ Axioms
    ∧ InferenceRules
    ∧ Semantics

Theory :=
    SetOfFormulae

SyntacticConsistency(T) :=
    ¬∃φ :
        T ⊢ φ
        ∧
        T ⊢ ¬φ

SemanticConsistency(T) :=
    ∃Model M :
        M ⊨ T

ModelExistence :=
    Mod(T) ≠ ∅

Inconsistency(T) :=
    ∃φ :
        T ⊢ φ
        ∧
        T ⊢ ¬φ

SemanticInconsistency(T) :=
    Mod(T) = ∅

Contradiction :=
    φ ∧ ¬φ

ExplicitContradiction :=
    T contains_or_derives
    both(φ,¬φ)

ImplicitContradiction :=
    Constraints(T)
    jointly_unsatisfiable

SatisfiableSet :=
    Γ
    iff
    ∃M :
        M ⊨ Γ

UnsatisfiableSet :=
    Γ
    iff
    ¬∃M :
        M ⊨ Γ

ConsistencyRelation :=
    Consistent(T)
    ↔ Satisfiable(T)
    under sound_and_complete classical systems

Soundness :=
    T ⊢ φ
    ⇒
    T ⊨ φ

Completeness :=
    T ⊨ φ
    ⇒
    T ⊢ φ

Explosion :=
    φ
    ∧ ¬φ
    ⊢ ψ

PrincipleOfExplosion :=
    InconsistentClassicalTheory
    ⇒ EveryFormulaDerivable

TrivialTheory :=
    ∀φ :
        T ⊢ φ

NonTriviality :=
    ∃φ :
        T ⊬ φ

ConsistencyImpliesNonTriviality :=
    Consistent(T)
    ⇒ NonTrivial(T)

ParaconsistentLogic :=
    Logic
    ∧ ContradictionDoesNotNecessarilyImplyExplosion

ParaconsistentConsistency :=
    contradiction_tolerant
    ∧ inference_controlled

LocalConsistency :=
    SubsetOfTheory
    ∧ internally_consistent

GlobalConsistency :=
    EntireTheory
    ∧ no_global_contradiction

PairwiseConsistency :=
    ∀φ,ψ ∈ T :
        satisfiable({φ,ψ})

JointConsistency :=
    satisfiable(T)

PairwiseDoesNotAlwaysImplyJoint :=
    PairwiseConsistency
    ↛ JointConsistency

FiniteConsistency :=
    EveryFiniteSubset(T)
    is_consistent

CompactnessRelation :=
    EveryFiniteSubset(T) satisfiable
    ⇒ T satisfiable
    for compact logics

RelativeConsistency :=
    Consistent(T1)
    ⇒ Consistent(T2)

Equiconsistency :=
    Consistent(T1)
    ↔ Consistent(T2)

ConsistencyStrength :=
    TheoryA
    proves
    ConsistencyStatement(TheoryB)

ConsistencyStatement :=
    Con(T)

MetaTheory :=
    Theory
    used_to_reason_about
    ObjectTheory

ObjectTheory :=
    TheoryBeingAnalysed

MetaConsistency :=
    Consistency(MetaTheory)

ProofTheoreticConsistency :=
    NoDerivationOf(⊥)

Bottom :=
    ⊥

ProofConsistency :=
    T ⊬ ⊥

ModelTheoreticConsistency :=
    ∃M :
        M ⊨ T

ConstraintConsistency :=
    ∃Assignment a :
        a satisfies(AllConstraints)

OntologyConsistency :=
    OntologyAxioms
    ∧ ClassConstraints
    ∧ PropertyConstraints
    ∧ IndividualAssertions
    are_jointly_satisfiable

KnowledgeBaseConsistency :=
    Facts
    ∧ Rules
    ∧ Constraints
    do_not_entail(Contradiction)

