Inductive Reasoning Ontology

FLOW_ALGEBRA InductiveReasoningOntology v1.0

TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/logical-reasoning/inductive-reasoning-ontology.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > LogicalReasoning > InductiveReasoningOntology

BEGIN_FLOW

ENTITY InductiveReasoning {
    TYPE := AmpliativeReasoningProcess
    DEFINITION := "A form of reasoning that moves from observed cases, evidence or samples toward broader generalisations, hypotheses or predictions whose conclusions may be well supported without being deductively entailed by the premises."
}

ENTITY Observation {
    TYPE := EvidenceItem
}

ENTITY Sample {
    TYPE := CollectionOfObservedCases
}

ENTITY Generalisation {
    TYPE := InductiveConclusion
}

ENTITY Prediction {
    TYPE := InductiveConclusion
}

RELATIONSHIP InductiveReasoning --uses--> Observation
RELATIONSHIP InductiveReasoning --uses--> Sample
RELATIONSHIP InductiveReasoning --produces--> Generalisation
RELATIONSHIP InductiveReasoning --produces--> Prediction
RELATIONSHIP InductiveReasoning --evaluates--> EvidenceStrength
RELATIONSHIP InductiveReasoning --supports--> HypothesisFormation

INDUCTIVE_PATTERN {
    OBSERVE cases C1,...,Cn
    IDENTIFY regularity R
    FORM hypothesis H
    ESTIMATE support_for_H
    TEST H against new evidence
    UPDATE H_or_support
}

AMPLIATIVE_CHARACTER {
    Conclusion
        may contain information
        not deductively guaranteed by premises
}

NON_ENTAILMENT_WARNING {
    StrongInductiveSupport(H)
        DOES_NOT imply
    Premises ⊨ H
}

ENUMERATIVE_INDUCTION {
    OBSERVED:
        a1 has property P
        a2 has property P
        ...
        an has property P

    INFER:
        unobserved_or_general_members
        likely_or_plausibly have P

    REQUIRE :=
        scope_and_sampling_assumptions
}

UNIVERSAL_GENERALISATION_BY_INDUCTION {
    SAMPLE :=
        observed_members_of_population

    CANDIDATE_GENERALISATION :=
        FOR_ALL x in target_population:
            P(x)

    STATUS :=
        inductive_hypothesis,
        not deductive theorem from finite positive observations alone
}

STATISTICAL_GENERALISATION {
    SAMPLE_STATISTIC
        -> estimate
        -> POPULATION_PARAMETER

    REQUIRE := {
        SamplingModel,
        PopulationDefinition,
        MeasurementDefinition
    }
}

STATISTICAL_SYLLOGISM {
    MOST members of class C have property P
    a is member of C
    THEREFORE
        a is plausibly P

    STATUS :=
        defeasible/statistical
}

PREDICTIVE_INDUCTION {
    PAST_OR_CURRENT_OBSERVATIONS
        -> inferred_pattern
        -> prediction_about_future_or_unobserved_case
}

CAUSAL_INDUCTION {
    OBSERVATIONS
        -> candidate_causal_relationship

    REQUIRE :=
        confounding,
        temporal_order,
        alternative_explanations,
        intervention_or_design_evidence
        considered where relevant
}

CAUSATION_WARNING {
    Correlation
        DOES_NOT by itself establish
    Causation
}

ANALOGICAL_INDUCTION {
    SOURCE and TARGET share relevant properties
    SOURCE has property P
    INFER TARGET may have P

    STRENGTH depends_on:
        relevance_of_similarities,
        absence_of_critical_differences,
        domain_knowledge
}

INDUCTIVE_ARGUMENT {
    PREMISES := Evidence E
    CONCLUSION := H
    SUPPORT_RELATION := E supports H
}

EVIDENCE_STRENGTH {
    TYPE := ContextDependentAssessment

    MAY_DEPEND_ON := {
        SampleSize,
        Representativeness,
        MeasurementQuality,
        Independence,
        Replication,
        PriorKnowledge,
        AlternativeHypotheses,
        PredictivePerformance
    }
}

