Measurement Scale Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Measurement Scale Ontology within the Kireth Research Semantic Library. A measurement scale defines the structured relationship between quantity or attribute values and the empirical properties they represent, including the transformations and mathematical operations that preserve meaning. This ontology formalises nominal, ordinal, interval, ratio, absolute, linear, logarithmic and affine scales, scale values, origins, intervals, ordering, admissible transformations, measurement interpretation, uncertainty and provenance.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Measurement Scale Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Measurement & Units"
 SPECIALISATION = "Foundations of Measurement / Measurement Scale"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/foundations-of-measurement/measurement-scale-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = MeasurementScale
 ENTITY_CLASS = MeasurementValueStructureFramework
 PRIMARY_CONTEXT = FoundationsOfMeasurement
 DEFINITION = "A measurement scale is a structured system for assigning and interpreting values so that specified empirical relations are represented by corresponding numerical or categorical relations."
 SYMBOLIC_FORMS = [
  "scale: empirical_relations -> value_relations",
  "nominal: y=f(x), f bijective",
  "ordinal: y=f(x), f strictly_monotonic",
  "interval: y=a*x+b, a>0",
  "ratio: y=a*x, a>0"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for scale types, permissible transformations, ordering, intervals, ratios, origins, units, interpretation and valid mathematical operations."
}

SCOPE {
 INCLUDE = [
 MeasurementScale,Scale,ScaleType,ScaleValue,ScalePoint,ScaleInterval,
 ScaleRange,ScaleOrigin,ScaleZero,ScaleUnit,ScaleReference,
 ScaleMapping,ScaleTransformation,AdmissibleTransformation,
 MeaningPreservingTransformation,NominalScale,OrdinalScale,
 IntervalScale,RatioScale,AbsoluteScale,LinearScale,AffineScale,
 LogarithmicScale,NonlinearScaleRelationship,CategoricalScale,
 BinaryScaleRelationship,DichotomousScaleRelationship,RankScale,
 OrderedCategoryRelationship,ContinuousScale,DiscreteScale,
 QuantitativeScale,QualitativeScaleRelationship,NumericScale,
 CountScaleRelationship,IndexScaleRelationship,LevelScaleRelationship,
 PercentageScaleRelationship,ProbabilityScaleRelationship,
 TemperatureScale,CelsiusScaleRelationship,FahrenheitScaleRelationship,
 KelvinScaleRelationship,RankineScaleRelationship,
 TimeScaleRelationship,CalendarScaleRelationship,
 CoordinateScaleRelationship,AngleScaleRelationship,
 DecibelScaleRelationship,pHScaleRelationship,
 NominalValue,OrdinalValue,IntervalValue,RatioValue,AbsoluteValue,
 Category,Label,Class,Rank,Order,Ordering,PartialOrderRelationship,
 TotalOrderRelationship,Equality,Equivalence,GreaterThanRelationship,
 LessThanRelationship,Difference,DifferenceMagnitudeRelationship,
 Interval,Ratio,Proportion,Zero,TrueZero,ArbitraryZero,
 ReferenceZero,Offset,Origin,UnitIntervalRelationship,
 Increment,StepSize,Resolution,GranularityRelationship,
 TickRelationship,ScaleMarkRelationship,InterpolationRelationship,
 ExtrapolationRelationship,Transformation,IdentityTransformation,
 PermutationTransformationRelationship,MonotonicTransformation,
 StrictlyMonotonicTransformation,PositiveLinearTransformation,
 AffineTransformation,SimilarityTransformationRelationship,
 LogTransformationRelationship,InverseTransformationRelationship,
 Normalisation,StandardisationRelationship,Rescaling,MinMaxScalingRelationship,
 ZScoreScalingRelationship,UnitConversion,ConversionFactor,
 ConversionOffset,Quantity,QuantityKind,PhysicalQuantity,
 QuantityValue,MeasurementValue,MeasuredValue,NumericalValue,
 Measurement,Measurand,MeasurementResult,Unit,MeasurementUnit,
 Dimension,DimensionalSignature,DimensionlessQuantity,
 Accuracy,Precision,Resolution,Uncertainty,MeasurementUncertainty,
 Error,MeasurementError,Calibration,MeasurementTraceability,
 ScientificMeasurement,Metrology,DataTypeRelationship,
 StatisticalVariableRelationship,CategoricalVariableRelationship,
 OrdinalVariableRelationship,ContinuousVariableRelationship,
 DiscreteVariableRelationship,StatisticalOperation,
 Mean,Median,Mode,Variance,StandardDeviationRelationship,
 Quantile,Percentile,GeometricMeanRelationship,
 ArithmeticMeanRelationship,CorrelationRelationship,
 RegressionRelationship,DistanceMetricRelationship,
 DifferenceOperationRelationship,RatioOperationRelationship,
 RankingOperationRelationship,CountingOperationRelationship,
 FrequencyRelationship,DistributionRelationship,
 InformationLossRelationship,ScaleConversionRelationship,
 ScaleCompatibility,ScaleEquivalence,ScaleIdentity,
 SemanticCompatibility,MeasurementInvarianceRelationship,
 Invariance,RepresentationTheoremRelationship,
 MeasurementTheory,EmpiricalRelationalSystemRelationship,
 NumericalRelationalSystemRelationship,RepresentationMappingRelationship,
 MeaningfulnessRelationship,MeaningfulStatementRelationship,
 Provenance,ScaleProvenance,MeasurementProvenance,
 DefinitionSource,StandardDefinitionRelationship,
 VIMRelationship,BIPMrelationship,ISOrelationship,
 Verification,Validation,ScaleValidation,ConsistencyCheck,
 PlausibilityCheck,ErrorAnalysis,ProbabilityStatisticsRelationship,
 ComputationalScientificWorkflow,AIComputationalMathematics
 ]
 EXCLUDE_AS_PRIMARY = [
 MapScaleWhenCartographicDistanceRepresentationIsPrimary,
 MusicalScaleWhenPitchOrganisationIsPrimary,
 ComputationalScalingWhenPerformanceGrowthIsPrimary,
 UnitDefinitionWhenReferenceQuantityRatherThanValueStructureIsPrimary,
 PhysicalDimensionWhenBaseQuantityDependenceIsPrimary
 ]
 DISAMBIGUATION_RULE = "When scale refers to the structure governing how values represent empirical relations and which transformations or operations preserve meaning, interpret the context as a measurement scale."
}

