What is Measured?

Abstract

This chapter aims to explore some key components of an ontology and an epistemology of properties. What is evaluated, and more specifically, measured, are properties of objects, such as lengths of rigid bodies and reading comprehension abilities of individuals, and the results of evaluations, and thus of measurements, are values of properties. Hence, a study of the nature of properties and of our ways of securing knowledge of them is a pivotal component of measurement science. We start from the hypothesis that properties of objects are associated with modes of empirical interaction of the objects with their environment. Consistently with the model-dependent realism introduced in Chap. 4, the Basic Evaluation Equation

$${\text{property}}\,{\text{of}}\,{\text{a}}\,{\text{given}}\,{\text{object}} = {\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{property}}$$

of which the relation

$${\text{measurand}} = {\text{measured}}\,{\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{property}}$$

is a specific case, is interpreted as a claim of an actual referential equality, which conveys information on the measurand because the measurand and the measured value remain conceptually distinct entities.

5.1 Introduction

Measurement is a specific kind of evaluation of empirical properties of objects. A measurement-oriented ontology and epistemology of properties is a complex subject, in which the ontic dimension (what properties are) and the epistemic dimension (what we can know about properties and how we can know it) are deeply intertwined. Hence, a discussion of properties is an important part of a discourse on measurement. Let us restart from our discussion about properties in Sect. 2.2.

An empirical property of an object is associated with a mode of empirical interaction of the object with its environment, where this association is understood such that

• an object empirically interacts with its environment in multiple modes, and each mode of interaction is considered to correspond to a property of the object,¹ and

• some objects are comparable with respect to some of their properties, and sometimes distinct objects are discovered to have empirically indistinguishable properties.²

By considering properties of objects as associated with modes of empirical interaction of objects with their environment, we avoid taking a position on

• the nature of properties of objects, other than that we consider them to be entities able to produce at least in principle observable effects and to support the comparison of objects,³

• the difference between inherent (or essential) and contingent (or accidental) properties, where the identity of the bearer, i.e., the object, is supposed to be affected by a change of its inherent properties (a vase might be considered a different object than the amorphous amount of clay from which it was shaped), but not by a change of its contingent properties (a vase may still be considered the same object, even if it is chipped), and

• the issue of the possible distinction between object-specific (“primary”, e.g., mass) and observer-related (“secondary”, e.g., perceived color) properties.⁴

This is consistent with a pragmatic stance which, we believe, is appropriate to ground a measurement-oriented discussion on properties:

• we consider a property of an object to exist insofar as the object somehow interacts with its environment, though we accept that everything we consider as known about a property is always revisable, and could turn out to be wrong;

• we acknowledge that empirical interactions may be physical, but also psychological, sociological, etc., thus admitting the existence of non-physical properties;

• we consider a property of an object to be associated with a mode of interaction of the object, but we acknowledge that the existence of a property may be hypothesized also independently of a mode of interaction, and we do not say anything further about what a property is per se.⁵

Furthermore, we consider that phrases such as “the objects a and b are comparable with respect to a given property” and “a given property of the object a and a given property of the object b are comparable” refer to the same empirical situation.⁶ Whenever this happens, we say that the given properties of a and b are of the same kind (as in JCGM, 2012: 1.2), and thus we assume that a general property exists of which the given properties of a and b, which we call individual properties, are instances (see also an introduction of these concepts in Sect. 2.2). General and individual properties—such as mass and any given mass, respectively—are sometimes called, particularly in the philosophical literature, “determinables” and “determinates”, respectively; see, e.g., Wilson, 2017). Hence, length and reading comprehension ability are examples of general properties, and the length of a given rod and the reading comprehension ability of a given individual are examples of individual properties. The length of a given rod and the wavelength of a given radiation are comparable, being individual properties of the same kind, i.e., the general property length, whereas the length and the mass of any two objects are not comparable, being individual properties of different kinds.

Even from this philosophically modest perspective, several important issues remain open for consideration, regarding in particular the distinction between the existence of a property and the knowledge that we can have of it. This chapter is devoted to an analysis of this subject, and to providing an interpretation of the (measurement) relation

$${\text{measurand}} = {\text{measured}}\,{\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{property}}$$

as introduced in Sect. 2.2.4, a specific case⁷ of the Basic Evaluation Equation

$${\text{property}}\,{\text{of}}\,{\text{a}}\,{\text{given}}\,{\text{object}} = {\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{property}}$$

formalized as⁸

$$P\left[ a \right] = p$$

for example

$$length\left[ {rod\,a} \right] = {1}.{2345}\,{\text{m}}$$

or

$$\begin{aligned} & reading\,comprehension\,ability\left[ {student\,b} \right] \\ & = {\text{1}}.{\text{23}}\,{\text{logits }}({\text{on}}\,{\text{a}}\,{\text{specific}}\,{\text{RCA}}\,{\text{scale)}} \\ \end{aligned}$$

or

$$blood\,type\left[ {patient\,c} \right] = {\text{A}}\,{\text{in}}\,{\text{the}}\,{\text{ABO}}\,{\text{system}}$$

where in the first case the relation is about a ratio (and more specifically an additive) quantity (length) and the value is reported as the product of a number and a quantity unit (the metre), in the second case the relation is about an interval quantity (reading comprehension ability) and the value is reported as a number in an interval scale (here denoted as logits, on a specific RCA scale) (Maul et al., 2019), and in the third case the relation is about a nominal property (blood type) and the value is reported as an identifier for a class in a specified classification system (the ABO system) (Mari, 2017).

In the case of ratio quantities, which is the common situation in the measurement of physical properties, the relation becomes

$${\text{quantity}}\,{\text{of}}\,{\text{a}}\,{\text{given}}\,{\text{object}} = {\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{quantity}}$$

formalized more specifically as

$$Q\left[ a \right] = \left\{ Q \right\}\left[ Q \right]$$

where the value is the product of a number {Q} and a quantity unit [Q] (which is a different usage of “[]” than on the left-hand side of the equation): thus, in the example above, {length[rod a]} = 1.2345 and [length[rod a]] = m.⁹ While most of what follows apply to all properties, independently of their type, we often refer more specifically to quantities, due to their widespread use in the tradition of measurement science and their richer algebraic structure, which makes examples easier to present and understand.