SAMPLE_SIZE {
    larger_sample
        MAY reduce sampling_variability
        under suitable sampling assumptions

    WARNING :=
        size alone does not remove bias
}

REPRESENTATIVENESS {
    Sample
        should reflect
        target_population
        for intended generalisation
}

SELECTION_BIAS {
    sampling_or_selection_process
        systematically distorts observed evidence
        relative to target inference
}

SURVIVORSHIP_BIAS {
    analysis_conditioned_on_surviving/visible_cases
        may omit relevant failures
}

MEASUREMENT_BIAS {
    measurement_process
        systematically shifts recorded values
}

CONFUNDING {
    variable Z
        associated_with candidate_cause X
        AND outcome Y
        may distort causal inference
}

INDEPENDENCE {
    repeated observations
        MAY not provide independent evidence
        if generated by shared source/process
}

REPLICATION {
    independent_or_relevant_repeat_studies
        MAY strengthen robustness
        when results reproduce
}

COUNTEREVIDENCE {
    Evidence E2
        conflicts_with
        Hypothesis H

    EFFECT :=
        reduce_or_revise support
        according to inference framework
}

FALSIFYING_INSTANCE {
    FOR universal claim ∀xP(x):

    OBSERVE a WITH ¬P(a)

    THEN
        universal claim is deductively false
        if observation and classification are correct
}

CONFIRMING_INSTANCE {
    OBSERVE a WITH P(a)

    EFFECT :=
        may support universal claim
        but does not deductively prove it
}

HYPOTHESIS {
    TYPE := CandidateGeneralisationOrModel
}

HYPOTHESIS_GENERATION {
    Data
        -> PatternDetection
        -> CandidateHypotheses
}

HYPOTHESIS_TESTING {
    Candidate H
        -> derive predictions
        -> compare with new observations
        -> retain/revise/reject according to method
}

OUT_OF_SAMPLE_TEST {
    TRAINING_EVIDENCE
        used_to_form_model

    TEST_EVIDENCE
        held_out_or_new

    PURPOSE :=
        assess generalisation beyond fitting data
}

OVERFITTING {
    Model
        captures noise_or_sample_specific_structure
        rather than stable_pattern

    INDICATOR :=
        strong training fit
        + weak out_of_sample performance
}

UNDERFITTING {
    Model
        insufficiently captures relevant structure
}

INDUCTIVE_BIAS {
    TYPE := AssumptionsEnablingGeneralisation

    EXAMPLES := {
        SimplicityPreference,
        Smoothness,
        Locality,
        Stationarity,
        ModelClassRestriction
    }
}

NO_FREE_GENERALISATION {
    finite_data
        alone generally insufficient
        to determine unique unrestricted generalisation

    REQUIRE :=
        inductive assumptions or model constraints
}

OCCAMS_RAZOR_RELATIONSHIP {
    SimplerHypothesis
        MAY be preferred
        under explicit simplicity criterion

    WARNING :=
        simplicity is methodological preference,
        not automatic proof of truth
}

BAYESIAN_INDUCTION_RELATIONSHIP {
    PriorBelief
        + Likelihood
        -> PosteriorBelief

    BayesRule:
        P(H|E)
        =
        P(E|H)P(H)/P(E)

    REQUIRE :=
        probabilistic model and prior
}

BAYESIAN_WARNING {
    PosteriorProbability
        is not identical to
        fuzzy_degree_of_truth
}

LIKELIHOOD {
    P(E|H)
    TYPE := EvidenceModelGivenHypothesis
}

PRIOR {
    P(H)
    TYPE := PreEvidenceProbability
}

POSTERIOR {
    P(H|E)
    TYPE := UpdatedProbability
}

BAYES_FACTOR {
    BF_12 :=
        P(E|H1)/P(E|H2)

    PURPOSE :=
        compare relative evidence under two hypotheses
}

FREQUENTIST_INDUCTION_RELATIONSHIP {
    SampleData
        -> EstimationOrTestingProcedure
        -> inferential_statement

