178
example, the VIM defines (JCGM, 2012: 1.19), but does not
define even though it deals with nonquantitative properties,
termed “nominal properties” (JCGM, 2012: 1.30). Hence the problem for us to consider here is whether the Basic Evaluation Equation is meaningful only in the specific case of quantities, i.e., in the case
quantity of an object value of a quantity
=
or is also able to convey knowledge about nonquantitative properties.
Our construction of values of quantitative properties, presented in Sect. 6.3,
relies on their empirical additivity, in the example of length, or on the invariance of
their empirical difference, in the examples of temperature and reading comprehension ability, conditions which do not hold for nonquantitative properties. Our concept of shape, for example, is indeed such that the ideas of “adding objects by their
shape” or “subtracting objects by their shape” are meaningless, in the sense that the
shape of an object obtained by somehow composing two other objects is in general
not additively related to the shapes of the composing objects (in fact, shapes are not
even ordered: for example, it is meaningless to say that a cube is more or less than
a cylinder). Hence, the interpretation of the Basic Evaluation Equation for quantities, according to the Q-notation, such that Q[a]/[Q] is a number (see Sect. 6.2.2)
does not apply to nonquantitative properties: there are no “shape units” in this case,
nor can shapes be compared by their ratio.
At a more fundamental level, however, the idea of conveying information on
properties like blood types of individuals or shapes of objects by means of “values
of blood type” and “values of shape” maintains its meaning and relevance, as when
we say that a given rod is cubic or cylindric, phraseological means for “whose shape
is cube” and “whose shape is cylinder”, which immediately leads to a formalization
such as shape[rod a] = cube and shape[rod a] = cylinder. Given their analogy in
structure with length[rod a] = 1.2345 m, where 1.2345 m is a value of length, the
conclusion seems to be that cube, cylinder, and so forth may be considered to be
values of shape.
This is not completely correct, given an important difference between the two
cases. Indeed, the value 1.2345 m includes, via the unit metre and the reported number of significant digits, information on the set of possible values from which
1.2345 m has been chosen, i.e., the set is the nonnegative multiples of the metre: it
is 1.2345 m and not 1.2346 m, and so on; it is 1.2345 m but we are unable to distinguish between 1.2345 m and 1.23451 m, and so on. Choosing a unit and a (finite)
number of significant digits corresponds to introducing a classification on the set of
the lengths, in which each value identifies one class. Hence, selecting a value of
length conveys both the information that (1) the class of lengths identified by that
value has been selected, and (2) all other classes (identified by all other multiples of
6 Values, scales, and the existence of properties
example, the VIM defines
define
termed “nominal properties” (JCGM, 2012: 1.30). Hence the problem for us to consider here is whether the Basic Evaluation Equation is meaningful only in the specific case of quantities, i.e., in the case
quantity of an object value of a quantity
=
or is also able to convey knowledge about nonquantitative properties.
Our construction of values of quantitative properties, presented in Sect. 6.3,
relies on their empirical additivity, in the example of length, or on the invariance of
their empirical difference, in the examples of temperature and reading comprehension ability, conditions which do not hold for nonquantitative properties. Our concept of shape, for example, is indeed such that the ideas of “adding objects by their
shape” or “subtracting objects by their shape” are meaningless, in the sense that the
shape of an object obtained by somehow composing two other objects is in general
not additively related to the shapes of the composing objects (in fact, shapes are not
even ordered: for example, it is meaningless to say that a cube is more or less than
a cylinder). Hence, the interpretation of the Basic Evaluation Equation for quantities, according to the Q-notation, such that Q[a]/[Q] is a number (see Sect. 6.2.2)
does not apply to nonquantitative properties: there are no “shape units” in this case,
nor can shapes be compared by their ratio.
At a more fundamental level, however, the idea of conveying information on
properties like blood types of individuals or shapes of objects by means of “values
of blood type” and “values of shape” maintains its meaning and relevance, as when
we say that a given rod is cubic or cylindric, phraseological means for “whose shape
is cube” and “whose shape is cylinder”, which immediately leads to a formalization
such as shape[rod a] = cube and shape[rod a] = cylinder. Given their analogy in
structure with length[rod a] = 1.2345 m, where 1.2345 m is a value of length, the
conclusion seems to be that cube, cylinder, and so forth may be considered to be
values of shape.
This is not completely correct, given an important difference between the two
cases. Indeed, the value 1.2345 m includes, via the unit metre and the reported number of significant digits, information on the set of possible values from which
1.2345 m has been chosen, i.e., the set is the nonnegative multiples of the metre: it
is 1.2345 m and not 1.2346 m, and so on; it is 1.2345 m but we are unable to distinguish between 1.2345 m and 1.23451 m, and so on. Choosing a unit and a (finite)
number of significant digits corresponds to introducing a classification on the set of
the lengths, in which each value identifies one class. Hence, selecting a value of
length conveys both the information that (1) the class of lengths identified by that
value has been selected, and (2) all other classes (identified by all other multiples of
6 Values, scales, and the existence of properties