Box 5.1 A very short introduction to ontology

Given our statement that this chapter is mainly devoted to exploring a measurement-oriented ontology of properties, a few preliminary words might be useful about what we consider to be an ontology. As Willard V. O. Quine wrote, “A curious thing about the ontological problem is its simplicity. It can be put in three Anglo-Saxon monosyllables: ‘What is there?’ It can be answered, moreover, in a word—‘Everything’—and everyone will accept this answer as true. However, this is merely to say that there is what there is. There remains room for disagreement over cases; and so the issue has stayed alive down the centuries.” (1948: p. 21).

Cases about which there can be or has been disagreement include the sphere of fixed stars, phlogiston, and continuous flows of electricity, not to mention nearly every proposed property in the human sciences, perhaps most famously general intelligence: it was not at all trivial to arrive at the conclusion that, for example, phlogiston does not exist, and in fact this required a radical revision of several related bodies of knowledge. But of course even today we can talk in a meaningful way about the sphere of fixed stars, phlogiston, and continuous flows of electricity; otherwise a sentence such as “all visible stars are fixed to a celestial sphere” would be meaningless rather than false (on par with a phrase like “all qwerty uiop are fixed to a celestial sphere”).

Hence, the fact that x appears in meaningful sentences is not sufficient to conclude that x exists. It should be acknowledged that there are indeed different modes of existence: for example, both paper books and prime numbers greater than 1 million exist, but their modes of existence differ. The “disagreement over cases” to which Quine refers is related to the existence in a given mode, not to the generic situation of any possible mode of existence. As another well-known example, unicorns do not exist as physical entities, but do exist as literary entities. Thus, our ability to talk in a grammatically correct way of an entity x is not sufficient to guarantee that x exists as an entity of the kind Y, nor is the fact that x exists as an entity of the kind Z is sufficient to guarantee that it exists also as an entity of the kind Y. As a consequence, a claim of existence of an entity x is interesting only if it is specified as a claim of Y-existence, i.e., existence in the mode Y, for a given Y. The ambiguity is avoided if instead of the generic “do unicorns exist?” we ask “do unicorns exist as physical entities?”, i.e., “do unicorns have Y-existence?”, where Y = physical, which according to our current knowledge has a different answer from “do unicorns have Z-existence?”, where Z = literary.

Furthermore, it is important to recognize that such a claim is about the Y-existence of x as an object, not about the meaning of the term “x”, or about the existence of the concept < x > . As it was put by Quine: “The phrase ‘Evening Star’ names a certain large physical object of spherical form, which is hurtling through space some scores of millions of miles from here. The phrase ‘Morning Star’ names the same thing, as was probably first established by some observant Babylonian. But the two phrases cannot be regarded as having the same meaning; otherwise that Babylonian could have dispensed with his observations and contented himself with reflecting on the meanings of his words. The meanings, then, being different from one another, must be other than the named object, which is one and the same in both cases.” (1948, p.28). In Gottlob Frege’s terminology (1892), this is effectively presented by acknowledging that two terms (such as “Evening Star” and “Morning Star”) can have different senses and nevertheless the same reference, and also that a term can have Y-sense but no Y-reference, i.e., a term can be intended as referring to an entity x of the kind Y even though x has no Y-existence (e.g., we may well understand the claim that some unicorns exist as biological entities, and nevertheless—or, in fact, exactly because we understand exactly its meaning—consider it false according to the best currently available knowledge). Furthermore, this explains the difference between the two relations

$${\text{Evening}}\,{\text{Star}} = {\text{Morning}}\,{\text{Star}}$$

and

$${\text{Evening}}\,{\text{Star}} = {\text{Evening}}\,{\text{Star}}$$

The former required a lot of astronomical ingenuity and knowledge for its discovery, while the latter is a trivial, logical truth, as is any identity x = x, independent of astronomical facts.

It is then worth emphasizing that, as used here, “ontology” is not a synonym of “metaphysics”. Rather, “it refers to the set of ‘things’ a person believes to exist, or the set of things defined by, or assumed by, some theory. What’s in your ontology? Do you believe in ghosts? Then ghosts are in your ontology, along with tables and chairs and songs and vacations, and snow, and all the rest.” (Dennett, 2017: p. 60) (for a wide presentation of a “scientific perspective” on ontology, see Bunge, 1977, possibly starting from his “list of ontological principles occurring in scientific research”, p. 16). Of course, at least some of the things we believe to exist are physical, and some are psychosocial.

A key problem in ontology—perhaps the key problem of any ontology of properties—is whether things such as a given mass and mass as such are existing (though possibly abstract) entities or are just concepts that we produce for organizing our knowledge. An intermediate position, called extensionalism, is that they are not entities as such but sets (or possibly mereological sums: Varzi, 2019) of them: a given mass would then just be a set of masses of objects, and mass the set of all given masses, and therefore a set of sets.

The answer to such a problem depends on whether one’s ontology has room for abstract entities or only for concrete entities, a distinction that is sometimes presented in terms of universals and particulars (the possible differences between <abstract> and <universal> and between <concrete> and <particular> are beyond the scope of our purposes here), and grounds the opposition between realism and nominalism: “The realist’s ontology represents a two-category ontology; it postulates entities of two irreducibly different types: particulars and universals. According to the nominalist, however, all the theoretical work done by the two-category ontology of the realist can be done by an ontological theory that commits us to the existence of entities of just one category, particulars.” (Loux & Crisp, 2017: p. 50).

In what follows we try to remain as neutral as possible about the alternative between realism and nominalism, and mention the position that we believe is more appropriate for effectively accounting for the key facts of measurement results only in Sect. 5.3.

5.1.1 The Possible Meanings of the Basic Evaluation Equation

The Basic Evaluation Equation

$${\text{property}}\,{\text{of}}\,{\text{a}}\,{\text{given}}\,{\text{object}} = {\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{property}}$$

conveys the core information obtained by measurement (neglecting measurement uncertainty, for the moment). Despite the fact that information of this sort is commonly produced and used, the apparent simplicity of the relation hides the question: is this relation an actual equality, or is the “ = ” sign in it just a placeholder for a different relation?

The problem is mostly immaterial in day-to-day practice and is thus usually left in the background, so that one sometimes encounters claims such as Gary Price’s (2001: p. 294) that the relation “‘equals’ means ‘is expressed, modeled, or represented by’”. Since <expression> , <modeling> , and <representation> are distinct concepts, and none of them is the same as <equality> , it seems that such a statement only informs us of a lack of interest in understanding what kind of information a Basic Evaluation Equation actually conveys. In distinction to this vagueness, it is our position that an answer to this problem is indeed critical for a measurement-related ontology and epistemology of properties. Note that different positions are possible (Mari, 1997), the two extremes being

• a strong ontology, which assumes that properties of objects inherently have values, so that if they are known it is because they have been discovered by means of experimental activities, and

• a weak ontology, which assumes that values are assigned as a means of representation.

