Instrumentation & Measurement Magazine 26-1 - 14

fundamentalsmeasurement continued
of
achievable in any measurement of a given measurand " , in the
sense that using a measuring instrument with a better resolution
would produce a value for a measurand that is not the
length of the rod anymore, but, say, the distance between two
extreme points on the opposite faces of the rod. Definitional
uncertainty is then the uncertainty about our possibility to
identify the property as it is defined and, as such, it depends
on both the object and the way we have defined the property.
Scales, Values of Properties, and Types of
(Kinds of) Properties
Relation (1) is about indistinguishability of comparable properties,
and as such it is in principle applicable to any kind of
property. For example, we could discover that individuals x
and y are indistinguishable with respect to their blood group
(i.e., " they have the same blood group " , as mentioned above),
and then write:
blood group of x = blood group of y
(3)
where the " = " sign is customarily adopted to indicate that the
blood of the two individuals belongs to the same group (i.e., indistinguishability
class), and that that group is labeled as A in
the ABO System:
blood group of x (and of y) = A in the ABO System (4)
Both (3) and (4) require a classification criterion of blood
groups, but the second one conveys a more sophisticated information,
due to presence on its right-hand side of a reference
to a value of a property (a value of blood group, in this case). As it
is clear, values of properties operate as selectors in given classifications.
In this example, we first learn how to compare blood
groups, then we agree upon a comparison-based classification
of blood groups, and finally we assign an information entity as
unique identifier (A, B, AB, O) to each class. Thus, we create a
scale, i.e., a mapping from classes of indistinguishable properties
to identifiers (someone might object against using the term
" scale " about structures that are only classificatory, and not at
least ordered; let us say that we are using it by extension).
The construction of a scale generates a value for each
property of the considered kind, where in (4) the identifier
A corresponds to the value (A in the ABO System) and so on.
While A is simply an identifier of an equivalence class of blood
groups, what is a value like (A in the ABO System) is a more
delicate issue. Indeed, the common view that values are " symbols "
to represent properties is not informative, as everything
can be used to symbolize everything.
What are values of properties then? To better discuss this
issue, let us come back to more usual cases of physical quantities.
Even still lacking a unit, or a scale, of length, we could
discover that:
14
Hence, a new issue arises: what are units? Of course, an
extensional, reductionist stance can be adopted also in this
case, by taking for example the metre as an equivalence class
of lengths (how such class is established, and therefore how
the metre is defined, is not relevant here: it could be identified
(the dates that follow refer to the Metre Convention [8]) as
the length of an artifact (from 1889 to 1960); or as a multiple or
submultiple of the length of a natural phenomenon (from 1960
to 2019); or by assuming a given numerical value for a length
of a natural phenomenon when expressed in metres (since
2019)). But this simplicity generates quite complex interpretations:
for example, is 1.2345 m really the multiple of a class of
equivalence of lengths? We believe that the traditional, realist
stance is well justified here: the metre is not an equivalence
class of lengths but what the lengths in the class have in common,
i.e., an abstract length, and of course the same applies to
any other unit.
Everything becomes simple and consistent in this way: if
a unit like the metre is a length, a value like 1.2345 m is a multiple
of a length and therefore a length in turn. Accordingly,
relation (1) is an equation of two properties, one of them identified
concretely as the property of an object and the other identified
abstractly as an element of a classification, so that the information
actually conveyed by such a relation is that the considered
property of the object belongs to an indistinguishability class.
In the example about the length of a rod, claiming that the
IEEE Instrumentation & Measurement Magazine
February 2023
A value of a property is a property
identified abstractly as an
element of a classification
corresponding to a scale.
length of a given steel rod = width of a given door (5)
(of course, always within the limits of the identification of
these properties, as discussed above), and then, when the scale
generated by the unit metre is available, that:
length of a given steel rod (and width of a given door) =
1.2345 m
(6)
Different from the value (A in the ABO System), that is only
about a classification, the value 1.2345 m reveals the underlying
structure of lengths, that may be additively composed
( " concatenated " as sometimes it is said) so that, if it is true that
length of a given steel rod = width of a given door, then their
composition is twice as long as each of them, and so on. This
supports the usual interpretation of the value 1.2345 m as what
is obtained by taking 1.2345 times the metre, whatever the metre
is.
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