147
values.
3
(Henceforth we occasionally use the short term “value”, rather than “value
of a property” or “value of a quantity”, if this does not create ambiguity.)
6.2.1 Values of properties: what they are not
Values of properties have such a critical role that it is perhaps not surprising that
there are multiple and even incompatible positions on what they are. According to
two common stereotypes, they are expressions, or they are symbols. Let us start our
analysis by showing that neither of these positions is correct. These stereotypes are
usually related to quantitative properties rather than properties as such; hence, pars
pro toto, we refer to quantities in the discussion that follows.
First, for example, according to the first edition of the VIM the value of a quantity is “the expression of a quantity in terms of a number and an appropriate unit of
measurement” (ISO, 1984: 1.17, emphasis added). The first definition that the
Oxford English Dictionary (OED) gives of is “things that people say,
write or do in order to show their feelings, opinions and ideas”: thus, in general
usage (according to the OED), expressions (including mathematical expressions)
are linguistic entities, i.e., in the sense of terminology, neither concepts nor objects
(see Sect. 2.1). But it should be clear that here, in discussing measurement, values
are not linguistic entities. Consider the difference, e.g., between the rod a, which is
an object, and the five-character (space included) term “rod a”, which is a linguistic
entity: the object has a weight, a color, etc., whereas the term does not. The term
“rod a” refers to a given rod, but is not that rod. Analogously, values are communicated by means of terms but they are not terms.
4
And in fact the same value, e.g.,
1.2345 m, can be expressed linguistically in multiple ways, e.g., “one point two …
meters”, “1.2345 m”, and “1,2345 m” (for most non-English-speaking people),
showing that 1.2345 m and “1.2345 m” are different entities. Certainly, values must
3 In fact, the analysis that follows may be easily generalized to the case of properties and values of
properties, as we do later on in this chapter, where we also discuss the characterization of quantities as specific kinds of properties. We start by presenting the more specific case of values of
quantities because the very concept of is not widely used, and some would
consider it controversial. It should be noted that the boundary between quantitative and nonquantitative properties is not uniquely defined, and in particular there are controversies whether ordinal
properties are quantitative or not. However, that additive properties are quantities is not an issue,
and we start our discussion from them.
4 One example of a term used to communicate a value is a numeral, which is a term for a number;
for example, “4” and “IV” are both numerals that stand for the number 4. As was previously discussed, Campbell (1920) defined measurement as “the assignment of numerals to represent properties”, and Stevens (1959) defined measurement as “the assignment of numerals to objects or
events according to rule”; such statements may have inadvertently contributed to the confusion
between values and terms. It may be worth noting that even though both Campbell and Stevens are
associated with representational theories of measurement, the wider literature on representationalism emphasizes the mapping of objects or properties to numbers, not numerals, as discussed further below.
6.2 Towards values of properties
values.
3
(Henceforth we occasionally use the short term “value”, rather than “value
of a property” or “value of a quantity”, if this does not create ambiguity.)
6.2.1 Values of properties: what they are not
Values of properties have such a critical role that it is perhaps not surprising that
there are multiple and even incompatible positions on what they are. According to
two common stereotypes, they are expressions, or they are symbols. Let us start our
analysis by showing that neither of these positions is correct. These stereotypes are
usually related to quantitative properties rather than properties as such; hence, pars
pro toto, we refer to quantities in the discussion that follows.
First, for example, according to the first edition of the VIM the value of a quantity is “the expression of a quantity in terms of a number and an appropriate unit of
measurement” (ISO, 1984: 1.17, emphasis added). The first definition that the
Oxford English Dictionary (OED) gives of
write or do in order to show their feelings, opinions and ideas”: thus, in general
usage (according to the OED), expressions (including mathematical expressions)
are linguistic entities, i.e., in the sense of terminology, neither concepts nor objects
(see Sect. 2.1). But it should be clear that here, in discussing measurement, values
are not linguistic entities. Consider the difference, e.g., between the rod a, which is
an object, and the five-character (space included) term “rod a”, which is a linguistic
entity: the object has a weight, a color, etc., whereas the term does not. The term
“rod a” refers to a given rod, but is not that rod. Analogously, values are communicated by means of terms but they are not terms.
4
And in fact the same value, e.g.,
1.2345 m, can be expressed linguistically in multiple ways, e.g., “one point two …
meters”, “1.2345 m”, and “1,2345 m” (for most non-English-speaking people),
showing that 1.2345 m and “1.2345 m” are different entities. Certainly, values must
3 In fact, the analysis that follows may be easily generalized to the case of properties and values of
properties, as we do later on in this chapter, where we also discuss the characterization of quantities as specific kinds of properties. We start by presenting the more specific case of values of
quantities because the very concept of
consider it controversial. It should be noted that the boundary between quantitative and nonquantitative properties is not uniquely defined, and in particular there are controversies whether ordinal
properties are quantitative or not. However, that additive properties are quantities is not an issue,
and we start our discussion from them.
4 One example of a term used to communicate a value is a numeral, which is a term for a number;
for example, “4” and “IV” are both numerals that stand for the number 4. As was previously discussed, Campbell (1920) defined measurement as “the assignment of numerals to represent properties”, and Stevens (1959) defined measurement as “the assignment of numerals to objects or
events according to rule”; such statements may have inadvertently contributed to the confusion
between values and terms. It may be worth noting that even though both Campbell and Stevens are
associated with representational theories of measurement, the wider literature on representationalism emphasizes the mapping of objects or properties to numbers, not numerals, as discussed further below.
6.2 Towards values of properties