179
the unit) have not been selected. In a relation such as shape[rod a] = cube this second component is missing.
39
In order to improve the structural analogy between length[rod a] = 1.2345 m and
shape[rod a] = cube, the set of the possible shapes, of which cube is one element,
needs to be declared: it might be, e.g., {cube, any other shape} or {cube, cylinder,
cone, sphere, any other shape}, thus showing that the report that rod a is cubic conveys different information in the two cases. We may call the set of the possible
shapes a reference set, R, so that an example of the Basic Evaluation Equation in the
case of a nominal property such as shape is
40
shape rod a
[
]= cube in R
where cube in R is then the example of a value of a property. Indeed, the same structure may also be used for quantities, e.g.,
length rod a
[
]= 1 2345
.
,
in metres
i.e., the value is 1.2345 in the classification of lengths generated by the metre and its
multiples, but in this case additivity of length permits the more informative interpretation that the class identified as 1.2345 in that classification corresponds to a length
which is 1.2345 times the metre.
Hence the concept is not bound to quantities: nonquantitative properties
also have values, and any such value is an individual property identified as an element of a given classification of comparable individual properties,
41
such that if the
classification changes, and therefore a different reference set is used, another value
may be obtained for the same property under evaluation. Under these conditions the
previous considerations about values of quantities can be correctly generalized to
values of properties: first, choosing a set of values for blood type or shape corresponds to introducing a classification on the set of the blood types or the shapes, in
which each value identifies one class, and second, Basic Evaluation Equations also
apply to nonquantitative properties and, if true, they convey much richer information than just representation: they state that the property of an object and the value
of a property are the same individual property.
39 This lack of a context—seen, for example, in that reporting that the shape of a given object is
cube does not in itself provide any hint about what other shapes the object might have had—is a
problem in particular for computing the quantity of information obtained by a value. According to
Claude Shannon (1948), this is related to the probability of selecting that value, which in turn supposes knowledge of the underlying probability distribution. We further discuss this fundamental
idea by Shannon in Sect. 8.1, in terms of quantity of (syntactic) information conveyed by
measurement.
40 This is clearly analogous to the way information is reported in ordinal cases, such as Mohs’ hardness, e.g., hardness(given sample) = 5 on the Mohs’ scale.
41 As a consequence, the possible concern that only numbers (or numerals) count as values of properties is unjustified. This also shows that the values of nonquantitative properties are not merely
“symbols” or “names”. The discussion in Sect. 6.2.1 about values of quantities applies more generally to values of properties.
6.5 Generalizing the framework to nonquantitative properties
the unit) have not been selected. In a relation such as shape[rod a] = cube this second component is missing.
39
In order to improve the structural analogy between length[rod a] = 1.2345 m and
shape[rod a] = cube, the set of the possible shapes, of which cube is one element,
needs to be declared: it might be, e.g., {cube, any other shape} or {cube, cylinder,
cone, sphere, any other shape}, thus showing that the report that rod a is cubic conveys different information in the two cases. We may call the set of the possible
shapes a reference set, R, so that an example of the Basic Evaluation Equation in the
case of a nominal property such as shape is
40
shape rod a
[
]= cube in R
where cube in R is then the example of a value of a property. Indeed, the same structure may also be used for quantities, e.g.,
length rod a
[
]= 1 2345
.
,
in metres
i.e., the value is 1.2345 in the classification of lengths generated by the metre and its
multiples, but in this case additivity of length permits the more informative interpretation that the class identified as 1.2345 in that classification corresponds to a length
which is 1.2345 times the metre.
Hence the concept
also have values, and any such value is an individual property identified as an element of a given classification of comparable individual properties,
41
such that if the
classification changes, and therefore a different reference set is used, another value
may be obtained for the same property under evaluation. Under these conditions the
previous considerations about values of quantities can be correctly generalized to
values of properties: first, choosing a set of values for blood type or shape corresponds to introducing a classification on the set of the blood types or the shapes, in
which each value identifies one class, and second, Basic Evaluation Equations also
apply to nonquantitative properties and, if true, they convey much richer information than just representation: they state that the property of an object and the value
of a property are the same individual property.
39 This lack of a context—seen, for example, in that reporting that the shape of a given object is
cube does not in itself provide any hint about what other shapes the object might have had—is a
problem in particular for computing the quantity of information obtained by a value. According to
Claude Shannon (1948), this is related to the probability of selecting that value, which in turn supposes knowledge of the underlying probability distribution. We further discuss this fundamental
idea by Shannon in Sect. 8.1, in terms of quantity of (syntactic) information conveyed by
measurement.
40 This is clearly analogous to the way information is reported in ordinal cases, such as Mohs’ hardness, e.g., hardness(given sample) = 5 on the Mohs’ scale.
41 As a consequence, the possible concern that only numbers (or numerals) count as values of properties is unjustified. This also shows that the values of nonquantitative properties are not merely
“symbols” or “names”. The discussion in Sect. 6.2.1 about values of quantities applies more generally to values of properties.
6.5 Generalizing the framework to nonquantitative properties