The common ground of these positions is the acknowledgment that (i) in performing measurement, the starting point is the identification of a property of an object to be measured, i.e., the measurand, and the discovery (according to a strong ontology) or the selection (according to a weak ontology) of a set of possible values of that property, and that (ii) the outcome of the process is that one value in the set (or a subset of them, as in JCGM, 2012: 2.1, if measurement uncertainty is taken into account) is attributed to the measurand. The issue is about how to interpret such an attribution: is the value established because it exists in the object before and independently of any experimental activity, or is it (just) chosen to suitably report the acquired information on the object? Is then measurement akin to discovery or invention? Even more fundamentally, this issue is grounded upon the issue of the very existence of properties: do entities such as length and reading comprehension ability exist in the world, or are they just constructs by which we organize our knowledge?

Positions like the one underlying the representational theories of measurement (see Sect. 4.2.3) emphasize the representational aspect of measurement, plainly stating that the task of measurement is “to construct numerical representations of qualitative structures” (Krantz et al., 1971: p. xviii), and from their beginnings have acknowledged that “the major source of difficulty in providing an adequate theory of measurement is to construct relations which have an exact and reasonable numerical interpretation” (Scott & Suppes, 1958: p. 113). Such positions are plausibly based on weaker, less demanding ontologies, and this may make their practical consequences applicable also to those who accept a stricter position: a value may be chosen to represent a property of an object exactly because that property has that value.¹⁰ If it is maintained that properties of objects do not inherently have values, the representation may be chosen according to different criteria and is required to be at least consistent: if properties of objects are observed to be ordered then their assigned values should be ordered in turn, but any ordered set would be suitable to perform such a purely symbolic task, and so on. However, a stronger ontology invites interpretation of advancements in measurement-related knowledge and practices as an evolutionary process: at the beginning the available information could be so “meager and unsatisfactory” (quoting Lord Kelvin; see the related discussion in Kuhn, 1961) that the evaluation results are more or less everything that is known of the considered property, and therefore consistency in the representation is the only condition that can be sought. Such an approach could be later abandoned with the acquisition of more and better information, leading to corroboration of the hypothesis of the very existence of the property, up to the extreme position that the measurand has a knowledge-independent true value, to be estimated through measurement.¹¹

5.1.2 A Pragmatic Introduction to the Problem

Basic Evaluation Equations are at the core of any measurement, and therefore an understanding of them is a requirement for a well-grounded measurement science. However, “equality gives rise to challenging questions which are not altogether easy to answer” (Frege, 1892: p. 25): quoting again Price (2001: p. 294), in a Basic Evaluation Equation does the equality sign mean “is expressed, modeled, or represented by”, or (in some sense to be specified) equality, or something else?

In what follows, we introduce the problem only in terms of ordinal comparisons, thus about what could be called a “Basic Evaluation Inequality”: this property of this object is less than this value. Our claim is that this does not affect the generality of the presentation and hopefully makes it clearer by referring to a less controversial relation than equality, thus avoiding the “challenging questions” to which Frege alluded—including the ones connected with the possible role of uncertainty—that equality brings.

Let us consider the case of mass.

• 1. In measurement we deal with entities such as masses of given objects, e.g., mass[rod a], and values of mass, e.g., 1.23 kg. For the sake of argument, let us call the former “O-entities” (i.e., related to objects) and the latter “V-entities” (i.e., related to values), by noting that different terms may (but do not necessarily) correspond to different kinds of entities, and the conclusion that we might well reach is that properties of objects, i.e., O-entities, and values of properties, i.e., V-entities, are different ways of referring to the same kind of entity (O-entities and V-entities are more specifically termed “addressed quantities” and “classifier quantities”, respectively, by Mari & Giordani, 2012).

• 2. O-entities and V-entities are such that we can compare

  • O-entities among themselves (the mass of rod a is less than the mass of rod b),

  • V-entities among themselves (1.23 kg is less than 2.34 kg), and

  • O-entities and V-entities (the mass of rod a is less than 1.23 kg).

• 3. In particular, the chain of inequalities

  $$mass\left[ {rod\,a} \right] < {1}.{23}\,{\text{kg}} < {2}.{34}\,{\text{kg}} < mass\left[ {rod\,b} \right]$$

  is understandable and does not pose problems of interpretation, also about the transitivity of the relation (e.g., from mass[rod a] < 1.23 kg and 1.23 kg < 2.34 kg the conclusion is unproblematically obtained that mass[rod a] < 2.34 kg), thus justifying the hypothesis that, at least at a sufficiently abstract level, what is designated here by “<” is actually the same relation applied to both O-entities and V-entities.

• 4. The comparison of O-entities among themselves and the comparison of V-entities among themselves are different processes:

  • for comparing O-entities among themselves we compare properties of objects, thus by means of an empirical process, such as the one performed by means of a balance and leading to the possible conclusion that the mass of rod a is less than the mass of rod b;

  • for comparing V-entities among themselves we compare numbers (assuming their unit is the same), thus by means of a mathematical process, such as the one that leads to the analytical conclusion that 1.23 kg is less than 2.34 kg.

• 5. A correspondence can be established between O-entities and V-entities:

  • O-entities can be made to correspond to V-entities through a process that can be generically called “evaluation” (see Chap. 2, and particularly Sect. 2.2.4, for discussion of this lexical choice), leading to the association of a value with the given property of an object; measurement is then a specific kind of evaluation: the mass of rod a can be evaluated as 0.12 kg;

  • V-entities can be made to correspond to O-entities through a process that can be generically called “realization”, leading to the selection or construction of an object such that one of its properties is associated with the given value: 0.12 kg can be realized by the mass of rod a.

• 6. Via these correspondences, if O-entities are evaluated, then the obtained V-entities can be compared among themselves, and this comparison conveys information about the relevant O-entities, so that the O-entities do not themselves need to be directly compared (as in stage (3) above): if the mass of rod a is evaluated as 0.12 kg and the mass of rod b is evaluated as 3.45 kg, the comparison that 0.12 kg is less than 3.45 kg leads to the inference that the mass of rod a is less than the mass of rod b where such an inference is valid if the two evaluations are valid in turn. Nevertheless, comparing O-entities among themselves and comparing V-entities among themselves remain different processes: the former is an empirical process, the latter is an analytical process.