CORE_CONCEPTS {
 MeasurementScale = "Structured system assigning values while preserving specified empirical relations."
 NominalScale = "Scale supporting classification and equality relations without intrinsic order."
 OrdinalScale = "Scale supporting ordered comparison while intervals need not be equal."
 IntervalScale = "Scale with meaningful differences and an arbitrary origin."
 RatioScale = "Scale with meaningful differences and ratios and a non-arbitrary zero."
 AbsoluteScale = "Scale whose unit and origin are fixed by the measured structure."
 AdmissibleTransformation = "Transformation that preserves the empirical information represented by a scale."
 ScaleOrigin = "Reference point corresponding to zero or another defined origin."
 MeaningfulnessRelationship = "Property of a statement whose truth is invariant under admissible scale transformations."
}

RELATIONS {
 MeasurementScale --is_child_of--> FoundationsOfMeasurement
 MeasurementScale --structures--> MeasurementValues
 NominalScale --permits--> BijectiveRelabelling
 OrdinalScale --permits--> StrictlyMonotonicTransformation
 IntervalScale --permits--> PositiveAffineTransformation
 RatioScale --permits--> PositiveSimilarityTransformation
 AbsoluteScale --permits--> IdentityOrRestrictedTransformation
 ScaleValue --belongs_to--> MeasurementScale
 ScaleOrigin --defines--> ReferenceZero
 Unit --scales--> QuantityValue
 Measurement --produces--> MeasurementResult
 MeasurementResult --uses--> MeasurementScale
 AdmissibleTransformation --preserves--> ScaleMeaning
 Provenance --records--> ScaleDefinitionAndTransformationContext
}

