176
this case, although it is representable by means of ordered entities, nationality is not
itself ordinal.
37
These two examples show why we do not see the category of internal representations as relevant to measurement.
Hence, in our view, the evaluated property exists and has features that are independent of its possible representations: an evaluation is expected to preserve the
structure of the property, not to induce a structure on the property.
38
Given the controversial nature of Stevens’ framework, it may be worth noting
that this has nothing to do with setting constraints on ways of representation and of
related data processing, such as, say, proscribing against computing the mean value
of a set of numbers that encode nationalities. Along the same lines as Lord (1953),
Velleman and Wilkinson (1993) emphasized the importance of not imposing such
constraints, given that “experience has shown in a wide range of situations that the
application of proscribed statistics to data can yield results that are scientifically
meaningful, useful in making decisions, and valuable as a basis for further research”
(p. 68) (“proscribed statistics” are those statistics that are not “permissible” in the
vocabulary of Stevens, 1946: p. 678). In fact, measurement science does include
some “proscriptions”, such as the condition of dimensional analysis that only values
of quantities of the same kind can be added. Nevertheless, the idea that through data
analysis something can be discovered also about the structure of the evaluated properties is not problematic per se. The point is that if the property under consideration
is evaluated based on (for example) purely ordinal comparisons (as in the case of
hardness), the values that are obtained cannot be expected to convey more than
ordinal information, exactly as prescribed by Stevens’ framework and its refinements (an example of which is mentioned in Footnote 32). In this view, what Stevens
introduced as the set of “permissible” functions is better understood as a matter of
algebraic invariance and meaningfulness under scale transformation (Narens, 2002),
and therefore of uniqueness of the scale itself.
A summary can be presented simply as follows:
• The representation of properties of objects, or the representation of objects as
such, is an unconstrained process, and anything could in principle be used as a
means of representation.
• The evaluation of properties of objects is a process that is expected to produce
values of the evaluated properties.
• The measurement of properties of objects is a specific kind of evaluation.
ence” (JCGM, 2012: 1.1)—the last part “that can be expressed as a number and a reference” is not
an actual specification, and could be removed.
37 Of course, nationality-related orderings can be defined; for example, names of nations are
ordered alphabetically and nations are ordered by the size of their population, but the former is a
relation among linguistic terms and the latter is a relation among cardinalities of sets: neither of
them involve the property of nationality, in the sense that one’s nationality is not a linguistic entity,
nor is it a number.
38 This is thus in sharp contrast with radically constructionist presentations, such as Kaplan’s view
that “the order of a set of objects is something which we impose on them. We take them in a certain
order; the order is not given by or found in the objects themselves” (1964: p. 180).
6 Values, scales, and the existence of properties
this case, although it is representable by means of ordered entities, nationality is not
itself ordinal.
37
These two examples show why we do not see the category of internal representations as relevant to measurement.
Hence, in our view, the evaluated property exists and has features that are independent of its possible representations: an evaluation is expected to preserve the
structure of the property, not to induce a structure on the property.
38
Given the controversial nature of Stevens’ framework, it may be worth noting
that this has nothing to do with setting constraints on ways of representation and of
related data processing, such as, say, proscribing against computing the mean value
of a set of numbers that encode nationalities. Along the same lines as Lord (1953),
Velleman and Wilkinson (1993) emphasized the importance of not imposing such
constraints, given that “experience has shown in a wide range of situations that the
application of proscribed statistics to data can yield results that are scientifically
meaningful, useful in making decisions, and valuable as a basis for further research”
(p. 68) (“proscribed statistics” are those statistics that are not “permissible” in the
vocabulary of Stevens, 1946: p. 678). In fact, measurement science does include
some “proscriptions”, such as the condition of dimensional analysis that only values
of quantities of the same kind can be added. Nevertheless, the idea that through data
analysis something can be discovered also about the structure of the evaluated properties is not problematic per se. The point is that if the property under consideration
is evaluated based on (for example) purely ordinal comparisons (as in the case of
hardness), the values that are obtained cannot be expected to convey more than
ordinal information, exactly as prescribed by Stevens’ framework and its refinements (an example of which is mentioned in Footnote 32). In this view, what Stevens
introduced as the set of “permissible” functions is better understood as a matter of
algebraic invariance and meaningfulness under scale transformation (Narens, 2002),
and therefore of uniqueness of the scale itself.
A summary can be presented simply as follows:
• The representation of properties of objects, or the representation of objects as
such, is an unconstrained process, and anything could in principle be used as a
means of representation.
• The evaluation of properties of objects is a process that is expected to produce
values of the evaluated properties.
• The measurement of properties of objects is a specific kind of evaluation.
ence” (JCGM, 2012: 1.1)—the last part “that can be expressed as a number and a reference” is not
an actual specification, and could be removed.
37 Of course, nationality-related orderings can be defined; for example, names of nations are
ordered alphabetically and nations are ordered by the size of their population, but the former is a
relation among linguistic terms and the latter is a relation among cardinalities of sets: neither of
them involve the property of nationality, in the sense that one’s nationality is not a linguistic entity,
nor is it a number.
38 This is thus in sharp contrast with radically constructionist presentations, such as Kaplan’s view
that “the order of a set of objects is something which we impose on them. We take them in a certain
order; the order is not given by or found in the objects themselves” (1964: p. 180).
6 Values, scales, and the existence of properties