DatabaseConsistency :=
    DataState
    satisfies
    IntegrityConstraints

SpecificationConsistency :=
    Requirements
    ∧ Assumptions
    ∧ Guarantees
    ∧ Constraints
    are_jointly_satisfiable

ConsistencyCheck :=
    Theory
    → Normalise
    → GenerateConstraints
    → TestSatisfiability
    → {
        Consistent,
        Inconsistent,
        Unknown
    }

ConsistencyStatus := {
    Consistent,
    Inconsistent,
    LocallyInconsistent,
    Unknown,
    Undetermined
}

SATConsistency :=
    PropositionalTheory
    → SATSolver

SATResult :=
    SAT
    ⇒ Consistent

SATUnsatResult :=
    UNSAT
    ⇒ Inconsistent

SMTConsistency :=
    Theory
    ∧ BackgroundTheories
    → SMTSolver

ModelFinding :=
    Theory
    → SearchForModel

ModelFound :=
    Model M
    ⇒ SemanticConsistency

Refutation :=
    Theory
    → Derive(⊥)

RefutationFound :=
    DerivationOfBottom
    ⇒ SyntacticInconsistency

ResolutionConsistency :=
    Clausify(Theory)
    → Resolution
    → {
        EmptyClauseFound,
        NoRefutationFound,
        Unknown
    }

EmptyClause :=
    □
    := Contradiction

UnsatCore :=
    MinimalOrNonMinimalSubset
    ∧ itself_unsatisfiable

MinimalUnsatisfiableSubset :=
    MUS ⊆ T
    ∧ Unsatisfiable(MUS)
    ∧ ∀ProperSubset S ⊂ MUS :
        Satisfiable(S)

MaximalConsistentSubset :=
    MCS ⊆ T
    ∧ Consistent(MCS)
    ∧ no_strict_superset_within_T_is_consistent

ConflictSet :=
    SetOfStatements
    ∧ jointly_inconsistent

ContradictionDetection :=
    Formula
    ∧ NegationOfFormula
    → Conflict

SemanticConflict :=
    ConstraintsA
    ∧ ConstraintsB
    are_unsatisfiable

TypeConflict :=
    Entity
    assigned_to
    MutuallyDisjointTypes

CardinalityConflict :=
    MinimumCardinality
    >
    MaximumCardinality

EqualityConflict :=
    a = b
    ∧ a ≠ b

OrderingConflict :=
    a < b
    ∧ b ≤ a

TemporalConflict :=
    RequiredBefore(A,B)
    ∧ RequiredBefore(B,A)
    under strict_order

IntegrityConflict :=
    Assertion
    violates
    IntegrityConstraint

AssumptionConflict :=
    AssumptionA
    ∧ AssumptionB
    cannot_both_hold

RuleConflict :=
    SameConditions
    derive
    incompatible_conclusions

ConsistencyPreservation :=
    Transformation(T,T')
    ∧ Consistent(T)
    ⇒ Consistent(T')

ConservativeExtension :=
    T' extends T
    ∧ no_new_theorems_in_old_language
    beyond those already entailed by T

DefinitionalExtension :=
    AddDefinitions
    ∧ preserve(Consistency)
    under valid definitional conditions

TheoryExtension :=
    T' = T ∪ NewAxioms

SafeTheoryExtension :=
    Consistent(T')
    ∧ preserves_required_properties(T)

ConsistencyUnderUnion :=
    Consistent(T1)
    ∧ Consistent(T2)
    does_not_alone_imply
    Consistent(T1 ∪ T2)

Compatibility :=
    Consistent(T1 ∪ T2)

TheoryMerge :=
    T1
    ∪ T2
    → DetectConflicts
    → ResolveOrAnnotate
    → MergedTheory

BeliefRevision :=
    InconsistentKnowledgeBase
    → RemoveOrModifyBeliefs
    → RestoreConsistency