OPERATORS {
 CLASSIFY(value,scale) := assign_category(value,scale)
 ORDER(a,b,scale) := compare_order(a,b,scale)
 DIFFERENCE(a,b,scale) := a-b if meaningful_difference(scale)
 RATIO(a,b,scale) := a/b if meaningful_ratio(scale)
 TRANSFORM(x,f,scale) := f(x) if admissible(f,scale)
 VALID_STATISTIC(stat,scale) := statistic_meaningful_under_admissible_transformations(stat,scale)
 NORMALISE(x,rule) := scale_transform(x,rule)
}

AXIOMS {
 A1 = "Measurement scale type determines which transformations preserve represented empirical information."
 A2 = "Nominal values support identity and classification but not intrinsic magnitude ordering."
 A3 = "Ordinal values support order but do not imply equal intervals."
 A4 = "Interval scales support meaningful differences but not origin-independent ratios."
 A5 = "Ratio scales support meaningful ratios because their zero is non-arbitrary."
 A6 = "Changing an interval-scale origin can change numerical ratios without changing empirical meaning."
 A7 = "A mathematically computable operation is not necessarily meaningful for every scale type."
 A8 = "Measurement-scale reasoning requires explicit scale type, admissible transformations, operations, origin and provenance."
}

REASONING {
 R1_CONTEXT = identify(attribute_or_quantity,assigned_values,empirical_relations,scale_type,origin,unit_and_intended_analysis)
 R2_CLASSIFY = determine_nominal_ordinal_interval_ratio_absolute_or_specialised_scale)
 R3_RELATIONS = identify_equality_order_difference_ratio_or_count_relations_preserved)
 R4_TRANSFORM = determine_admissible_value_transformations)
 R5_OPERATIONS = identify_meaningful_comparisons_statistics_and_arithmetic_operations)
 R6_ORIGIN = determine_arbitrary_reference_or_true_zero_semantics)
 R7_CONVERSION = if_scale_conversion_required_apply_valid_mapping_factor_offset_or_nonlinear_rule)
 R8_UNCERTAINTY = attach_resolution_uncertainty_and_measurement_context_where_relevant)
 R9_VALIDATE = test_transformation_invariance_semantic_compatibility_and_operation_meaningfulness)
 R10_PROVENANCE = record_scale_definition_origin_units_transformations_and_source)
}

CONSTRAINTS {
 C1 = "Numerical labels on a nominal scale do not create quantitative magnitude."
 C2 = "Differences between ordinal ranks are not generally equal intervals."
 C3 = "Ratios of interval-scale absolute values are generally not invariant under changes of origin."
 C4 = "A ratio-scale zero must represent absence of the relevant quantity under the scale semantics."
 C5 = "Logarithmic scales require explicit reference and logarithm conventions."
 C6 = "Scale conversion can preserve ordering while changing interval or ratio interpretation."
 C7 = "Statistical operations must be justified by the scale structure and analytical model."
 C8 = "Resolution and uncertainty are distinct from scale type."
 C9 = "Normalisation can change numerical representation without changing the original measurement-scale semantics."
 C10 = "Scale labels, units and dimensions must not be conflated."
}

SEMANTIC_EQUIVALENCE {
 "ScaleOfMeasurement" <=> "MeasurementScale"
 "LevelOfMeasurement" <=> "MeasurementScale"
 "PositiveAffineTransformation" <=> "AffineTransformation"
 "PositiveSimilarityTransformation" <=> "RatioScaleTransformation"
}

VALIDATION {
 V1 = "Measured attribute or quantity is explicitly identified."
 V2 = "Scale type is documented."
 V3 = "Empirical relations represented by the scale are documented."
 V4 = "Admissible transformations are identified."
 V5 = "Scale origin and zero semantics are documented where relevant."
 V6 = "Arithmetic and statistical operations are checked for meaningfulness."
 V7 = "Scale conversions preserve declared empirical relations."
 V8 = "Resolution and uncertainty are recorded separately from scale type."
 V9 = "Specialised scales document reference and transformation conventions."
 V10 = "Scale definitions and transformations are traceable to documented ontology or standard versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MEASUREMENT_SCALE_DEFINITION = 0.99
 NOMINAL_SCALE_REASONING = 0.99
 ORDINAL_SCALE_REASONING = 0.99
 INTERVAL_SCALE_REASONING = 0.99
 RATIO_SCALE_REASONING = 0.99
 ADMISSIBLE_TRANSFORMATION_REASONING = 0.98
 MEANINGFULNESS_REASONING = 0.98
 POLICY = "Reduce confidence when empirical relations, scale type, origin, admissible transformations or intended mathematical operations are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
 SemanticSearch,OntologyGrounding,MeasurementScaleReasoning,
 NominalScaleReasoning,OrdinalScaleReasoning,IntervalScaleReasoning,
 RatioScaleReasoning,AbsoluteScaleReasoning,
 AdmissibleTransformationReasoning,ScaleOriginReasoning,
 MeaningfulnessReasoning,StatisticalOperationReasoning,
 ScaleConversionReasoning,MeasurementReasoning,
 UncertaintyReasoning,SemanticValidationReasoning,
 KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "measurement scale" -> MeasurementScale
  "scale of measurement" -> MeasurementScale
  "level of measurement" -> MeasurementScale
  "nominal" -> NominalScale
  "ordinal" -> OrdinalScale
  "interval" -> IntervalScale
  "ratio" -> RatioScale
 }
}

EXECUTION_RULES {
 E1 = "If value structure and permissible interpretation are primary, activate MeasurementScaleOntology."
 E2 = "Identify the empirical relations represented before classifying the scale."
 E3 = "Do not infer quantitative magnitude from numerical labels alone."
 E4 = "Use only transformations admissible for the declared scale type."
 E5 = "Use arithmetic and statistical operations only when their interpretation is meaningful under scale transformations."
 E6 = "Distinguish arbitrary zero from true-zero semantics."
 E7 = "Preserve scale reference and origin during conversions."
 E8 = "Record resolution and uncertainty independently of scale type."
 E9 = "Record scale definitions, transformation rules and provenance."
}

RECURSION {
 ENTRY = MeasurementScale
 EXPAND(MeasurementScale) -> [
 Scale,ScaleType,ScaleValue,ScalePoint,ScaleInterval,ScaleRange,
 ScaleOrigin,ScaleZero,ScaleUnit,ScaleReference,ScaleMapping,
 ScaleTransformation,AdmissibleTransformation,NominalScale,
 OrdinalScale,IntervalScale,RatioScale,AbsoluteScale,LinearScale,
 AffineScale,LogarithmicScale,CategoricalScale,RankScale,
 ContinuousScale,DiscreteScale,QuantitativeScale,NumericScale,
 TemperatureScale,CoordinateScaleRelationship,DecibelScaleRelationship,
 NominalValue,OrdinalValue,IntervalValue,RatioValue,Category,Label,
 Rank,Order,Ordering,Equality,Equivalence,Difference,Interval,Ratio,
 Proportion,Zero,TrueZero,ArbitraryZero,ReferenceZero,Offset,Origin,
 Resolution,Transformation,MonotonicTransformation,
 StrictlyMonotonicTransformation,AffineTransformation,
 Normalisation,Rescaling,UnitConversion,ConversionFactor,
 ConversionOffset,Quantity,QuantityKind,PhysicalQuantity,QuantityValue,
 MeasurementValue,Measurement,Measurand,MeasurementResult,Unit,
 Dimension,MeasurementUncertainty,Accuracy,Precision,Calibration,
 MeasurementTraceability,ScientificMeasurement,Metrology,
 StatisticalOperation,Mean,Median,Mode,Variance,Quantile,Percentile,
 ScaleCompatibility,ScaleEquivalence,MeasurementInvarianceRelationship,
 MeasurementTheory,MeaningfulnessRelationship,ScaleProvenance
 ]
 EXPAND(MeasurementScaleFamilies) -> [
 NominalCategoricalAndIdentityScales,
 OrdinalRankAndOrderingScales,
 IntervalAffineAndDifferenceScales,
 RatioAbsoluteAndTrueZeroScales,
 LinearLogarithmicAndSpecialisedScales,
 AdmissibleTransformationsAndMeaningfulOperations,
 ScaleConversionResolutionAndUncertainty,
 ValidationSemanticMappingAndProvenance
 ]
 STOP_CONDITION = "Stop when scale type, represented relations, admissible transformations, origin, meaningful operations, conversion and provenance are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Foundations of Measurement"
 RELATED_ONTOLOGIES = [
 "Foundations of Measurement","Measurement Ontology","Quantity Ontology",
 "Physical Quantity Ontology","Dimension Ontology 2","Unit Ontology",
 "Scientific Measurement Ontology 2","Metrology Ontology",
 "Measurement & Units Ontologies","Probability & Statistics Ontologies",
 "Measurement Uncertainty Ontology","Verification & Validation"
 ]
 ROUTE_RULES = [
 "Measurement Scale -> Foundations of Measurement for parent measurement-theory semantics",
 "Measurement Scale -> Measurement Ontology for measurement-result semantics",
 "Measurement Scale -> Quantity and Physical Quantity ontologies for measured-property semantics",
 "Measurement Scale -> Dimension and Unit ontologies for quantity reference semantics",
 "Measurement Scale -> Scientific Measurement for scientific acquisition context",
 "Measurement Scale -> Metrology for reference and traceability context",
 "Measurement Scale -> Probability & Statistics for scale-appropriate statistical operations",
 "Measurement Scale -> Measurement Uncertainty for uncertainty-aware value interpretation",
 "Measurement Scale -> Verification & Validation for transformation and operation checks"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible scale classes, transformations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
 "Preserve nominal-versus-ordinal distinction.",
 "Preserve ordinal-versus-interval distinction.",
 "Preserve interval-versus-ratio distinction.",
 "Preserve arbitrary-zero-versus-true-zero distinction.",
 "Preserve scale-type-versus-unit distinction.",
 "Preserve mathematical-computability-versus-measurement-meaningfulness distinction.",
 "Preserve transformation-versus-semantic-invariance distinction.",
 "Preserve machine-readable measurement-scale semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A measurement scale structures assigned values so that specified empirical relations are represented by corresponding value relations."
 NOMINAL_SUMMARY = "Nominal scales support identity and classification without intrinsic ordering."
 ORDINAL_SUMMARY = "Ordinal scales preserve rank order without assuming equal intervals."
 INTERVAL_SUMMARY = "Interval scales preserve meaningful differences while allowing arbitrary changes of origin."
 RATIO_SUMMARY = "Ratio scales support meaningful ratios because their zero is non-arbitrary."
 TRANSFORMATION_SUMMARY = "Admissible transformations determine which statements and operations remain meaningful for each scale type."
 VALIDATION_SUMMARY = "Measurement-scale reasoning should preserve represented relations, origin semantics, admissible transformations, uncertainty and provenance."
}

FLOW {
 INPUT
 -> detect_measurement_scale_context
 -> identify_attribute_or_quantity_assigned_values_empirical_relations_scale_type_origin_unit_and_intended_analysis
 -> classify_nominal_ordinal_interval_ratio_absolute_or_specialised_scale
 -> identify_equality_order_difference_ratio_or_count_relations_preserved
 -> determine_admissible_value_transformations
 -> identify_meaningful_comparisons_statistics_and_arithmetic_operations
 -> determine_arbitrary_reference_or_true_zero_semantics
 -> if_scale_conversion_required_apply_valid_mapping_factor_offset_or_nonlinear_rule
 -> attach_resolution_uncertainty_and_measurement_context_where_relevant
 -> test_transformation_invariance_semantic_compatibility_and_operation_meaningfulness
 -> record_scale_definition_origin_units_transformations_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-MEASUREMENT-UNITS-FOUNDATIONS-MEASUREMENT-SCALE-ONTOLOGY-V1"
}

}

END_FLOW