# Instrumentation & Measurement Magazine 26-1 - 15

```equation is true implies that the length identified as the length
of the rod and the length identified as 1.2345 times the length
that is the metre are the same length. Basically, this applies
also to blood groups, with only some differences, due to the
fact that the classification of blood groups is not generated by
a unit. In relation (4) above, claiming that the equation is true
implies that the blood group identified as the blood group of
a given individual and the blood group identified as the value
(A in the ABO System) are the same blood group. With this
broad, encompassing interpretation, relation (1) can be called
the Basic Evaluation Equation [9], and on this ground the framework
of property (or scale) types is developed.
Indeed, relations other than indistinguishability or operations
can be often applied to properties of a given kind.
For example, some properties can be ordered, like temperatures,
whereas there is not a general criterion to order others,
like blood groups and shapes, and only an indistinguishability
relation holds for them. As another example, for some
kinds of properties, two or more objects can be combined in
such a way that the relation between their properties and the
property of the resulting object fulfills the conditions of the addition
between numbers, as happens for example for lengths
and masses in non-relativistic conditions. Thus, in the set of
kinds of properties different types can be distinguished based
on which relations or operations apply to the properties of that
kind ( " kind " and " type " , that in everyday language are often
taken as synonyms, have then specifically distinct meanings
here). Claiming that a given kind of properties is of a given
type means that the properties of that kind are comparable according
to the relations and can be combined according to the
operations that are characteristic of that type. For example,
stating that the type of mass is ordinal and additive means that
masses can be ordered and combined additively.
Stevens [10] set the framework for the types of (kinds of)
properties (that in fact he operationally characterized as types
of scales), by identifying four types. The first two, that he
called " nominal " and " ordinal " , assume that the properties
of the considered kind are only comparable with each other,
as either indistinguishable or distinguishable in the nominal
type (as for blood groups and shapes), or also ordered in the
ordinal type (as for hardness of minerals in the case of Mohs
scale). The other two types, that Stevens called " interval " and
" ratio " , assume that on the concerned properties some empirical
operations are also possible, and this makes them richer
in the information they convey but also more complex in their
characterization.
A basic issue comes from the traditional distinction between
intensive and extensive properties, as exemplified by
positions in space (let us say, a one-dimensional space for the
sake of simplicity) and lengths. Positions can be operated by
difference, and the difference of two positions is their distance
and therefore a length, but cannot be operated by sum in
February 2023
the sense that there is not an operation between pairs of positions
with the features of an arithmetic addition. Lengths can
instead be operated by both sum and difference. Exactly the
same applies to positions in time, sometimes called " calendar
times " , and durations: durations can be added and subtracted,
whereas positions in time can only be subtracted ( " today plus
yesterday " is meaningless), and the result is their distance in
time, i.e., a duration.
On this matter Stevens focused on the condition that interval
scales / properties, like positions in space and in time, do
not have an intrinsic zero: a zero can be set (what in physics is
done by defining a spatiotemporal frame of reference) but remains
conventional. Vice versa, we consider the existence of
a zero property as an integral part of what we know of ratio
properties (or ratio scales according to Stevens), like length
and duration.
However, things are more complex than the possible association
intensive = interval and extensive = ratio could suggest,
as the historical development of temperature scales shows.
Plausibly thought of as only ordinal in a far past, thermometric
scales, like Celsius and Fahrenheit, upgraded temperature
to an interval property, and in fact such scales do not identify
an intrinsic zero, as witnessed by the fact that the temperatures
mapped to the identifier 0 in Celsius and Fahrenheit scales are
not the same, at the same time recognizing the intensive nature
of temperature, such that the temperature of a body obtained
by putting two bodies in thermal contact is not the sum of their
temperatures but a weighted mean of them. The development
of thermodynamic scales, like Kelvin, led to recognition of an
intrinsic ( " absolute " ) zero temperature, thus upgrading temperature
to a ratio property, while maintaining its intensive
nature. This shows that the correct association is extensive =
The four types originally proposed by Stevens are by no
means the only ones. Stevens himself later introduced a fifth
type, called " absolute " , related to properties that are countable
and therefore are considered to have both an intrinsic zero and
an intrinsic unit, but examples of other types can be found, like
in the case of angle amplitude, that is " circularly additive " , i.e.,
Furthermore, note that the type-related structure of a kind
of properties is observed under specified empirical conditions.
For example, electrical resistances of resistors combine additively
if combined in series but not in parallel (in a hypothetical
world in which only parallel combination of electrical resistances
were possible or known, electrical resistance would be
supposed to be an intensive ratio property, analogous to temperature).
Hence, stating that a given kind is of a given type is
a shorthand for something like: given the currently best available
knowledge, at least one procedure is known to operate on
the properties of that kind according to the type-related structure
[12].
IEEE Instrumentation & Measurement Magazine
15
```

# Instrumentation & Measurement Magazine 26-1

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