173
A basic framework on this matter was proposed by Stanley Smith Stevens (1946),
with his well-known classification of what he called “scale types”, and since then
his distinction between nominal, ordinal, interval, and ratio scales has been widely
adopted (for an early, extended, and clear presentation, see Siegel, 1956: ch. 3), and
variously refined.
33
Such a framework was conceived as dealing with scales of measurement, given the perspective that “measurement, in the broadest sense, is […] the
assignment of numerals to objects or events according to rules” (p. 677), explicitly
drawing from Campbell’s seminal representationalist statement that “measurement
is the assignment of numerals to represent properties” (Campbell, 1920: p. 267; see
also the related discussion in Sect. 4.2). From the perspective of the present analysis
Stevens’ “broadest sense” is indeed too broad, if considered to be specifically
related to measurement. Rather, what is interesting in his classification is more correctly understood by considering it as related to scales of property evaluation, thus
disentangled from issues about measurability. We have then to deal with two interrelated issues:
• To which entities does the feature of being quantitative or nonquantitative apply?
• How should the condition of being quantitative or nonquantitative be defined?
But are the terms “nominal”, “ordinal”, and so forth best understood as referring to
types of properties, or of evaluations? And, in consequence, how should such types
be defined?
6.5.1 The scope of the quantitative/nonquantitative distinction
The first question for us to consider is about the scope of the classification between
nominal, ordinal, interval, and ratio—let us call it NOIR, from the initials of the four
adjectives—that is, what does it apply to, and therefore what does NOIR classify?
choosing among them is not relevant here. On this matter a general issue is whether order is sufficient for a property to be considered a quantity. While the sources just cited all answer this question in the negative, more encompassing positions are possible, such as Ellis’, according to whom
“a quantity is usually conceived to be a kind of property. It is thought to be a kind of property that
admits of degrees, and which is therefore to be contrasted with those properties that have an all-ornone character” (1968: p. 24). By using the term “ordinal quantity”, the VIM adopted the same
stance (JCGM, 2012: 1:26): ordinal properties are considered to be quantities. This multiplicity is
one more reason not to fall in the trap of what Abraham Kaplan called the “mystique of quantity”
(1964: p. 172).
33 For example, Nicholas Chrisman mentions the following ten “levels [which] are by no means
complete”, where for each level the “information required” is specified (1998: p. 236): (1) Nominal
(definition of categories). (2) Graded membership (definition of categories plus degree of membership or distance from prototype). (3) Ordinal (definition of categories plus ordering). (4) Interval
(definition of unit and zero). (5) Log interval (definition of exponent to define intervals). (6)
Extensive ratio (definition of unit—additive rule applies). (7) Cyclic ratio (unit and length of
cycle). (8) Derived ratio (units—formula of combination). (9) Counts (definition of objects
counted). (10) Absolute (type: probability, proportion, etc.).
6.5 Generalizing the framework to nonquantitative properties
A basic framework on this matter was proposed by Stanley Smith Stevens (1946),
with his well-known classification of what he called “scale types”, and since then
his distinction between nominal, ordinal, interval, and ratio scales has been widely
adopted (for an early, extended, and clear presentation, see Siegel, 1956: ch. 3), and
variously refined.
33
Such a framework was conceived as dealing with scales of measurement, given the perspective that “measurement, in the broadest sense, is […] the
assignment of numerals to objects or events according to rules” (p. 677), explicitly
drawing from Campbell’s seminal representationalist statement that “measurement
is the assignment of numerals to represent properties” (Campbell, 1920: p. 267; see
also the related discussion in Sect. 4.2). From the perspective of the present analysis
Stevens’ “broadest sense” is indeed too broad, if considered to be specifically
related to measurement. Rather, what is interesting in his classification is more correctly understood by considering it as related to scales of property evaluation, thus
disentangled from issues about measurability. We have then to deal with two interrelated issues:
• To which entities does the feature of being quantitative or nonquantitative apply?
• How should the condition of being quantitative or nonquantitative be defined?
But are the terms “nominal”, “ordinal”, and so forth best understood as referring to
types of properties, or of evaluations? And, in consequence, how should such types
be defined?
6.5.1 The scope of the quantitative/nonquantitative distinction
The first question for us to consider is about the scope of the classification between
nominal, ordinal, interval, and ratio—let us call it NOIR, from the initials of the four
adjectives—that is, what does it apply to, and therefore what does NOIR classify?
choosing among them is not relevant here. On this matter a general issue is whether order is sufficient for a property to be considered a quantity. While the sources just cited all answer this question in the negative, more encompassing positions are possible, such as Ellis’, according to whom
“a quantity is usually conceived to be a kind of property. It is thought to be a kind of property that
admits of degrees, and which is therefore to be contrasted with those properties that have an all-ornone character” (1968: p. 24). By using the term “ordinal quantity”, the VIM adopted the same
stance (JCGM, 2012: 1:26): ordinal properties are considered to be quantities. This multiplicity is
one more reason not to fall in the trap of what Abraham Kaplan called the “mystique of quantity”
(1964: p. 172).
33 For example, Nicholas Chrisman mentions the following ten “levels [which] are by no means
complete”, where for each level the “information required” is specified (1998: p. 236): (1) Nominal
(definition of categories). (2) Graded membership (definition of categories plus degree of membership or distance from prototype). (3) Ordinal (definition of categories plus ordering). (4) Interval
(definition of unit and zero). (5) Log interval (definition of exponent to define intervals). (6)
Extensive ratio (definition of unit—additive rule applies). (7) Cyclic ratio (unit and length of
cycle). (8) Derived ratio (units—formula of combination). (9) Counts (definition of objects
counted). (10) Absolute (type: probability, proportion, etc.).
6.5 Generalizing the framework to nonquantitative properties