139
ontology of modern science [comprises] material objects (or, alternatively, spacetime points), sets of material objects, sets of sets of material objects, …, but no
properties” (p. 305).
39
Were the extensionalist objection against properties to be accepted, any discourse
about the ontology and epistemology of properties should be deemed to be extrinsic, a purely linguistic shorthand reducible to a set theoretical analysis. Again from
Rozeboom we take a general reply (1966: p. 172):
What is objectionable about this […] is that properties are really distinguishing features of
the entities which possess them, as that in principle properties can be coextensive even
though non-identical. Thus if all crystalline rocks were translucent and conversely, we
should deny that crystallinity and translucency are the same property of rocks even though
the class of crystalline rocks would be identical with the class of translucent rocks.
A more specific and non-hypothetical example is provided by any physical law
which connects two quantities via a constant, as is the case of the Planck–Einstein
relation E = hν, stating that the photon energy E is proportional to its frequency ν,
via the Planck constant h. According to this law, in the case of photons the individual property having energy e, for any given e, and the individual property having
frequency e/h are coextensive (i.e., the set of photons a i such that E[a i ] = e and the
set of photons a j such that ν[a j ] = e/h are the same). Nevertheless, the quantities
energy and frequency remain distinct. Moreover, though not logically contradictory,
the idea that laws of physics establish relations among sets, and that products and
powers of sets can be somehow considered (as in the case of kinetic energy, which
depends on the square of velocity), is counterintuitive, to say the least. Finally,
another well-known counterexample was provided by W.O. Quine (1951): “creatures with a heart” and “creatures with a kidney” have the same extension, but their
meanings are clearly not interchangeable.
This whole discussion seems to provide sufficient reasons to refuse the extensionalist objection, and more generally not to endorse a nominalist position, and to
continue exploring a measurement-oriented ontology and epistemology of properties under a realist perspective, according to which the Basic Evaluation Equation
erties. For example, according to Earl Babbie (who uses a peculiar lexicon: “variable” for general
property and “attribute” for value), “variables … are logical sets of attributes. Thus, for example,
male and female are attributes, and sex or gender is the variable composed of those two attributes.
The variable occupation is composed of attributes such as farmer, professor, and truck driver.
Social class is a variable composed of a set of attributes such as upper class, middle class, and
lower class” (2013: p. 13). Along the same line, Michell claims that “the variable of length is simply the class of all lengths” (1990: p. 51).
39 Indeed, representational theories of measurement usually formalize measurement as a mapping
from objects to numbers: “the first problem … the analysis of any procedure of measurement must
consider … is justification of the assignment of numbers to objects or phenomena” (Suppes &
Zinnes, 1962: p. 3). See also Sect. 6.2.2. Interestingly, extensionalism models logical properties
and properties of measurement science (i.e., what we above designated as P
# and P, respectively)
in exactly the same way, as mappings from sets of objects to sets of values (whatever they are), the
only difference being that the cardinality of the codomain of logical properties is 2 (Lawvere &
Rosebrugh, 2003: 1.2).
5.3 A philosophical interlude
ontology of modern science [comprises] material objects (or, alternatively, spacetime points), sets of material objects, sets of sets of material objects, …, but no
properties” (p. 305).
39
Were the extensionalist objection against properties to be accepted, any discourse
about the ontology and epistemology of properties should be deemed to be extrinsic, a purely linguistic shorthand reducible to a set theoretical analysis. Again from
Rozeboom we take a general reply (1966: p. 172):
What is objectionable about this […] is that properties are really distinguishing features of
the entities which possess them, as that in principle properties can be coextensive even
though non-identical. Thus if all crystalline rocks were translucent and conversely, we
should deny that crystallinity and translucency are the same property of rocks even though
the class of crystalline rocks would be identical with the class of translucent rocks.
A more specific and non-hypothetical example is provided by any physical law
which connects two quantities via a constant, as is the case of the Planck–Einstein
relation E = hν, stating that the photon energy E is proportional to its frequency ν,
via the Planck constant h. According to this law, in the case of photons the individual property having energy e, for any given e, and the individual property having
frequency e/h are coextensive (i.e., the set of photons a i such that E[a i ] = e and the
set of photons a j such that ν[a j ] = e/h are the same). Nevertheless, the quantities
energy and frequency remain distinct. Moreover, though not logically contradictory,
the idea that laws of physics establish relations among sets, and that products and
powers of sets can be somehow considered (as in the case of kinetic energy, which
depends on the square of velocity), is counterintuitive, to say the least. Finally,
another well-known counterexample was provided by W.O. Quine (1951): “creatures with a heart” and “creatures with a kidney” have the same extension, but their
meanings are clearly not interchangeable.
This whole discussion seems to provide sufficient reasons to refuse the extensionalist objection, and more generally not to endorse a nominalist position, and to
continue exploring a measurement-oriented ontology and epistemology of properties under a realist perspective, according to which the Basic Evaluation Equation
erties. For example, according to Earl Babbie (who uses a peculiar lexicon: “variable” for general
property and “attribute” for value), “variables … are logical sets of attributes. Thus, for example,
male and female are attributes, and sex or gender is the variable composed of those two attributes.
The variable occupation is composed of attributes such as farmer, professor, and truck driver.
Social class is a variable composed of a set of attributes such as upper class, middle class, and
lower class” (2013: p. 13). Along the same line, Michell claims that “the variable of length is simply the class of all lengths” (1990: p. 51).
39 Indeed, representational theories of measurement usually formalize measurement as a mapping
from objects to numbers: “the first problem … the analysis of any procedure of measurement must
consider … is justification of the assignment of numbers to objects or phenomena” (Suppes &
Zinnes, 1962: p. 3). See also Sect. 6.2.2. Interestingly, extensionalism models logical properties
and properties of measurement science (i.e., what we above designated as P
# and P, respectively)
in exactly the same way, as mappings from sets of objects to sets of values (whatever they are), the
only difference being that the cardinality of the codomain of logical properties is 2 (Lawvere &
Rosebrugh, 2003: 1.2).
5.3 A philosophical interlude