• 7. Hence, O-entities and V-entities are at the same time

  • analogous in some respects, because they are comparable: both O-entities and V-entities can be thought of as properties, but

  • different in some other respects, because the ways in which we compare them among themselves are different: O-entities are properties identified through objects that have them, whereas V-entities are properties identified through numbers that multiply units.¹²

Figure 5.1 summarizes the relations among these entities.

Fig. 5.1

Graphical representation of the relations among object-related entities, such as the mass of some given object, and value-related entities, such x kg for some given positive number x

What follows in this and the following chapter is an exploration and an analysis of these fundamental issues.

5.1.3 Anticipating the Main Outcomes

As we have already seen, a measurement-oriented ontology and epistemology of properties is definitely a non-trivial subject. To help follow and orient the analysis that follows, we start by anticipating here some of the main conclusions.

For any given property, say mass, there are four interrelated but conceptually distinct kinds of entities that can be taken into account:

• (i) the general property (e.g., mass), M;

• (ii) individual properties (e.g., given masses), m;

• (iii) properties of given objects (e.g., the masses of given objects a), M[a];

• (iv) values of the property (e.g., x kg for any given appropriate x): 1.2345 kg.

Our basic claim is that all four of these kinds of entities are required in a sufficiently well-structured discourse on measurement.

• (i) General properties are the entities that measuring instruments are designed to measure, so that for example balances are designed to measure masses, not the mass of any given object in particular; moreover, scales (and therefore, in particular, units) are about general properties. Scientific laws, when they are invoked, pertain first to general properties, and the same holds for dimensional analysis (e.g., when stating that density has the dimension of mass times length to the minus three we refer to density, mass, and length as such, and not to any given density, any given mass, and any given length).

• (ii) Individual properties are the entities whose relations characterize the mathematical structure of the general property of which they are instances: mass is an additive quantity because the set of masses, independently of the objects that can have such masses and their relation with any possible unit of mass, has an additive structure (e.g., when stating that for any two positive masses their addition/composition/concatenation is greater than either of them we may refer to individual masses as such, and not necessarily to the masses of given objects or to some values of mass).

• (iii) Properties of given objects are the entities that are measured in any actual measurement and to which values of properties are attributed: a given balance in a given situation is an instrument for measuring the mass of a given object.

• (iv) Finally, values of properties are the entities that report the information acquired by means of calibrated measuring instruments applied to properties of objects: 1.2345 kg and 2.7216 lb are values that could be attributed to the mass of a given object.

While all this could well be taken for granted, the structure of the relations among these entities is important for a measurement-oriented ontology and epistemology of properties. Starting from the uncontroversial assumption that individual properties (ii)¹³ are instances of general properties (i) (so that, e.g., any given mass is an instance—i.e., an example, a case, …—of mass, i.e., more customarily and trivially, any given mass is a mass), we acknowledge that individual properties are entities which need to be somehow identified in order to be handled, and in particular to be compared with each other. On this basis, we develop here two basic arguments.

First, individual properties (ii) are identified as properties of given objects (iii) or as values of properties (iv), so that for example a given mass can be identified as the mass of a given object a or as x kg for a given non-negative number x, as it is explicit in the case of the Basic Evaluation Equation. In fact, properties of objects and values of properties are complementary modes of identification of instances of general properties: by identifying a mass as the mass of a given object a, the reference is to the object a that bears that mass; if a mass is instead identified as x kg for a given non-negative number x, the reference is to an element of the structure that, via the choice of a mass unit and the construction of its multiples, includes all masses.¹⁴

Figure 5.2 summarizes the relations among these kinds of entities and highlights the pivotal role of individual properties in the conceptual framework we are developing. An individual property p is an instance of a general property P and can be identified as the property P[a] of an object a or as the value p of property P.

Fig. 5.2

Graphical representation of the relations among the four kinds of entities related to properties: the generic model (top) and an example (bottom)

Second, the complementarity of the two modes of identification of individual properties is exploited in measurement. The information conveyed by a Basic Evaluation Equation

$${\text{property}}\,{\text{of}}\,{\text{a}}\,{\text{given}}\,{\text{object}} = {\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{property}}$$

is indeed that an individual property p₁, identified as a property of a given object (i.e., p₁ = P[a] for a given object a), and an individual property p₂, identified as a value of a property (i.e., p₂ = p for some value p of P), are reported to be the same individual property, p₁ = p₂ and therefore P[a] = p, as the result of the evaluation. Any Basic Evaluation Equation is then interpreted to be a mere identity from an ontic point of view, but a significant relation from an epistemic point of view.¹⁵ This reveals the fundamental meaning of the Basic Evaluation Equation:

• the property P[a] of the object a is an individual property p (the mass of any given object is a given mass);

• by means of a measurement the individual property p that was known as P[a] is identified also as a given value p of P (the mass that was known as the mass of a given object is identified also as 1.2345 kg).

These two basic arguments need to be carefully presented, explained, and justified and to this purpose the balance of this chapter and the next one are devoted.

5.2 Some Clarifications About Properties

The concept <property> has some ambiguities that we need to discuss before proceeding with our analysis.

5.2.1 Properties of Objects as Entities of the World

First, by considering properties of objects as associated with modes of interaction of the objects with their environments we acknowledge that properties of objects exist in the empirical world, and thus not only in our minds.¹⁶ A given object generally has multiple modes of interaction with its environment, and each corresponds to a property of the object. In fact, against radical operationalism, it may be discovered that the same property is the cause of distinct modes of interaction.¹⁷

According to the tripartition introduced in Sect. 2.1, properties of objects are then, at least preliminarily, claimed to be entities of the world, not conceptual entities and not linguistic entities (as discussed further in Sect. 5.3). For example, that an object floats in water is a fact that can be observed or experimentally assessed and is independent of the information that we may have on the object and its properties. That is, objects floated before Archimedes’ explanation in terms of the relation between the weight and the shape of the object and the density of the water. In other words, a property of an object can be conceptualized and given a term, but it is not itself a concept or a linguistic expression. Of course, there can be disputable observations and mistaken reports of observations: what we assume here is just that at least some interactions are uncontroversially observed, and that there must be something in the empirical world, thus independent of our conceptions, which causes those interactions.