RevisionOperator :=
    K * φ

Contraction :=
    K - φ

Expansion :=
    K + φ

ConsistencyRestoration :=
    IdentifyConflictSets
    → RankOrAnalyseStatements
    → ModifyTheory
    → RecheckConsistency

ProvenanceAwareConsistency :=
    Conflict
    → PreserveSources
    → PreserveDerivationPaths
    → SupportResolution

ContradictionProvenance :=
    Contradiction
    → SetOfSupportingAxiomsAndRules

DependencyGraph :=
    G = (Statements, DerivationDependencies)

ConflictPropagation :=
    ChangedAxiom
    → DependentConclusions
    → PotentialConsistencyChanges

IncrementalConsistencyChecking :=
    TheoryUpdate
    → IdentifyAffectedRegion
    → RecheckAffectedConstraints
    → UpdateConsistencyStatus

ConsistencyCertificate :=
    Model
    ∨ ProofOfRelativeConsistency
    ∨ SolverCertificate

InconsistencyCertificate :=
    Refutation
    ∨ UnsatCore
    ∨ ContradictoryDerivation

ConfidenceInConsistency :=
    depends_on(
        LogicCoverage,
        SolverCompleteness,
        ModelCoverage,
        AxiomQuality,
        Provenance
    )

GödelConsistencyBoundary :=
    SufficientlyExpressiveConsistentFormalSystems
    have_limits_on
    internal_proof_of_own_consistency

SecondIncompletenessRelation :=
    UnderRequiredConditions,
    ConsistentTheory T
    cannot_prove
    Con(T)
    using_only_its_own_resources

ConsistencyVsCompleteness :=
    Consistency
    ≠ Completeness

ConsistencyVsValidity :=
    Consistency(Theory)
    ≠ Validity(Formula)

ConsistencyVsSatisfiability :=
    CloselyRelated
    but scoped_to
    TheoryOrFormulaSet

LogicalConsistencyOntology :=
    Classes
    ∧ Relations
    ∧ Constraints
    ∧ Axioms
    ∧ InferenceRules

CoreClasses := {
    LogicalConsistency,
    Theory,
    Formula,
    Axiom,
    Model,
    Contradiction,
    Conflict,
    Constraint,
    Proof,
    Refutation,
    UnsatCore,
    MaximalConsistentSubset,
    FormalSystem,
    MetaTheory,
    ConsistencyStatement,
    Solver,
    ConsistencyCertificate
}

CoreRelations := {
    consistentWith,
    contradicts,
    entails,
    derives,
    satisfies,
    violates,
    conflictsWith,
    dependsOn,
    extends,
    preserves,
    refutes,
    supports,
    hasModel,
    hasUnsatCore
}

TypedRelations :=
    Theory --hasModel--> Model
    Formula --contradicts--> Formula
    Model --satisfies--> Theory
    Constraint --conflictsWith--> Constraint
    Refutation --refutes--> ConsistencyClaim
    UnsatCore --supports--> InconsistencyClaim
    TheoryExtension --extends--> Theory
    Transformation --preserves--> Consistency
    ConsistencyCertificate --supports--> ConsistencyClaim

InferenceRules := {
    ∃M : M ⊨ T
        ⇒ SemanticallyConsistent(T),

    T ⊢ ⊥
        ⇒ SyntacticallyInconsistent(T),

    T ⊢ φ
        ∧ T ⊢ ¬φ
        ⇒ Inconsistent(T),

    Unsatisfiable(T)
        ⇒ SemanticallyInconsistent(T),

    Sound(FormalSystem)
        ∧ SyntacticallyConsistent(T)
        does_not_by_itself_imply_model_existence_without_relevant_completeness_conditions,

    Sound(FormalSystem)
        ∧ Complete(FormalSystem)
        ⇒ (
            SyntacticConsistency(T)
            ↔ SemanticConsistency(T)
        ),

