Tautology Ontology

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

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

BEGIN_FLOW

Tautology :=
    Formula
    ∧ TrueUnderEveryPermittedValuation

PropositionalTautology(φ) :=
    ∀Valuation v :
        v(φ) = True

SemanticValidity(φ) :=
    ∀Interpretation I :
        I ⊨ φ

TautologyPurpose :=
    identify(LogicalTruthIndependentOfAtomicAssignments)
    ∧ support(ValidityTesting)
    ∧ support(EquivalenceReasoning)
    ∧ support(TheoremProving)
    ∧ support(FormalVerification)

Formula :=
    AtomicFormula
    ∨ CompoundFormula

AtomicFormula :=
    PropositionalVariable

CompoundFormula :=
    Negation
    ∨ Conjunction
    ∨ Disjunction
    ∨ Implication
    ∨ Biconditional

Valuation :=
    v : PropositionalVariables → {True,False}

TruthEvaluation :=
    Formula
    ∧ Valuation
    → TruthValue

TruthValue := {
    True,
    False
}

Negation :=
    v(¬P) = ¬v(P)

Conjunction :=
    v(P∧Q) = v(P) ∧ v(Q)

Disjunction :=
    v(P∨Q) = v(P) ∨ v(Q)

Implication :=
    v(P→Q) = ¬v(P) ∨ v(Q)

Biconditional :=
    v(P↔Q) = (v(P)=v(Q))

LawOfExcludedMiddle :=
    P ∨ ¬P

NonContradictionLaw :=
    ¬(P ∧ ¬P)

IdentityLaw :=
    P → P

DoubleNegation :=
    P ↔ ¬¬P

ImplicationExpansion :=
    (P → Q)
    ↔
    (¬P ∨ Q)

Contraposition :=
    (P → Q)
    ↔
    (¬Q → ¬P)

DeMorgan1 :=
    ¬(P ∧ Q)
    ↔
    (¬P ∨ ¬Q)

DeMorgan2 :=
    ¬(P ∨ Q)
    ↔
    (¬P ∧ ¬Q)

CommutativityAnd :=
    (P ∧ Q)
    ↔
    (Q ∧ P)

CommutativityOr :=
    (P ∨ Q)
    ↔
    (Q ∨ P)

AssociativityAnd :=
    ((P ∧ Q) ∧ R)
    ↔
    (P ∧ (Q ∧ R))

AssociativityOr :=
    ((P ∨ Q) ∨ R)
    ↔
    (P ∨ (Q ∨ R))

DistributivityAnd :=
    P ∧ (Q ∨ R)
    ↔
    (P∧Q) ∨ (P∧R)

DistributivityOr :=
    P ∨ (Q ∧ R)
    ↔
    (P∨Q) ∧ (P∨R)

AbsorptionAnd :=
    P ∧ (P ∨ Q)
    ↔ P

AbsorptionOr :=
    P ∨ (P ∧ Q)
    ↔ P

IdempotenceAnd :=
    P ∧ P
    ↔ P

IdempotenceOr :=
    P ∨ P
    ↔ P

TautologyDetection :=
    Formula
    → Parse
    → IdentifyVariables
    → GenerateValuations
    → EvaluateAllValuations
    → DetermineTautologyStatus

TruthTableMethod :=
    Variables(φ)
    → 2^n Valuations
    → Evaluate(φ)

TruthTableTautology :=
    all_rows(φ) = True

TruthTableCounterexample :=
    ∃row :
        φ = False

Countervaluation :=
    v
    ∧ v(φ) = False

TautologyCriterion :=
    no(Countervaluation)

ValidityByUnsatisfiability :=
    Tautology(φ)
    iff
    Unsatisfiable(¬φ)

SATBasedTautologyCheck :=
    SAT(¬φ)

SATResult :=
    SAT(¬φ)
    ⇒ NotTautology(φ)

UNSATResult :=
    UNSAT(¬φ)
    ⇒ Tautology(φ)

ResolutionTautologyCheck :=
    Negate(φ)
    → CNF
    → Resolution
    → EmptyClause

EmptyClause :=
    □
    := Contradiction

ResolutionValidity :=
    NegationRefuted
    ⇒ OriginalFormulaValid

SemanticTableauCheck :=
    Tableau(¬φ)
    → {
        AllBranchesClosed: Tautology,
        OpenBranch: Countermodel
    }

NaturalDeductionCheck :=
    ⊢ φ
    ∧ SoundProofSystem
    ⇒ Valid(φ)

