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Preliminary notes, very much a draft as of 01/04/10.     (1B)

Some main categories     (1D)

Quantity : a physical property of an object, material, location or phenomenon which can be expressed numerically, measured or computed. Examples: length of a finite path in space; weight of a loaded freight container; Rockwell hardness of a metal alloy; instantaneous temperature at a point in a body of liquid.     (1E)

Formalized as a function from physical entities to quantity values. Quantities and quantity values are understood to exist independently of, and prior to, any method of giving them a numerical value.     (1F)

(Quantity length) (domain length SpatialPath) (= (length (across MyTable)) (meter 0.85))     (1G)

Each quantity has an associated domain of physical entities, all of which possess the quantity in question. (Example: domain of length is all finite continuous paths in space; of mass, all pieces of physical stuff. Some quantities, such as temperature, can apply to points in space, objects and other entities.) Quantities are unique-valued and total on their domains.     (1H)

Note, it is not a primary purpose of this ontology to describe the various physical entities in all the possible domains of quantities. We will use the generic class name PhysicalEntity to refer to the class from which these are all drawn.     (1I)

Note, in order to describe "time-varying quantities", temporal variations must be incorporated into the domain. For example, the dynamic temperature of a cooling liquid is not a quantity, but rather a function from times to quantities, so that the domain of temperature is objects or points at a time. Some ontological frameworks may require this to be treated as a function from (entities x times) to quantity values, rather than as a quantity function applied to instantaneous moments of a temporally extended entity.     (1J)

(forall (x y)(if (and (Quantity x)(domain x y)) (subClass y PhysicalEntity) ))     (1K)

Measurement: an estimated value or value of a quantity, or a range of such values, produced by a measurement event. Measurements, unlike quantities, may be approximate and subject to error. Many issues in metrology, concerned with estimating and recording the accuracy of measurements, are beyond the scope of this ontology.     (1L)

In the first draft, we will ignore questions of accuracy or error in measurements.     (1M)

Note. Although quantities and measurements can often be expressed in the same way, they should be conceptually distinguished. We formalize a measurement as a pair of a scale and a number. Note the distinction between for example the measurement <meter, 3.2> and the quality value (meter 3.2).     (1N)

Note. Some metrological perspectives deny the existence of quantities other than measurements, or treat quantities as derivative upon measurements. In contrast, this ontology is based on a realist perspective which treats quantities as real, whether or not they have been, or even could be, measured.     (1O)

MeasurementEvent: an event or activity which yields a measurement of the numerical value of a quantity. Each measurement is the result of a single measurement event. MeasurementEvents are to be considered genuine events: they occur at a place and time, and may involve actors and instruments in various ways. We formalize events only by having the class MeasurementEvent and the relationship measures between measurement events, physical entities and measurements. For example, we can express that the measurement made during the event E is accurate by the assertion     (1P)

(forall (scale number)(if (measures E p <scale number>)(= (scale number) ((dimension scale) p)) ))     (1Q)

Example: (if (measures E (across MyTable) <meter 0.83> ) (= (meter 0.83)(length (width MyTable)) ))     (1R)

Scale: a structured set of numbers used to record quantity values. Scales come in several types, described below. A given quantity may be measured using a variety of scales, but usually all of the same type.     (1S)

Note, there is an issue about whether a scale can have vector values or more complex values (eg quarternions). We restrict ourselves to scalar scales.     (1T)

Scales are formalized as functions from numbers to quantity values, such as (meter 1.3). Note that scale conversions can then be expressed by equations, such as     (1U)

(= (meter 1)(foot 3.281))     (1V)

Note. It would be useful to express approximate conversions (like the above), but axiomatizing 'approximately equal' is a pain.     (1W)

OrderScale: a scale with a linear (total) order on its values. (Rockwell hardness, Beaufort storm scale)     (1X)

ContinuousOrderScale: Order scale with values in some continous segment of the real line which includes zero. (Centigrade temperature)     (1Y)

UnitScale: ContinuousOrderScale in which values can be meaningfully multiplied by real numbers. (Mass, force, distance, Kelvin temperature, etc.) Every UnitScale has an 'absolute zero' and a unit, so that all values can be expressed as multiples of the unit. We will use the conventional multiplication symbol times to write such multiples of a number and a quantity value. The unit of a unit scale s is the quantity (s 1).     (1Z)

Note, the use of a given scale type to represent a quantity value presupposes that the set of quantities themselves has the requisite structure, eg continuous variability and being multiplied. (For example, it makes physical sense to talk of one mass being twice another, but not of a Centigrade temperature being twice another.)     (1AA)


Dimension. The quantity associated with a scale. Example: (= (dimension meter) length)     (1AB)

System of Units (SOI). A collection of unit scales, each associated with a distinct dimension, these dimensions being called base or simple dimensions. Typically, the SOI allows for the definition of other dimensions by multiplying powers of the base dimensions, with associated composition of units to provide units for composed scales, such as meters per second for the derived scale velocity, with dimension (length.time|-1). We will write such derived units and dimensions using the composition functions times and power, so that for example (= velocity (times length (power time -1))) From which it follows that travelling at a velocity of (times 4 (times meter (power second -1))) for 3 seconds will move 12 meters. This can be abbreviated by defining for example the function per: (forall (x y)(= (per x y)(times x (power y -1)) )) and then the above is travelling at (times 4 (per meter second)) for 3 seconds.     (1AC)