175
The examples proposed by Michell are interesting, and useful for better understanding of what is at stake with this distinction. He exemplified external representations
(i.e., such that NOIR is a feature of properties) by means of hardness:
Minerals can be ordered according to whether or not they scratch one another when rubbed
together. The relation, x scratches y, between minerals, is transitive and asymmetric and
these [features] can be established prior to any numerical assignments being made.
The idea is then that once a property-related criterion of comparison has been identified (in this case, mutual scratching), the outcomes of property-related comparisons
do not depend on the way they are represented: the conclusion would be that hardness is ordinal (or, more correctly, that hardness is at least ordinal). As the example
suggests, this seems to be based on the assumption that, for an external representation to be possible, properties of objects must be empirically comparable according
to some given conditions, and the outcome of the comparison must be observable,
as in the paradigmatic case of mass via a two-pan balance. This condition was
embedded in the representational theories of measurement under the assumption
that the availability of an empirical relational system is a precondition of
measurement.
Michell proposes two examples of internal representations (i.e., such that NOIR
is a feature of representations rather than properties). The first one is about
an extreme case […] of assigning different numbers to each of a class of identical things
(say, white marbles) and on that basis defining a [property]. The [property] represented by
such assignments would not be logically independent of them and, so, had they not been
made, the [property] would not exist.
This is indeed the extreme case of an assignment claimed to be a representation but
that does not represent anything, being only a means of object identification: it is not
even a property evaluation, given that there is no property to evaluate, in the specific
sense that a Basic Evaluation Equation cannot be written because there is not a
general property to be evaluated of the considered objects.
35
We may then safely
ignore this case, and consider the second, “less extreme” example,
where an independent [property] may exist, but the structure that it is taken to have depends
upon numerical assignments made. For example, people may be assigned numbers according to nationality (say, Australian, 1; French, 2; American, 3; Belgian, 4; etc.) and then the
[property] of nationality may be taken to have the ordinal structure of the numbers assigned.
In this case, had numerical assignments not been made, the [property] (nationality) would
still exist but the supposed ordinal structure would not.
This is a case in which Stevens’ framework proves to be non-trivially applicable.
While it is always possible to adopt numbers for representational means, the numerical relations do not necessarily relate to empirical relations among the objects:
36
in
35 This is analogous to the unfortunate example given by Stevens about “the numbering of football
players for the identification of the individuals” (1946: p. 678): identification is not property evaluation, and so the mocking critique by Lord (1953) rightly applies to this example.
36 This is why in the definition of given by the VIM—“property of a phenomenon, body,
or substance, where the property has a magnitude that can be expressed as a number and a refer6.5 Generalizing the framework to nonquantitative properties
The examples proposed by Michell are interesting, and useful for better understanding of what is at stake with this distinction. He exemplified external representations
(i.e., such that NOIR is a feature of properties) by means of hardness:
Minerals can be ordered according to whether or not they scratch one another when rubbed
together. The relation, x scratches y, between minerals, is transitive and asymmetric and
these [features] can be established prior to any numerical assignments being made.
The idea is then that once a property-related criterion of comparison has been identified (in this case, mutual scratching), the outcomes of property-related comparisons
do not depend on the way they are represented: the conclusion would be that hardness is ordinal (or, more correctly, that hardness is at least ordinal). As the example
suggests, this seems to be based on the assumption that, for an external representation to be possible, properties of objects must be empirically comparable according
to some given conditions, and the outcome of the comparison must be observable,
as in the paradigmatic case of mass via a two-pan balance. This condition was
embedded in the representational theories of measurement under the assumption
that the availability of an empirical relational system is a precondition of
measurement.
Michell proposes two examples of internal representations (i.e., such that NOIR
is a feature of representations rather than properties). The first one is about
an extreme case […] of assigning different numbers to each of a class of identical things
(say, white marbles) and on that basis defining a [property]. The [property] represented by
such assignments would not be logically independent of them and, so, had they not been
made, the [property] would not exist.
This is indeed the extreme case of an assignment claimed to be a representation but
that does not represent anything, being only a means of object identification: it is not
even a property evaluation, given that there is no property to evaluate, in the specific
sense that a Basic Evaluation Equation cannot be written because there is not a
general property to be evaluated of the considered objects.
35
We may then safely
ignore this case, and consider the second, “less extreme” example,
where an independent [property] may exist, but the structure that it is taken to have depends
upon numerical assignments made. For example, people may be assigned numbers according to nationality (say, Australian, 1; French, 2; American, 3; Belgian, 4; etc.) and then the
[property] of nationality may be taken to have the ordinal structure of the numbers assigned.
In this case, had numerical assignments not been made, the [property] (nationality) would
still exist but the supposed ordinal structure would not.
This is a case in which Stevens’ framework proves to be non-trivially applicable.
While it is always possible to adopt numbers for representational means, the numerical relations do not necessarily relate to empirical relations among the objects:
36
in
35 This is analogous to the unfortunate example given by Stevens about “the numbering of football
players for the identification of the individuals” (1946: p. 678): identification is not property evaluation, and so the mocking critique by Lord (1953) rightly applies to this example.
36 This is why in the definition of
or substance, where the property has a magnitude that can be expressed as a number and a refer6.5 Generalizing the framework to nonquantitative properties