TheoremProvingCheck :=
    EmptyAssumptionContext
    ⊢ φ

ProofTheoreticTheorem :=
    ⊢ φ

SemanticTheorem :=
    ⊨ φ

SoundnessRelation :=
    ⊢ φ
    ⇒
    ⊨ φ

CompletenessRelation :=
    ⊨ φ
    ⇒
    ⊢ φ
    for complete proof systems

Contradiction :=
    Formula
    ∧ FalseUnderEveryValuation

ContradictionDefinition :=
    ∀v :
        v(φ) = False

Contingency :=
    Formula
    ∧ TrueUnderSomeValuations
    ∧ FalseUnderSomeValuations

Classification :=
    Formula
    → {
        Tautology,
        Contradiction,
        Contingency
    }

TautologyNegationRelation :=
    Tautology(φ)
    iff
    Contradiction(¬φ)

ContradictionNegationRelation :=
    Contradiction(φ)
    iff
    Tautology(¬φ)

LogicalEquivalence :=
    φ ≡ ψ
    iff
    Tautology(φ ↔ ψ)

SemanticEntailment :=
    Γ ⊨ φ

EntailmentAsTautology :=
    {γ1,...,γn} ⊨ φ
    iff
    Tautology(
        (γ1 ∧ ... ∧ γn) → φ
    )
    for finite Γ

ArgumentValidity :=
    Premises
    → Conclusion

ValidArgument :=
    Tautology(
        Conjunction(Premises)
        → Conclusion
    )

InferenceRuleValidity :=
    Premises
    semantically_entail
    Conclusion

SubstitutionInstance :=
    TautologicalSchema
    → ReplaceVariablesWithFormulae

SubstitutionPreservation :=
    Tautology(φ)
    ⇒
    Tautology(σ(φ))

TautologySchema :=
    FormulaPattern
    ∧ every_uniform_substitution_is_tautological

BooleanFunction :=
    {True,False}^n
    → {True,False}

TautologicalBooleanFunction :=
    constant(True)

NormalForm :=
    CNF
    ∨ DNF

CNF :=
    Conjunction(Clauses)

DNF :=
    Disjunction(ConjunctiveTerms)

TautologicalDNF :=
    DNF
    ∧ covers(AllValuations)

CNFTautology :=
    each_clause_is_tautological
    under standard propositional clause semantics

ClauseTautology :=
    ClauseContains(L)
    ∧ ClauseContains(¬L)

Simplification :=
    Formula
    → ApplyLogicalIdentities
    → EquivalentFormula

TautologySimplification :=
    φ ∧ ⊤ ↔ φ
    ∧
    φ ∨ ⊤ ↔ ⊤

TruthConstantTrue :=
    ⊤

TruthConstantFalse :=
    ⊥

IdentityWithTrue :=
    P ∧ ⊤ ↔ P

DominationByTrue :=
    P ∨ ⊤ ↔ ⊤

IdentityWithFalse :=
    P ∨ ⊥ ↔ P

DominationByFalse :=
    P ∧ ⊥ ↔ ⊥

SemanticPreservation :=
    Transform(φ,ψ)
    ∧ φ ≡ ψ
    ⇒
    Tautology(φ) ↔ Tautology(ψ)

TautologyUnderRenaming :=
    BijectiveVariableRenaming
    preserves
    TautologyStatus

PropositionalValidity :=
    Tautology

FirstOrderValidity :=
    FormulaTrueInEveryStructure
    under every assignment

FirstOrderTautologyRelation :=
    PropositionalTautologyInstances
    are
    FirstOrderValid
    when propositional atoms are replaced appropriately

LogicalTruth :=
    Formula
    ∧ true_in_all_admissible_interpretations

AnalyticTruthFormal :=
    TruthByLogicalForm
    within specified formal semantics

TautologyApplication := {
    CircuitSimplification,
    FormalVerification,
    ProofChecking,
    RuleValidation,
    SpecificationValidation,
    QueryOptimisation,
    ConstraintReasoning
}

VerificationCondition :=
    LogicalFormula
    ∧ correctness_requires(Validity)

VCAsTautology :=
    PropositionalVerificationCondition
    ∧ Tautology(VC)
    ⇒ Discharged

InvariantImplication :=
    Invariant
    ∧ Transition
    → NextInvariant

InvariantProofTautology :=
    Tautology(
        (Invariant ∧ Transition)
        → NextInvariant
    )
    under propositional abstraction

SafetyImplication :=
    ReachableConditions
    → SafeCondition

ContractValidity :=
    Precondition
    ∧ ProgramRelation
    → Postcondition

RedundancyDetection :=
    ConstraintSet Γ
    ∧ Constraint φ
    ∧ Γ ⊨ φ
    ⇒ φ is semantically redundant relative to Γ

TautologicalConstraint :=
    Constraint
    ∧ always_true
    ∧ imposes_no_restriction

KnowledgeRepresentation :=
    Tautology
    → LogicalTruthNode
    → ReusableInferencePattern