    UnsatCore(U)
        ∧ U ⊆ T
        ⇒ Inconsistent(T),

    Consistent(T1 ∪ T2)
        ⇒ Compatible(T1,T2)
}

SEMANTIC_EQUIVALENCE(TheoryA, TheoryB) :=
    Mod(TheoryA)
    =
    Mod(TheoryB)

ConsistencyEquivalence :=
    Consistent(TheoryA)
    ↔ Consistent(TheoryB)

ConflictSubsumption :=
    ConflictSetA ⊆ ConflictSetB
    ∧ Unsatisfiable(ConflictSetA)
    ⇒ ConflictSetB is Unsatisfiable

CanonicalConsistencyRepresentation :=
    ParseTheory
    → NormaliseFormulae
    → ResolveLogic
    → ExtractConstraints
    → BuildDependencyGraph
    → DetectDirectContradictions
    → TestSatisfiability
    → ExtractConflictEvidence

ConsistencyGraph :=
    G = (V,E)

V := {
    Theories,
    Formulae,
    Axioms,
    Constraints,
    Models,
    Contradictions,
    ConflictSets,
    UnsatCores,
    Proofs,
    Refutations,
    Certificates
}

E := {
    consistentWith,
    contradicts,
    entails,
    derives,
    satisfies,
    violates,
    conflictsWith,
    dependsOn,
    extends,
    preserves,
    supports
}

CONFIDENCE(LogicalConsistencyAssessment) :=
    weighted_mean(
        FormalisationQuality,
        LogicResolution,
        SolverCoverage,
        ModelEvidence,
        ConflictAnalysis,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectConsistencyConcept
    → ResolveTheoryOrKnowledgeBase
    → ResolveLogic
    → IdentifyAssertionsAndConstraints
    → IdentifyRequestedConsistencyRelation

AI_REASONING_FLOW :=
    Query
    → IdentifyFormalSystem
    → ParseTheory
    → NormaliseFormulae
    → ResolveAxioms
    → ResolveInferenceRules
    → BuildConstraintSet
    → DetectExplicitContradictions
    → DetectImplicitConflicts
    → TestSatisfiability
    → SearchForModel
    → SearchForRefutation
    → ExtractUnsatCoreIfInconsistent
    → DetermineConsistencyStatus
    → BuildConsistencyGraph
    → GenerateEvidence
    → ExplainResult

AI_CONSUMPTION := {
    ParseConsistencyEntities,
    ParseTypedRelations,
    ResolveFormalSystem,
    ResolveTheory,
    DetectContradictions,
    TestJointSatisfiability,
    ConstructModels,
    SearchForRefutations,
    ExtractUnsatCores,
    CompareTheoryCompatibility,
    FollowDependencies,
    PreserveConflictProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewLogicalEvidence
    → ValidateEvidence
    → IdentifyAffectedStatements
    → RecomputeDependencies
    → RecheckLocalConsistency
    → RecheckGlobalConsistency
    → UpdateConsistencyGraph
    → RecomputeConsistencyStatus
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

AxiomUpdate :=
    AddedOrRemovedAxiom
    ⇒ recheck(
        DependentDerivations,
        ModelClass,
        ConsistencyStatus
    )

ConstraintUpdate :=
    ChangedConstraint
    ⇒ recompute(
        ConflictSets,
        UnsatCores,
        Satisfiability
    )

ModelUpdate :=
    NewModel
    satisfying Theory
    ⇒ mark(Theory, SemanticallyConsistent)

RefutationUpdate :=
    ValidRefutation
    ⇒ mark(Theory, SyntacticallyInconsistent)

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        SyntacticConsistency,
        SemanticConsistency,
        ConsistencyCheck,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    ConsistencyEntityResolution,
    FormalSystem,
    Theory,
    ConstraintSet,
    ContradictionStatus,
    SatisfiabilityStatus,
    ModelEvidence,
    RefutationEvidence,
    UnsatCore,
    CompatibilityStatus,
    ConsistencyStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA