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An initial organization of some concepts defined within the VIM (International Vocabulary of Metrology) By David Leal (CAESAR Systems Limited) 24-Sep-2009 v1.0 1 Ontolog Panel Discussion: Strawman for a UoM_Ontology_Standard

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generic quantity VIM quantity: property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference with a generic meaning – middle/left column of table in note 1 EXAMPLES: radius, Rockwell C hardness (150 kg) VIM quantity: property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference meaning a particular trope – right hand column of table in note 1 EXAMPLES: radius of circle A, energy of particle i, Rockwell C hardness (150 kg) of steel sample i particular quantity has type particular phenomenon body or substance particular quantity is possessed by 1 * * 1 individual physical object EXAMPLES: circle A, particle i, steel sample i magnitude of quantityparticular quantity * 1 magnitude is within 1 * quantity value magnitude of quantity is expressed by particular quantity is of magnitude * 1 number VIM quantity value: number and reference together expressing magnitude of a quantity EXAMPLES: 5.34 m, 43.5 HRC(150 kg) ?????? * 1 reference for quantity value particular quantity has magnitude value * * kind of quantity VIM kind of quantity: aspect common to mutually comparable quantities also quantity with a generic meaning – left column of table in note 1 EXAMPLES: length, energy, Rockwell C hardness (150 kg) 1 * generic quantity has kind 2

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kind of quantity 1 quantity dimension factor has dimension power: number VIM quantity dimension: expression of the dependence of a quantity on the base quantities of a system of quantities as a product of powers of factors corresponding to the base quantities, omitting any numerical factor EXAMPLE: In the ISQ, the quantity dimension of force is denoted by dim F = LMT –2. * system of quantities VIM system of quantities: set of quantities together with a set of non-contradictory equations relating those quantities EXAMPLE: International System of Quantities (ISQ) quantity of dimension one VIM quantity of dimension one: dimensionless quantity quantity for which all the exponents of the factors corresponding to the base quantities in its quantity dimension are zero EXAMPLES: plane angle, solid angle, refractive index, mass permeability quantity dimension 1 * system of quantities for quantity dimension * base quantity in factor 1,* * quantity dimension has factor has base quantity has derived quantity * * 1,** VIM base quantity: quantity in a conventionally chosen subset of a given system of quantities, where no subset quantity can be expressed in terms of the others VIM derived quantity: quantity, in a system of quantities, defined in terms of the base quantities of that system 1 3

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VIM coherent system of units: system of units, based on a given system of quantities, in which the measurement unit for each derived quantity is a coherent derived unit EXAMPLE: International System of Units (SI) VIM measurement unit: real scalar quantity, defined and adopted by convention, with which any other quantity of the same kind can be compared to express the ratio of the two quantities as a number system of units has base unit has derived unit * * VIM base unit: measurement unit that is adopted by convention for a base quantity magnitude of quantity measurement unit VIM system of units: set of base units and derived units, together with their multiples and submultiples, defined in accordance with given rules, for a given system of quantities EXAMPLE: International System of Units (SI) system of quantities * 1 system of quantities for system of units VIM derived unit: measurement unit for a derived quantity coherent system of units 1,* * has off-system measurement unit * VIM off-system measurement unit: measurement unit that does not belong to a given system of units * 4

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magnitude of quantity measurement unit unit defined by phenomenon unit defined by derivation 5

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kind of quantity * 1 kind of quantity for quantity-value scale VIM conventional reference scale: quantity-value scale defined by formal agreement quantity-value scale VIM quantity-value scale: ordered set of quantity values of quantities of a given kind of quantity used in ranking, according to magnitude, quantities of that kind EXAMPLES: Celsius temperature scale, Rockwell C hardness scale ordinal quantity VIM ordinal quantity: quantity, defined by a conventional measurement procedure, for which a total ordering relation can be established, according to magnitude, with other quantities of the same kind, but for which no algebraic operations among those quantities exist EXAMPLES: Rockwell C hardness, octane number for petroleum fuel, earthquake strength on the Richter scale ordinal quantity-value scale conventional reference scale VIM ordinal quantity-value scale: quantity-value scale for ordinal quantities EXAMPLES: Rockwell C hardness scale, scale of octane numbers for petroleum fuel. space of values 1 space of values for quantity-value scale * 6

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length of my thumb height of my coffee cup distance from London to New York mass of my car mass of Barbar the elephant mass of the moon particular quantity 7

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length of my thumb height of my coffee cup distance from London to New York mass of my car mass of Barbar the elephant mass of the moon length – as class of particular quantity mass – as class of particular quantity particular quantity 8

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kind of quantity – as class of particular quantity length – as class of particular quantity mass – as class of particular quantity energy – as class of particular quantity 9

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particular quantity length of my thumb height of my coffee cup distance from London to New York lengths equal to the height of my coffee cup lengths equal to the distance from London to New York mass of my car mass of Barbar the elephant mass of the moon masses equal to the mass of my car masses equal to the mass of the moon 10

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lengths equal to the height of my coffee cup lengths equal to the distance from London to New York masses equal to the mass of my car masses equal to the mass of the moon magnitude of quantity 11

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kind of quantity – as class of magnitude of quantity length – as class of magnitude of quantity mass – as class of magnitude of quantity the metre the kilogram 12

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hasPowerClass hasPowerClass hasPowerClass length of my table at my instant Q1 (particular quantity) 1.3 metres length – as class of particular quantity Q2 (kind of quantity – as class of particular quantity) Q3 (magnitude of quantity) length – as class of magnitude Q1* L* Q3* the metre Q4 (kind of quantity – as class of magnitude) 0 1 2 3 13

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Quantity Those things we can measure is called quantity.

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