    REQUIRE :=
        sampling-distribution semantics
}

CONFIDENCE_INTERVAL {
    TYPE := FrequentistProcedureOutput

    WARNING :=
        interval interpretation MUST follow confidence-procedure semantics
}

HYPOTHESIS_TEST {
    NULL := H0
    ALTERNATIVE := H1
    TEST_STATISTIC := T

    REQUIRE :=
        declared significance/test procedure
}

P_VALUE {
    TYPE := TailProbabilityUnderNullOrMoreExtremeCriterion

    WARNING :=
        p_value != probability_that_H0_is_true
}

EFFECT_SIZE {
    TYPE := MagnitudeMeasure
    PURPOSE :=
        quantify substantive size of observed relationship/effect
}

UNCERTAINTY_QUANTIFICATION {
    MAY_USE := {
        Probability,
        ConfidenceIntervals,
        CredibleIntervals,
        PredictionIntervals,
        Bootstrap,
        SensitivityAnalysis
    }

    REQUIRE :=
        method-specific interpretation
}

INDUCTION_BY_ELIMINATION {
    CandidateHypotheses := {H1,...,Hn}

    USE evidence
        to reject/invalidate alternatives

    REMAINING hypotheses
        gain relative standing

    WARNING :=
        incomplete hypothesis space may leave unconsidered alternatives
}

MILL_METHODS_RELATIONSHIP {
    HISTORICAL_METHODS := {
        Agreement,
        Difference,
        JointMethod,
        Residues,
        ConcomitantVariation
    }

    ROLE :=
        heuristic structures for causal inquiry

    WARNING :=
        modern causal inference requires stronger explicit assumptions/designs
}

ROBUSTNESS {
    Conclusion remains similar
        across reasonable modelling/measurement choices
}

SENSITIVITY_ANALYSIS {
    VARY assumptions_or_parameters
    RECOMPUTE inference
    MEASURE conclusion_change
}

EVIDENCE_UPDATE {
    CURRENT := H with support S
    RECEIVE new evidence E

    IF E supports H:
        UPDATE support upward according_to framework

    IF E conflicts:
        UPDATE support downward_or_revise H

    IF E reveals bias:
        REASSESS earlier evidence
}

CONCEPT_DRIFT {
    DataGeneratingProcess changes over time

    CONSEQUENCE :=
        past inductive patterns may lose predictive validity
}

STATIONARITY_ASSUMPTION {
    relevant generating_process
        remains sufficiently stable
        across inference horizon
}

EXTRAPOLATION {
    APPLY learned relation
        outside observed data range

    RISK :=
        assumptions may fail beyond observed region
}

INTERPOLATION {
    INFER within observed domain/range

    GENERALLY requires fewer extrapolative assumptions
        but still depends on model
}

BASE_RATE {
    background_frequency_or_probability
        relevant to inductive assessment
}

BASE_RATE_NEGLECT {
    ignoring relevant prior prevalence
        may distort probabilistic inference
}

CONFIRMATION_BIAS {
    preferentially seek/interpret evidence
        supporting existing hypothesis

    MITIGATION :=
        actively test alternatives and counterevidence
}

HINDSIGHT_BIAS {
    after outcome known,
        outcome may appear more predictable than it was
}

AVAILABILITY_BIAS {
    salient/easily recalled examples
        may distort frequency judgement
}

INDUCTIVE_FALLACY_HASTY_GENERALISATION {
    small_or_unrepresentative_sample
        -> broad conclusion

    STATUS :=
        weak_inductive_support
}

CHERRY_PICKING {
    selectively include favourable evidence
        while omitting relevant contrary evidence
}

POST_HOC_CAUSAL_FALLACY {
    A precedes B
    THEREFORE A caused B

    STATUS :=
        insufficient causal inference
}

INDUCTIVE_STRENGTH_ASSESSMENT(Argument A) {
    IDENTIFY target_population_or_domain
    IDENTIFY evidence_set
    IDENTIFY sampling_process
    IDENTIFY measurement_process
    IDENTIFY candidate_hypothesis
    IDENTIFY alternative_hypotheses