There is an analogy in this between objects and properties (see Fig. 5.3). Concepts and linguistic terms can be associated with objects, such as rods and human beings, but rods and human beings remain something other than concepts and linguistic terms. Objects can exist without any associated concept or linguistic term (an obvious example being objects that existed before conscious beings evolved), and vice versa we can have concepts and linguistic terms of or about non-existing objects, as in the canonical cases of unicorns and phlogiston. Furthermore, while objects are subjected to empirical transformations (rods can rust, human beings grow old, etc.), concepts and linguistic terms of or about objects are unaffected by such transformations, but can be adjusted to better match objects (concepts and linguistic terms do not rust, but the concept of a rusted object is different from the concept of a polished object, etc.). Further, what can be properly defined is the concept of an object, not the object as such: objects are manipulated, designed, assembled, identified, etc., but not defined. By analogy, concepts can be provided of properties of objects, such as the length of a rod and the reading comprehension ability of an individual, but lengths of given rods and reading comprehension abilities of given individuals are not supposed to be themselves concepts. Indeed, while properties of objects can be subjected to empirical transformations (the length of a rod can change, the reading comprehension ability of an individual can improve, etc.), the concepts of properties of objects are unaffected by such transformations. Importantly, then, when expressions such as “unit definition” (e.g., throughout the SI Brochure, BIPM, 2019) and “measurand definition” are used (e.g., in the definition of <definitional uncertainty> given by the International Vocabulary of Metrology (VIM) (JCGM, 2012: 2.27), they are just shorthands for “definition of the concept of the unit” and “definition of the concept of the measurand”, or, more operationally, “definition of the mode of identification of the unit” and “definition of the mode of identification of the measurand”.

Fig. 5.3

Graphical representation of the relations between objects, properties, and their concepts

An exception to this is found in objects and properties of objects whose existence is dependent on (usually but not always shared) belief and social agreement, such as money, limited liability corporations, marriages, and beauty, which are often referred to as “social constructs” (see, e.g., Searle, 1995) to emphasize the role of human intentionality in their existence. But even in such cases, there is a distinction between the object or property and the concepts one may have of it (e.g., one may have a concept of money, separately from having money), and so the other comments given here about the distinction between objects and concepts remain applicable. In Sect. 6.6, we further discuss the existence of these kinds of properties.

A summary can be depicted as in Fig. 5.3, adapted from ISO (2009: 5.4.1, where the term “characteristic” is used to denote concepts of properties of objects).

In this context, the condition that properties of objects are associated with modes of interaction is not obvious. In Sect. 3.4.1, we mentioned the “hage” of a person, defined as the product of her height and age, and presented as an exemplary case of a supposedly non-existing property of human beings (Ellis, 1968: p. 31). In other words, this is a definition of a perfectly legitimate concept, but it might not correspond to any empirical property. However, this is not something that can be taken for granted: sooner or later, it might happen that a hage-related interaction of human beings is discovered. Again, the problem is meaningful because it relates to the claim that a property does exist as such, whereas that the concept of hage exists is only a matter of someone having ever thought about it, possibly even as a counterexample in a discourse about properties and their existence.

While the arrow in Fig. 5.3 points from property to concept of property, indicating that a concept of an empirical property must be derived from the property itself, the historical relationship may have well gone the other way; that is, one might develop a concept of a property before the conceived property is found empirically. This sort of historical sequence is far too many centuries in the past to be observed for common physical properties like length and weight, but this is not so for properties such as energy and temperature, and for many properties in the human sciences.

5.2.2 Properties and Predicates

In the philosophical tradition, and in formal logic in particular, a property is what a predicate designates¹⁸ and is therefore a Boolean entity that either applies or does not apply to a given object, or, as more commonly said, that a given object either has or does not have. The distinction between predicates (as well as characteristics) and properties is effectively depicted as in Fig. 5.4.

Fig. 5.4

Semiotic triangle (as in Fig. 2.1) applied to properties in the sense of formal logic

For example, properties in this sense are designated by the predicates “has a length”, “is longer than one metre”, and “is 1.2345 m long”: for any given object a that is the subject of these predicates, it is assumed that either it has a length or it has not, and so on. If a property in the sense of formal logic (hereafter designated P^# for maintaining a notational distinction with the properties as considered in measurement science, P) applies to an object, and therefore the corresponding predicate applied to a term that designates the object is true, we say that the object has that property: a rod a has the property of having a length, can have the property of being longer than one metre, etc. The proposition that the rod a is longer than one metre is then written as¹⁹

$$is\_longer\_than\_one\_metre\left( {rod \, a} \right) = {\text{true}}$$

whereas for example it might be that

$$is\_longer\_than\_one\_metre\left( {screw \, a^{\prime}} \right) = {\text{false}}$$

There is an ambiguity about the very concept that an object does not have a property. For example, if we consider now the object water a″ = the water in a given glass, should we simply accept that

$$is\_longer\_than\_one\_metre\left( {water\ a^{\prime \prime }} \right) = {\text{false}}$$

despite its obvious difference with the previous case? Hence the problem arises of maintaining a distinction between the case of objects that do not have a property P^# but could have it and the case of objects that even in principle cannot have a property P^#. This is based on the idea that only in the first case can P^# be experimentally assessed on the object, and thus that the second case would be better reported as

$$is\_longer\_than\_one\_metre\left( {water\ a^{\prime \prime }} \right) = {\text{undefined}}$$

We can account for this difference by restricting the application of each property P^# to a given set of objects, called the domain of P^#. Hence, the object screw a′ belongs to the domain of the property is_longer_than_one_metre and might not have the property, whereas the object water a″ does not belong to the domain of the property, because trying to assess whether some amount of water in a glass is longer than one metre is meaningless. The distinction between physical properties and psychosocial properties is then usually and first a distinction of domain: a rod has no reading comprehension ability (i.e., the domain of reading comprehension ability does not include rods), and a company has no length (i.e., the domain of length does not include companies). In summary, for each property P^# the set of objects is split into three subsets: the subset of objects that actually have P^#, the subset of objects that may have P^# but do not actually have it, and the subset of objects that cannot have P^#. The identification of the domain of a property, i.e., the union of the first two subsets, can be considered an essential component of the knowledge of that property.²⁰

5.2.3 Properties and Relations

In the philosophical tradition, and again in formal logic in particular, a distinction is also maintained between properties and relations, where the former are features of (i.e., apply to) single objects and the latter are features of two or more objects, and all are designated by predicates with either one or two or more arguments respectively. For example, ordering is a relation between pairs of objects—if a is less than a′ then order(a, a′) = true (more usually written a < a′)—and betweenness is a relation between triples of objects—if a is greater than a′ and less than a″ then between(a, a′, a″) = true (more usually written a′ < a < a″). In this sense, any physical quantity that is relative to a reference system is a relation, as for example is the case of the speed of an object, which is not a property of the object but a relation between the object and the system in reference to which speed is considered. Hence, according to this terminology, while has_a_given_length is a property that an object can have, has_a_given_speed is a relation, e.g., has_a_given_speed(rod a, reference system b).²¹

This distinction is not usually maintained in measurement science, in which the terms “property” and “quantity” are usually applied both to what would be considered properties and relations in formal logic. For a property to change it is then sufficient that one object, to which the property applies, changes.²²

We accept this terminological custom here, and—consistently with the current edition of the VIM (JCGM, 2012: 1.1)—use the term “property” for relations as well.²³ With this convention, the difference between properties in the sense of formal logic and properties in the sense of measurement science can be analyzed.

5.2.4 From Properties of Formal Logic to Properties of Measurement Science

Since a Basic Evaluation Equation such as

$$length\left[ {rod \, a} \right] = {1}.{2345}\,{\text{m}}$$

can be rewritten in the predicative form as

$$is\_{1}.{2345}\_{\text{m}}\_long\left( {rod \, a} \right) = {\text{true}}$$

one could conclude that these expressions convey exactly the same information. This is not the case, and a consideration of the differences allows us to highlight some fundamental features of properties (in the sense of measurement science, the meaning to which we implicitly refer henceforth).

Consider the three (logical) equations:

$$is\_{1}.{2345}\_{\text{m}}\_long\left( {rod\,a} \right) = {\text{true}}$$

(5.1)

$$is\_{2}.{3456}\_{\text{m}}\_long\left( {rod\,a} \right) = {\text{true}}$$

(5.2)

$$is\_{3}.{4567}\_{\text{kg}}\_heavy\left( {rod\,a} \right) = {\text{true}}$$

(5.3)

While Eqs. (5.1) and (5.3) can hold at the same time, Eqs. (5.1) and (5.2) cannot. However, the predicative form P^#(object) is unable to prevent both Eqs. (5.1) and (5.2) from being asserted as true at the same time. Indeed, consider rewriting the three predicates as P^#₁, P^#₂, and P^#₃ respectively: how could one know that, for a given x, both P^#₁(x) and P^#₃(x) can be true but that if P^#₁(x) is true then P^#₂(x) must be false?

In order to acknowledge that Eqs. (5.1) and (5.2) are incompatible, the involved properties must be recognized as having some sort of internal structure, such that the equations could be rewritten in a parametric form as

$$is\_long_{{{1}.{2345}\_{\text{m}}}} \left( {rod\,a} \right) = {\text{true}}$$

(5.1a)

$$is\_long_{{{2}.{3456}\_{\text{m}}}} \left( {rod\,a} \right) = {\text{true}}$$

(5.2a)

$$is\_heavy_{{{3}.{4567}\_{\text{kg}}}} \left( {rod\,a} \right) = {\text{true}}$$

(5.3a)

or in the relational form

$$is\_long\left( {rod\,a,{ 1}.{2345}\,{\text{m}}} \right) = {\text{true}}$$

(5.1b)

$$is\_long\left( {rod\,a,{ 2}.{3456}\,{\text{m}}} \right) = {\text{true}}$$

(5.2b)

$$is\_heavy\left( {rod\,a,{ 3}.{4567}\,{\text{kg}}} \right) = {\text{true}}$$

(5.3b)

that under the functional condition of uniqueness—for all x, P^#(x, y₁) = true and P^#(x, y₂) = true implies y₁ = y₂—corresponds to the more usual functional form

$$long\left[ {rod\,a} \right] = {1}.{2345}\,{\text{m}}$$

(5.1c)

$$long\left[ {rod\,a} \right] = {2}.{3456}\,{\text{m}}$$

(5.2c)

$$heavy\left[ {rod\,a} \right] = {3}.{4567}\,{\text{kg}}$$

(5.3c)

where “long” and “heavy” are not predicates anymore (as used in this way “long” is then different from the predicate “is long”²⁴), but examples of what Rudolf Carnap called functors (1937: p. 14; a discussion on predicates and functors in the context of measurement is in Mari, 1996). In short, once the set of possible values of the parameter x is given, one functor, “long”, which maps objects to values, corresponds to the whole set of predicates “is long_{x m}”. Hence, just as predicates are the linguistic counterparts of properties and relations in the sense of formal logic, functors are the linguistic counterparts of properties in the sense of measurement science.²⁵

With a formalization based on functors, the incompatibility of Eqs. (5.1b) and Eq. (5.2b) becomes explicit. However, this cannot be justified on the basis of the linguistic fact that the functor “long” is the same in these equations: they remain incompatible even if in Eq. (5.2b) “long” is translated into another language, e.g., into the Italian “lungo”. Such an incompatibility is an empirical fact, which calls for a justification, to be developed in the sections that follow. Interestingly, the basics of a measurement-oriented ontology and epistemology of properties can be first developed without recourse to values of properties, which will deserve a specific analysis on their own.

5.2.5 Context Dependence of Properties

Our analysis of properties and objects is grounded on the basic assumption that objects can persist in space and time even if one or more of their properties change.²⁶ In particular, by indexing properties of objects by time instant (so that, e.g., long[a, t] is the property designated by the functor “long” that the object a has at time instant t), a property-related comparability criterion is given such that, for distinguishable time instants t and t’, it may happen that

$$long[a,t] \approx long\left[ {a,t^{\prime }} \right]$$

$$heavy[a,t]{\not\approx}heavy\left[ {a,t^{\prime }} \right]$$

where ≈ denotes indistinguishability with respect to the given criterion:,^{27 28} it is a situation in which the object a has maintained its individuality from t to t’ because in particular its being long has not changed,²⁹ while, plausibly among other properties, its being heavy has changed.

A complementary basic assumption is that property-related comparability is applicable not only to the same object in different time instants, but also to different objects, in the same or different time instants: objects are comparable via the comparison of their properties. Hence, it may happen, for example, that

$$\begin{aligned}& long[a,t] \approx long\left[ {b,t} \right]~\\&\quad \left( {a\,{\text{and}}\,b\,{\text{are}}\,{\text{synchronously}}\,{\text{indistinguishable}}\,{\text{in}}\,{\text{their}}\,{\text{being}}\,{\text{long}}} \right)\end{aligned}$$

$$\begin{aligned}&long[a,t] \approx long\left[ {b,t^\prime} \right]~\\&\quad \left( {a\,{\text{and}}\,b\,{\text{are}}\,{\text{asynchronously}}\,{\text{indistinguishable}}\,{\text{in}}\,{\text{their}}\,{\text{being}}\,{\text{long}}} \right) \end{aligned}$$

but also, of course, that

$$\begin{aligned} &long[a,t] \not\approx long\left[ {b,t} \right]~\\&\quad \left( {a\,{\text{and}}\,b\,{\text{are}}\,{\text{synchronously}}\,{\text{distinguishable}}\,{\text{in}}\,{\text{their}}\,{\text{being}}\,{\text{long}}} \right) \end{aligned}$$

$$\begin{aligned}&long[a,t] \not\approx long\left[ {b,t^\prime} \right]~\\&\quad \left( {a\,{\text{and}}\,b\,{\text{are}}\,{\text{asynchronously}}\,{\text{distinguishable}}\,{\text{in}}\,{\text{their}}\,{\text{being}}\,{\text{long}}} \right) \end{aligned}$$

More generally, properties are acknowledged to be mutually related, so that the property of being long of an object may change not only with time but also, for example, with the temperature of the object, the pressure of the surrounding environment, etc. Hence, there is a context c that influences (and is influenced by) long[a]: we designate by long[a, c] the property of being long of the object a in the context c, with the understanding that time is part of the context, or possibly long[a, c, t] in order to emphasize the time instant when the property is taken into account.

The relation of experimental indistinguishability of properties of objects deserves some more analysis.

5.2.6 Indistinguishability of Properties of Objects

Let us assume that the indistinguishability of two properties of objects has been observed, as in the case

$$long[a] \approx long\left[ b \right]$$

How should such a relation be interpreted?³⁰ An aspect of the issue pertains to the fact that property comparisons relevant to measurement are empirical activities, as discussed in Sect. 2.3, and that any such activity is usually affected by factors that prevent its ideal realization. Hence, even when two properties are found to be indistinguishable, a more specific comparison might reveal that they are not equal, but only similar.

Like any generic similarity, indistinguishability is

• reflexive (any property is indistinguishable from itself: P[a] ≈ P[a]) and

• symmetric (if two properties are indistinguishable, then the order in which they are considered is immaterial: P[a_i] ≈ P[a_j] if and only if P[a_j] ≈ P[a_i]), but

• not transitive (given three properties, from the facts that the first and the second are indistinguishable and that the second and the third are indistinguishable, the conclusion that also the first and the third are indistinguishable does not follow: P[a_i] ≈ P[a_j] and P[a_j] ≈ P[a_k] do not imply that P[a_i] ≈ P[a_k]).

The non-transitivity of indistinguishability has at least one critical consequence: properties of objects could not be consistently represented, or even named, in any sufficiently simple form. Indeed, the observation that P[a_i] and P[a_j] are indistinguishable would lead one to represent them with the same symbol, and the observation that P[a_j] and P[a_k] are also indistinguishable would lead one to represent P[a_k] with the same symbol as P[a_j], and therefore as P[a_i]; but since P[a_i] and P[a_k] could be instead distinguishable, this would lead to the situation in which distinguishable properties are represented by the same symbol, a case of homonymy and therefore information loss in representation. Furthermore, since the comparison can be iterated, there might be a sequence of objects a₁, a₂, …, a_n such that P[a₁] ≈ P[a₂] and P[a₂] ≈ P[a₃] and … P[a_n–1] ≈ P[a_n] even though, for each i, P[a_i] ≉ P[a_i+2], with the consequence that all comparable but mostly distinguishable properties of objects are represented by the same symbol, and therefore that the representation conveys no information at all.³¹

This issue does not seem to have a general solution better than provisionally assuming the transitivity of indistinguishability, and therefore modeling the comparison of properties of objects as an equivalence relation, which could be then discovered to be not such by means of further and more refined comparisons. As we discuss below, providing information on the properties of objects in terms of traceable values is the specific solution to this problem adopted in measurement science (see also Mari & Sartori, 2007).

5.3 A Philosophical Interlude

The analysis developed so far about properties has been mainly technical, aimed at setting a conceptual framework for interpreting the Basic Evaluation Equation.

$${\text{property}}\,{\text{of}}\,{\text{a}}\,{\text{given}}\,{\text{object}} = {\text{value}}\,{\text{of}}\,{\text{a}}\,{\text{property}}$$

In fact, such an interpretation has an unavoidable, though possibly only implicit, philosophical background. Given a Basic Evaluation Equation such as

$$length\left[ {rod\,a} \right] = {1}.{2345}\,{\text{m}}$$

one may claim, for example, that this is just a sophisticated linguistic shorthand for reporting something about the relation between two objects, rod a and an object that in some primitive sense is attributed “to be one metre”, or possibly “to have one metre”, but that there is nothing in the world that corresponds to <being one metre> or <having one metre> , let alone the <length of rod a> and <one metre> . Accordingly, properties of objects and values of properties would be nothing more than conceptual tools created to organize our knowledge of the world.³² However, a Basic Evaluation Equation could be instead interpreted literally, as reporting a relation between entities—properties of objects and values of properties—that exist in the world, of course with respect to a given mode of existence, as discussed in Box 5.1.

The subject is so fundamental for any ontology (see references in Footnote 5) that we will not feign to provide any original contribution to this discussion: this chapter is concluded simply by discussing why we support a realist position about individual properties, while postponing to Sect. 6.6 the more complex analysis of the existence of general properties.

5.3.1 Do Individual Properties Exist?

Even if the experimental issues regarding the indistinguishability of properties of objects are somehow settled, there remains a general ontological problem. Let us suppose that the repeated application of the best available means of comparison has been unable to find any difference between the comparable properties of two objects, so that according to the available information the indistinguishability relation P[a_i] ≈ P[a_j] could be in fact considered an equality, P[a_i] = P[a_j]: how should such a relation be interpreted?

There are two, basically alternative, answers to this question (see Orilia & Swoyer, 2020).³³

• According to one position, no matter how similar P[a_i] and P[a_j] are, they are still distinct entities, as the objects a_i and a_j are distinct, because any property of an object is by definition a property that only that object has and can have. In this sense, the equality P[a_i] = P[a_j] is simply a convenient notation for an equivalence P[a_i] ≅ P[a_j]: while the objects a_i and a_j have indistinguishable modes of interaction with the environment, their properties remain different because they are of different objects. A plausibly unavoidable consequence of this position is that an object a at the time instant t_i and the same object at any other time instant t_j—as introduced in Sect. 5.2.5—cannot have the same property in turn, i.e., P[a, t_i] and P[a, t_j] must be distinct entities.

  Under this assumption, the ontology of properties is then straightforward, given that it may include only properties of objects at time instants as fundamental entities,³⁴ and it assumes their relation of indistinguishability as primitive. Thus, there are no individual properties independent of objects at time instants, and what we consider lengths, given reading comprehension abilities, and so forth are only concepts that we create to help organize our experience of and communication about the world. The price to be paid is that each property of each object at each time instant is a distinct entity, and therefore that at each new instant a bunch of properties is created, so that an ever-increasingly growing multitude of properties exists. This is an instance of a position that in the philosophical tradition is often called nominalism.

• According to another position, properties of objects are such that distinct objects can have one and the same property. Hence the indistinguishability of P[a_i] and P[a_j] suggests that they could be indeed the same property. In this sense, P[a_i] = P[a_j] is the theoretical counterpart of P[a_i] ≈ P[a_j]: the fact that the objects a_i and a_j have indistinguishable modes of interaction with the environment supports the hypothesis that they have the same property. In the lexicon of ontology, individual properties are then universals, that can be instantiated in, i.e., exemplified by, one or more objects.

  The ontology of properties in this case might be considered then more complex than in the previous case, given that it includes universal entities, such as any given length and any given reading comprehension ability, together with particulars such as the length of any given rod and the reading comprehension ability of any given individual. However, this position avoids assuming that each property of each object in each instant is a distinct entity, and accounts for the relation of indistinguishability of properties of objects in a simple way: two properties of objects are indistinguishable if they identify the same universal property, or if they identify distinct universal properties which prove to be indistinguishable according to the available empirical means. This is an instance of a position that in the philosophical tradition is often called realism.

The two positions share the common condition that properties of objects are particular entities, spatiotemporally situated. Whether things such as lengths and reading comprehension abilities are concepts, as nominalists assert, or universals that exist in an abstract world independently of the knowledge that we have of them, as realists assert, is an ontological alternative that affects the interpretation of some key components of measurement, in particular the meaning of the Basic Evaluation Equation, but cannot be decided by experimental activity.³⁵ Each position can be translated into the other one safely (and usually unproblematically, though sometimes in a cumbersome way—see the examples in Footnote 35; see also Mari & Giordani, 2012: p. 762).

We believe that most scientists, technologists, and practitioners, in both the physical and human sciences, are, perhaps unconsciously, at least moderate realists about properties, in that they consider individual properties such as lengths and reading comprehension abilities to exist in some sense (see also the discussion in Sect. 6.6). If explicitly asked, they might plausibly accept the more complex ontology in which properties are considered to be universals, given that the adoption of the more complex ontology does not affect day to day practice, and simplifies the reporting of experimental results by at least provisionally accounting for the observed indistinguishability of properties of objects in terms of their equality, such that properties of objects can be included in relations interpreted as actual equations. Furthermore, this ontology allows for a more flexible treatment of properties, in particular by admitting the possibility of formally handling properties that might currently not be instantiated by any existing object (and maybe even properties that are known not to be instantiated by any object, such as lengths greater than the diameter of the universe and masses greater than the mass of the universe).³⁶ Finally, while a realist ontology has a greater categorical complexity, it spares the nominalist requirement of an immensely great number of properties, immensely growing at each time instant with the creation of new properties. For these reasons, we maintain here the position that individual properties are universals.

5.3.2 Individual Properties as Universals: An Explanation

The idea that individual properties are universals is conceptually sophisticated: how can it be that indistinguishable properties of distinct objects may correspond in fact to the same individual property? Let us consider a mathematical relation such as ∑_i 1/(i 2ⁱ) = ln(2), where i is an integer ranging from 1 to infinity, an equation which is known to be true, given that both ∑ 1/(i 2ⁱ) = 0.693147… and ln(2) = 0.693147…. In terms of the involved numbers the relation ∑_i 1/(i 2ⁱ) = ln(2) is not different from 0.693147… = 0.693147…: but while the latter is a logical identity, which does not convey any information, the former implies some mathematical knowledge, so that in some respect the two entities, ∑_i 1/(i 2ⁱ) and ln(2), must be different. However, there is also a respect in which the equality actually holds, so that we can say that ∑_i 1/(i 2ⁱ) = ln(2) is true, while, for example, ∑_i 1/(i 2ⁱ) = 2 is false. This double interpretation—different but equal at the same time—accounts for the principled difference between ∑_i 1/(i 2ⁱ) = ln(2) and 0.693147… = 0.693147…, which can be explained in terms of the distinction between the sense and the reference of an expression (as noted in Box 5.1), where the sense of an expression is the concept it designates and the reference of an expression is the entity it refers to.³⁷

Hence “∑_i 1/(i 2ⁱ)” and “ln(2)”.

• are different expressions (the former starts with a symbol of summation, the latter with the name of a function, and so on),

• with different senses, since they designate different concepts (the former is a series, the latter is a function evaluated in a given argument),

• but with the same referent, since they refer to the same mathematical object (the number 0.693147…).

This is a possible interpretation of the relation P[a_i] ≈ P[a_j], and indeed the one we adopt here: when claiming, e.g., that the length of a_i is indistinguishable from (or 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 the 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 both a_i and a_j have and therefore both P[a_i] and P[a_j] instantiate.

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: p. 113–116). Accordingly, the natural number n is 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 about properties considers them 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, in analogy to the fact that if the sets A and B have the same elements then they are one and the same set. 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 ontology of modern science [comprises] material objects (or, alternatively, space–time points), sets of material objects, sets of sets of material objects, …, but no properties” (1990: 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 William 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 general and less 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 photon energy and photon 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 to the extensionalist position was provided by Quine (1951): according to the current knowledge, “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 conveys information about the relation between entities which have their own modes of existence in the world. On this basis, in the following chapter we develop the position that individual properties are universals by framing it in a metrological framework, in which values of properties also play a significant role, and then broaden the picture in order to consider the very problem of the existence of general properties.