138
even the same as) the length of a j , we interpret this as the hypothesis that there is one
individual length that is identified as the length of the two objects, and therefore is
known in two different ways. In fact
• while the expressions “P[a i ]” and “P[a j ]” have different senses (because they
convey information on properties of different objects),
• their referent could be the same, and actually is the same if the equation
P[a i ] = P[a j ] is true, i.e., they refer to the same individual length.
In other words, if P[a i ] = P[a j ] is true, it is because there is one individual length
which is a universal entity that a i and a j both have.
5.3.3 Do we really need properties?
The position developed in the previous section has some analogies with Bertrand
Russell’s conception of natural numbers: “Under what circumstances do two classes
have the same number? The answer is, that they have the same number when their
terms can be correlated one to one, so that any one term of either corresponds to one
and only one term of the other. […] When the relation holds between two [classes],
those two [classes] have a certain common property, and vice versa. This common
property we call their number. This is the definition of numbers by abstraction”
(1903: pp. 113–116). The natural number n is then what all classes of n elements
have in common, as identified by a one-to-one correspondence among their elements. Such a position is compatible with both
• an extensionalist position: the number n is the class of all classes of n elements,
and
• an intensionalist position: the number n is the property that all classes of n elements share.
Hence extensionalism considers properties to be nothing but “classes of the entities
whose properties they are [, so that] for example, human baldness (or being bald) is
to be identified with the class of all bald humans, while over the domain comprising
all chunks of minerals, the property crystalline is the class of all crystalline rocks”
(Rozeboom, 1966: p. 172). As Hilary Putnam discusses (1969), extensionalism on
properties assumes that if, for all x, x is P
#
if and only if x is Q
#
, then P
#
and Q
#
are
the same property. This applies not only to the properties in the sense of formal
logic, but also to the properties in the sense of measurement science. Any reference
to a property would be then just a convenient shorthand for a given (although usually unknown) set: “the object a has a given length” would precisely and only mean
“the object a belongs to a given set”—that is, the set of objects having the same
length as a—and so on.
38
According to Joel Michell, the consequence is that “the
38 This is about individual properties. There is also an extensionalist interpretation of general prop5 What is measured?
even the same as) the length of a j , we interpret this as the hypothesis that there is one
individual length that is identified as the length of the two objects, and therefore is
known in two different ways. In fact
• while the expressions “P[a i ]” and “P[a j ]” have different senses (because they
convey information on properties of different objects),
• their referent could be the same, and actually is the same if the equation
P[a i ] = P[a j ] is true, i.e., they refer to the same individual length.
In other words, if P[a i ] = P[a j ] is true, it is because there is one individual length
which is a universal entity that a i and a j both have.
5.3.3 Do we really need properties?
The position developed in the previous section has some analogies with Bertrand
Russell’s conception of natural numbers: “Under what circumstances do two classes
have the same number? The answer is, that they have the same number when their
terms can be correlated one to one, so that any one term of either corresponds to one
and only one term of the other. […] When the relation holds between two [classes],
those two [classes] have a certain common property, and vice versa. This common
property we call their number. This is the definition of numbers by abstraction”
(1903: pp. 113–116). The natural number n is then what all classes of n elements
have in common, as identified by a one-to-one correspondence among their elements. Such a position is compatible with both
• an extensionalist position: the number n is the class of all classes of n elements,
and
• an intensionalist position: the number n is the property that all classes of n elements share.
Hence extensionalism considers properties to be nothing but “classes of the entities
whose properties they are [, so that] for example, human baldness (or being bald) is
to be identified with the class of all bald humans, while over the domain comprising
all chunks of minerals, the property crystalline is the class of all crystalline rocks”
(Rozeboom, 1966: p. 172). As Hilary Putnam discusses (1969), extensionalism on
properties assumes that if, for all x, x is P
#
if and only if x is Q
#
, then P
#
and Q
#
are
the same property. This applies not only to the properties in the sense of formal
logic, but also to the properties in the sense of measurement science. Any reference
to a property would be then just a convenient shorthand for a given (although usually unknown) set: “the object a has a given length” would precisely and only mean
“the object a belongs to a given set”—that is, the set of objects having the same
length as a—and so on.
38
According to Joel Michell, the consequence is that “the
38 This is about individual properties. There is also an extensionalist interpretation of general prop5 What is measured?