    CHECK {
        SampleSize,
        Representativeness,
        Independence,
        Bias,
        Counterevidence,
        Replication,
        PredictiveAccuracy,
        AssumptionSensitivity
    }

    RETURN structured_support_assessment
}

GENERALISATION_TEST(H,TrainingData,TestData) {
    FIT_OR_FORM H USING TrainingData
    DERIVE predictions
    EVALUATE on TestData
    COMPARE predicted_vs_observed
    RETURN generalisation_performance
}

CAUSAL_REASONING_CHECK {
    IDENTIFY cause_candidate X
    IDENTIFY outcome Y
    IDENTIFY confounders
    IDENTIFY temporal_order
    IDENTIFY intervention_or_natural_experiment_if_any
    IDENTIFY alternative_causes
    RETURN causal_support_structure
}

INDUCTIVE_REASONING_VS_DEDUCTION {
    Deduction :=
        truth_preserving entailment

    Induction :=
        ampliative support beyond deductive entailment
}

INDUCTIVE_REASONING_VS_ABDUCTION {
    Induction :=
        infer pattern/generalisation from evidence

    Abduction :=
        infer candidate explanation for evidence

    OVERLAP :=
        real scientific reasoning may combine both
}

INDUCTIVE_REASONING_VS_PROBABILITY {
    InductiveReasoning :=
        broad reasoning category

    Probability :=
        one mathematical framework for quantifying uncertainty

    CONSTRAINT :=
        induction need not always be probabilistic
}

APPLICATION_RELATIONSHIPS {
    InductiveReasoning --supports--> ScientificReasoning
    InductiveReasoning --supports--> Statistics
    InductiveReasoning --supports--> MachineLearning
    InductiveReasoning --supports--> ArtificialIntelligence
    InductiveReasoning --supports--> Prediction
    InductiveReasoning --supports--> PatternRecognition
    InductiveReasoning --supports--> HypothesisFormation
    InductiveReasoning --supports--> EvidenceAssessment
}

ONTOLOGY_LINKS {
    PARENT := LogicalReasoningOntology

    RELATED := {
        DeductiveReasoningOntology,
        AbductiveReasoningOntology,
        AnalogicalReasoningOntology,
        ProbabilityOntology,
        StatisticsOntology,
        BayesianInferenceOntology,
        HypothesisTestingOntology,
        ScientificMethodOntology,
        MachineLearningOntology,
        CausalInferenceOntology
    }
}

SEMANTIC_EQUIVALENCE {
    InductiveReasoning ≈ Induction
    InductiveConclusion ≈ EvidenceSupportedGeneralisation
    PredictiveInduction ≈ InferenceToUnobservedCases
        WHEN prediction_context
}

CONSTRAINTS {
    C1 :=
        inductive conclusions MUST NOT be presented as deductively entailed unless an independent deduction establishes them

    C2 :=
        target population/domain MUST be identified for generalisation claims

    C3 :=
        sample representativeness MUST NOT be assumed without justification

    C4 :=
        sample size MUST NOT be treated as eliminating systematic bias

    C5 :=
        correlation MUST NOT automatically be interpreted as causation

    C6 :=
        p-values MUST NOT be interpreted as posterior probabilities of hypotheses

    C7 :=
        Bayesian conclusions MUST identify priors and likelihood assumptions where material

    C8 :=
        confirming instances MUST NOT be treated as deductive proof of unrestricted universal claims

    C9 :=
        counterevidence and alternative hypotheses MUST be considered when relevant

    C10 :=
        extrapolation MUST identify assumptions about stability beyond observed data

    C11 :=
        probability, fuzzy truth and inductive support MUST remain semantically distinct

    C12 :=
        conclusions MUST remain calibrated to evidence quality and modelling assumptions
}

REASONING_CHAIN Query Q {
    IDENTIFY evidence E
    IDENTIFY target_hypothesis H
    IDENTIFY target_population_or_domain
    IDENTIFY inference_type

    IF sample_based {
        IDENTIFY sampling_process
        CHECK representativeness
        CHECK sample_size
        CHECK independence
        CHECK selection_bias
    }

    IF predictive {
        IDENTIFY training_evidence
        IDENTIFY test_or_future_domain
        CHECK stationarity
        CHECK extrapolation
    }

    IF causal {
        CHECK confounding
        CHECK temporal_order
        CHECK intervention_evidence
        CHECK alternative_explanations
    }

    IF Bayesian {
        IDENTIFY prior
        IDENTIFY likelihood
        COMPUTE posterior_if_data_complete
    }

    IF frequentist {
        IDENTIFY sampling_model
        IDENTIFY estimator_or_test
        APPLY correct procedure
    }

    SEARCH counterevidence
    SEARCH alternative_hypotheses
    ASSESS robustness
    ASSESS sensitivity

    CALIBRATE conclusion_strength
    VALIDATE constraints
    RETURN inductive_result WITH evidence_trace
}

CONFIDENCE {
    CORE_INDUCTIVE_DEFINITION := 1.00
    DEDUCTION_DISTINCTION := 1.00
    SAMPLE_BIAS_PRINCIPLES := 1.00
    CAUSATION_WARNING := 1.00
    BAYESIAN_UPDATE_CORE := 1.00
    P_VALUE_DISTINCTION := 1.00

    SPECIFIC_INDUCTIVE_RESULT :=
        IF evidence_sampling_and_assumptions_complete
        THEN derive_framework_specific_support
        ELSE REQUIRE missing_information
}

UPDATE_MODEL {
    IMMUTABLE_CORE := {
        AmpliativeCharacter,
        EvidenceConclusionDistinction,
        NonEntailmentWarning,
        BiasAwareness,
        CounterevidenceRequirement
    }

    EXTENSIBLE := {
        StatisticalMethods,
        BayesianMethods,
        CausalInferenceMethods,
        PredictiveModels,
        SamplingMethods,
        RobustnessMethods,
        MachineLearningMethods,
        ApplicationDomains,
        RelatedOntologies
    }

    ON_NEW_KNOWLEDGE {
        IDENTIFY evidence_source
        VALIDATE sampling_or_observation_process
        CHECK measurement_quality
        CHECK bias
        CHECK alternatives
        CHECK counterevidence
        UPDATE hypothesis_support
        UPDATE confidence
    }
}

AI_CONSUMPTION {
    MACHINE_READABLE := TRUE
    HUMAN_INTERPRETABLE := TRUE
    RECURSIVE_REASONING := TRUE
    ENTITY_RESOLUTION := TRUE
    EVIDENCE_RESOLUTION := TRUE
    SAMPLE_REASONING := TRUE
    GENERALISATION_REASONING := TRUE
    PREDICTIVE_REASONING := TRUE
    CAUSAL_SUPPORT_REASONING := TRUE
    BAYESIAN_REASONING := TRUE
    STATISTICAL_REASONING := TRUE
    BIAS_DETECTION := TRUE
    COUNTEREVIDENCE_REASONING := TRUE
    ROBUSTNESS_REASONING := TRUE
    VALIDATION_REQUIRED := TRUE
}

FLOW_HASH {
    TARGET := TARGET_URL
    SUBJECT := "Inductive Reasoning Ontology"
    VERSION := "1.0"
    CANONICAL_SIGNATURE :=
        HASH(TARGET + SUBJECT + DOMAIN + VERSION)
}

OUTPUT {
    PRIMARY_ENTITY := InductiveReasoning

    INTERPRETATION :=
        "Inductive reasoning extends from observed evidence to broader generalisations, hypotheses and predictions. Its conclusions are evaluated by the quality, representativeness and independence of evidence, the assumptions linking observed and unobserved cases, competing explanations, counterevidence and predictive performance rather than by deductive entailment alone."
}

END_FLOW