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

CoreClasses := {
    Tautology,
    Formula,
    AtomicFormula,
    CompoundFormula,
    Valuation,
    TruthValue,
    TruthTable,
    Countervaluation,
    LogicalTruth,
    Contradiction,
    Contingency,
    LogicalEquivalence,
    SemanticEntailment,
    Proof,
    SATSolver,
    Tableau,
    VerificationCondition
}

CoreRelations := {
    evaluatesTo,
    validUnder,
    equivalentTo,
    entails,
    negates,
    contradicts,
    instantiatedBy,
    provedBy,
    checkedBy,
    simplifiesTo,
    preserves,
    dependsOn
}

TypedRelations :=
    Valuation --evaluatesTo--> TruthValue
    Tautology --validUnder--> EveryValuation
    Formula --equivalentTo--> Formula
    PremiseSet --entails--> Conclusion
    Countervaluation --falsifies--> Formula
    Proof --provedBy--> ProofSystem
    Formula --checkedBy--> SATSolver
    Formula --simplifiesTo--> Formula
    Transformation --preserves--> TautologyStatus

InferenceRules := {
    ∀v : v(φ)=True
        ⇒ Tautology(φ),

    ∃v : v(φ)=False
        ⇒ ¬Tautology(φ),

    Unsatisfiable(¬φ)
        ⇒ Tautology(φ),

    Tautology(φ↔ψ)
        ⇒ LogicalEquivalent(φ,ψ),

    Tautology(φ)
        ⇒ Contradiction(¬φ),

    Tautology(φ)
        ∧ UniformSubstitution(σ)
        ⇒ Tautology(σ(φ)),

    ⊢ φ
        ∧ Sound(ProofSystem)
        ⇒ Tautology(φ),

    Tautology((PremisesConjunction)→Conclusion)
        ⇒ ValidArgument(Premises,Conclusion)
}

SEMANTIC_EQUIVALENCE(TautologyA, TautologyB) :=
    same_truth_function(TautologyA,TautologyB)

TautologyEquivalence :=
    Tautology(A↔B)

CanonicalTautologyRepresentation :=
    ParseFormula
    → ResolveVariables
    → EliminateDerivedConnectives
    → NormaliseNegation
    → CanonicalBooleanRepresentation
    → EvaluateValidity

TautologyGraph :=
    G = (V,E)

V := {
    Formulae,
    Variables,
    Valuations,
    TruthValues,
    Tautologies,
    Contradictions,
    Contingencies,
    Proofs,
    Countervaluations,
    EquivalenceClasses
}

E := {
    evaluatesTo,
    validUnder,
    equivalentTo,
    entails,
    negates,
    contradicts,
    instantiatedBy,
    checkedBy,
    simplifiesTo,
    preserves
}

CONFIDENCE(TautologyClassification) :=
    weighted_mean(
        ParseConfidence,
        VariableResolution,
        EvaluationCoverage,
        SolverEvidence,
        ProofEvidence,
        SemanticConsistency
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectTautologyConcept
    → ResolveFormula
    → ResolveLogic
    → IdentifyVariables
    → IdentifyRequestedValidityRelation

AI_REASONING_FLOW :=
    Query
    → ParseFormula
    → BuildAbstractSyntax
    → IdentifyAtomicVariables
    → ResolveConnectiveSemantics
    → SelectValidityMethod
    → GenerateTruthAssignmentsOrSymbolicConstraints
    → EvaluateFormula
    → SearchForCountervaluation
    → TestNegationForSatisfiability
    → DetermineTautologyStatus
    → CompareLogicalEquivalenceIfRequested
    → ProduceProofOrCounterexampleEvidence
    → BuildTautologyRelations
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseTautologyEntities,
    ParseTypedRelations,
    ResolveFormulaSyntax,
    ResolveTruthFunctions,
    GenerateValuations,
    EvaluateTruthTables,
    PerformSATValidityCheck,
    DetectCountervaluations,
    CompareLogicalEquivalence,
    ValidateInferencePatterns,
    FollowOntologyRelations,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewFormulaOrEvidence
    → ValidateInput
    → ReparseFormula
    → RecomputeVariables
    → ReevaluateTruthFunction
    → RecheckTautologyStatus
    → UpdateTautologyGraph
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

FormulaUpdate :=
    ChangedFormula
    ⇒ invalidate_or_recompute(
        TruthTable,
        SATResult,
        ProofResult,
        TautologyStatus
    )

SemanticRuleUpdate :=
    ChangedLogicOrTruthFunction
    ⇒ reevaluate(
        DependentFormulae,
        Equivalences,
        Classifications
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        PropositionalTautology,
        ValidityByUnsatisfiability,
        LogicalEquivalence,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    TautologyEntityResolution,
    Formula,
    Variables,
    TruthFunction,
    TruthTableStatus,
    CountervaluationStatus,
    SATValidityStatus,
    ProofStatus,
    TautologyStatus,
    LogicalEquivalence,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA