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 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
of which the relation
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.
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Notes
- 1.
This does not preclude the possibility that in some cases distinct modes of interaction do not correspond to distinct properties, i.e., one property may correspond to multiple modes of interaction. A well-known example in physics is mass, which is associated with both inertial and gravitational phenomena, and therefore to distinct modes of interaction. This is consistent with Brian Ellis’ insight that “our quantity concepts are generally cluster concepts” (1968: p. 32). More generally, the interaction of an object with each different instrument might be interpreted as a different mode of interaction: from an operationalist perspective, the interaction with each instrument corresponds, by definition, to a different property (see Sect. 4.2.2).
- 2.
The plausibility of these conditions is confirmed also in a philosophical context. For example, to the question What is a property? Baron et al. (2013: p. 35) answer: “This is a thorny issue. For present purposes we conceive of properties as the entities that ground causal powers and similarity relations.”
- 3.
That “the familiar objects of the everyday world agree in their characteristics, features, or attributes” is considered “a prephilosophical truism” (Loux & Crisp, 2017: p. 18).
- 4.
For an analysis on this distinction, see for example Heil (2003: ch. 8), who also presents the idea that properties manifest themselves through empirical interactions of objects: “all there really is to a concrete entity is its power to affect and be affected by other entities. Assuming that an entity’s powers depend on its properties, this suggests that there is no more to a property than powers or dispositionalities it confers on its possessors. […] The business of science is to tease out fundamental properties of objects. Properties are what figure in laws of nature, and laws govern the behaviour of objects. Properties, then, are features of the world that make a difference in how objects behave or would behave” (p. 75).
- 5.
The subject of what properties are is complex, and out of the scope of this book, in which a (black box) characterization of how properties manifest themselves is sufficient. For a philosophically oriented introduction to this subject, see for example the articles by Orilia and Swoyer (2020), Weatherson and Marshall (2018), and Wilson (2017) in the Stanford Encyclopedia of Philosophy. Concepts such as <property>, <attribute>, <feature>, and <characteristic> are so fundamental in human knowledge that it is not clear how they could be defined without circularity. For example, René Dybkaer (2004: p. 51) defines <property> as an “inherent state- or process-descriptive feature of a system including any pertinent components”, while leaving <feature> as a primitive (i.e., undefined) concept. Were he requested to define <feature>, he plausibly might have included the term “property” in the definition, thus showing that the concept is ultimately defined in terms of itself. Not surprisingly, these concepts are also sometimes used in a somewhat confusing way in pivotal texts of measurement science, for example Foundations of Measurement (Krantz et al., 1971: p. 1) which states in its opening sentence: “when measuring some attribute of a class of objects or events, we associate numbers (or other familiar mathematical entities, such as vectors) with the objects in such a way that the properties of the attribute are faithfully represented as numerical properties”. The idea that an attribute has properties represented as properties is not exactly obvious, to say the least.
- 6.
In fact, the first phrase, while seemingly better reporting what empirically happens (“in the comparison we handle objects, right?”, as a colleague of ours told us), assumes a greater ontic burden, given that the entity with respect to which objects are compared is a kind of property, whereas kinds of properties do not explicitly appear in the second phrase. Furthermore, it is surely possible that properties of the same object are compared, for example the height and the width of a rigid body: in terms of comparison of objects this would require a cumbersome phrasing like “an object is compared with itself with respect to a (kind of) property”.
- 7.
Measurands are “quantities intended to be measured” (as defined in JCGM, 2012: 2.3): hence, while measurands are properties of objects, a property of an object becomes a measurand for us only when we are interested in obtaining a value for it via a measurement. Several aspects of our analysis apply to the generic case, thus for example also to Basic Evaluation Equations which describe specifications, instead of reporting results of measurements.
- 8.
As noted in Sect. 2.2.3, the notation P[a] is aimed at highlighting that P can be formalized as a function but it is not a mathematical entity as such.
- 9.
As inspired by the seminal work of James Clerk Maxwell (1873), this relation is commonly written as
$$ Q=\left\{Q\right\}\;\left[Q\right] $$which we call “Q-notation” for short. Despite its success (see, e.g., de Boer, 1995: p. 405 and Emerson, 2008: p. 134, but also JCGM, 2012: 1.20 Note 2 and ISO, 2009b), this notation is not completely clear, as it does not maintain the distinction between general quantities and individual quantities. By writing the left-hand-side entity as Q[a] we make explicit the reference to the quantity Q of the object a. This also highlights that, while the unit is a feature of the general quantity Q, and therefore writing it as [Q] is correct (recalling that brackets in “Q[a]” and “[Q]” have different meanings: “Q[a]” stands for the Q of a, e.g., the length of a given rod; “[Q]” stands for the unit of Q), the numerical value depends on both the quantity of the object and the chosen unit. A more complete form of the expression is then {Q[a]}[Q], to be read “the numerical value of Q[a] in the unit [Q]” (see also ISO, 2009b: 6.1).
- 10.
Interestingly, the form of the Basic Evaluation Equation, in which the property of the object and the value of the property are related by an “=” sign, suggests the interpretation that the property is the value: we discuss this delicate point in Sect. 6.4.
- 11.
The history of the concept of true value is complex, and definitely still not settled (see also Sects. 3.2.2 and 4.2.1). While sometimes considered to be a useless metaphysical residual, in some contexts the reference to true values is maintained and emphasized. As an example, the current version of the NIST Quality Manual for Measurement Services (the National Institute of Standards and Technology, NIST, is the US National Metrology Institute) defines <measurement> as (emphasis added) “an experimental or computational process that, by comparison with a standard, produces an estimate of the true value of a property of a material or virtual object or collection of objects, or of a process, event, or series of events, together with an evaluation of the uncertainty associated with that estimate, and intended for use in support of decision-making” (11th version, 2019, www.nist.gov/system/files/documents/2019/04/09/nist_qm-i-v11_controlled_and_signed.pdf, including the note that “the NIST Measurement Services Council approved [this] definition of measurement to include value assignments of properties using qualitative techniques”) (Possolo, 2015: p. 12).
- 12.
We have not been able to find a term to designate the entities obtained by multiplying or dividing a unit by a number which is not necessarily integer (a multiple of a unit is a “measurement unit obtained by multiplying a given measurement unit by an integer greater than one” (JCGM, 2012: 1.17), and a submultiple of a unit is a “measurement unit obtained by dividing a given measurement unit by an integer greater than one” (JCGM, 2012: 1.18, emphasis added)). Hence we maintain the term “multiple” with this broader meaning: if u is a unit and x is a nonnegative real number, x u is a multiple of u.
- 13.
The label “(ii)” refers to the four assertions immediately above, and similarly for the labels below.
- 14.
It is not controversial that individual properties are instances of general properties, but there is a delicate ontological issue about how properties of objects and values of properties are related to individual properties (see also the discussion in Sect. 5.3.1). With the aim of remaining as independent as possible of ontological presuppositions, we only assume that properties of objects and values of properties identify individual properties.
- 15.
- 16.
The concept <property> is so general that this condition needs to be specified. For example, one could consider that number 2 has the property of being even and of having 4 as its square: considering these as modes of interaction would be peculiar, at least because the very idea that numbers interact with something empirical is peculiar in turn. Hence our analysis actually relates to empirical properties and empirical modes of interaction. We use the adjective “empirical” for referring to a feature of something in opposition to the possibility that that something is purely conceptual, informational, or linguistic (see also Sect. 2.2.1).
- 17.
According to Abraham Kaplan, “We do not first identify some magnitude, then go about devising some way to measure it. As operationists have long insisted, what is measured and how we measure it are determined jointly. Operationists may have given undue emphasis to the ‘how’ as against the ‘what’, but this emphasis is a healthy corrective to the naive idea that magnitudes can be conceived quite independently of procedures for determining their measure in particular cases” (1964: p. 177). Hence, we endorse such a “naive [!] idea”, and consider then that individual quantities, and individual properties more generally, are conceived independently of procedures for measuring them.
- 18.
There are many excellent books that can be used as reference on formal logic. The textbook by Hodges (1977), for example, is interesting for its explicit emphasis on the relations between natural languages and logic and the absence of required mathematical pre-competences.
- 19.
The relations P#(a) = true and P#(a) = false are usually written as P#(a) and not(P#(a)) for short, respectively. In what follows we use the same symbols and expressions to denote properties and the corresponding predicates. This notational choice, of using the same symbol for a property and the mathematical entity that models the property, is usual—for example, the Guide to the expression of uncertainty in measurement (GUM) adopts it with this justification: “For economy of notation, in this Guide the same symbol is used for the physical quantity (the measurand) and for the random variable that represents the possible outcome of an observation of that quantity” (JCGM, 2008: 4.1.1, Note 1). Nevertheless, it is a possible source of confusion: even though properties are not notationally differentiated from their concepts and expressions, as previously noted, properties are not concepts and are not expressions.
- 20.
If the position is assumed that every predicate designates a property (in the sense of formal logic), things can become tricky. Consider, e.g., the predicate “is a length”: if is_a_length(a) = true, then it is because a is a property, and it is in fact a length. The domain of is_a_length is then a set of properties—so that is_a_length(a given rod) is undefined, not false—and is_a_length is a higher order property, i.e., a property of properties. It is doubtful that such kinds of properties can be assessed via empirical interactions (on the other hand, is_a_quantity is an example of a second-order property, which instead admits an empirical validation—see the related discussion in Sect. 6.3.2).
- 21.
This is what the Galilean relativity principle asserts. According to Einstein’s relativity theory, the length of a body observed from a frame of reference in motion with respect to the body depends on the relative velocity of the two systems: in this view length is also a relation. Moreover, even in classical physics is_1.2345_m_long(a) could be reinterpreted as the relation is_1.2345-fold_long(a, s) between the object a and any measurement standard s which materializes the metre: from this perspective, all ratio properties treated in measurement are relations.
- 22.
This avoids the need of specifically dealing with the so-called Cambridge changes, “such as when I change from having ‘non-brother’ true of me to having ‘brother’ true of me, just when my mother gives birth to a second son”, the problem being of course that “it might seem faintly paradoxical that there need be no (other) changes in me (height, weight, colouring, memories, character, thoughts) in this circumstance” (Mortensen, 2020: ch. 2). Consider an example closer to our context, like
$$ is\_ at\_a\_ distance\_ of\_1.2345\_\mathrm{m}\_ from\left(\mathrm{body}\;a,\mathrm{reference}\;b\right)=\mathrm{true} $$If instead we assumed that the property is
$$ is\_ at\_a\_ distance\_ of\_1.2345\_\mathrm{m}\_ from\_ reference\_b\left(\mathrm{body}\;a\right)=\mathrm{true} $$then changes of the position of b would need to be considered also as changes of a property of a, even though a itself did not move.
- 23.
Sometimes the term “attribute” is used to encompass properties and relations. This was the choice of the second edition of the VIM, which defines <(measurable) quantity> as an “attribute of a phenomenon, body or substance that may be distinguished qualitatively and determined quantitatively” (ISO et al., 1993: 1.1).
- 24.
Admittedly, a form such as “long[rod a]” is clearly awkward, and is introduced here only as an intermediate step from properties in the sense of logic, e.g., is long[rod a], to properties in the sense of measurement science, e.g., length(rod a).
- 25.
There is in fact another functional form for conveying the information brought by a Basic Evaluation Equation:
$$ \mathrm{R}{1}_{\mathrm{d}}: long\_ in\_\mathrm{m}\left[ rod\;a\right]=1.2345 $$$$ \mathrm{R}{2}_{\mathrm{d}}: long\_ in\_\mathrm{m}\left[ rod\;a\right]=2.3456 $$$$ \mathrm{R}{3}_{\mathrm{d}}: heavy\_ in\_\mathrm{kg}\left[ rod\;a\right]=3.4567 $$We further discuss it in particular in Sect. 6.2.2, in the context of the analysis of the way representational theories of measurement deal with values.
- 26.
For example, on the one hand, at least in the broadly Western tradition, each of us admits our own persistence in time as individuals even though we change, say, our height and weight and, less trivially, our cognitive abilities. On the other hand, an object can change to another one if one or more of its properties change, as in the case of an informous amount of clay that is modeled and finally becomes a jar. An extreme case of the dilemma of the conditions of object persistence is known since the classical world as the Theseus paradox (see Korman, 2016: 2.4): Does a ship remain the same even if, one by one, all its wooden boards are substituted? And therefore, is an object characterized by the matter of which it is constituted, or by its shape? The assumption of some basic persistence is intrinsic to our concept of object. Even just imagining how to avoid it is challenging. In one of his tales, “Funes el memorioso” (“Funes the Memorious”), Jorge Luis Borges (1944) tried imagining what it would be like to avoid it, by telling of an individual of prodigious memory: “In the seventeenth century, Locke postulated (and condemned) an impossible language in which each individual thing—every stone, every bird, every branch—would have its own name; Funes once contemplated a similar language, but discarded the idea as too general, too ambiguous. The truth was, Funes remembered not only every leaf of every tree in every patch of forest, but every time he had perceived or imagined that leaf.” Compare this with: “All is impermanent, because all is in a state of perpetual change. A thing does not remain the same during two consecutive ksanas (the ksana being the shortest period of time in Buddhism). It is because things transform themselves ceaselessly that they cannot maintain their identity, even during two consecutive ksanas” (Nhat Hanh, 1974: p. 35).
- 27.
By acknowledging that time instant is also a property (of the reference system shared by the considered objects), the condition that t and t′ are distinguishable time instants should be written as t ≉ t′, not t ≠ t′, thus emphasizing that two time instants could be indistinguishable (because, e.g., they are within 1 ns of one another and the quality of the available instrumentation is not able to detect this difference) even though they are not necessarily exactly the same.
- 28.
A relation such that object a at time instant t and object a′ at time instant t′ are indistinguishable in their being long may be differently interpreted. If objects are considered to be entities with time instances, then the relation could be written as long[a(t)] ≈ long[a′(t′)] (see Mortensen, 2020: ch. 5). By focusing even more explicitly on objects and considering to be long as a feature of the way objects are compared, the relation could be written instead as a(t) ≈ long[a′(t′)]. The focus in measurement science on properties—what is measured is the property of an object, not an object per se—justifies our choice of adopting the form long[a] ≈ long[a′], and possibly its time explicit version long[a, t] ≈ long[a′, t′] whenever appropriate. The alternative position of focusing on objects is taken in particular in the representational theories of measurement, which usually develop their formalization on empirical relations among objects; see, e.g., how the concept <relational structure> is introduced in Krantz et al. (1971: p. 8).
- 29.
“Its being long has not changed” is awkward phrasing, and could be changed to the more usual “its length has not changed”. We maintain it at the moment, given that the relation of the adjective “long” with the corresponding noun “length” is discussed afterwards.
- 30.
In what follows, it is immaterial whether the comparison is synchronous or asynchronous, and whether it depends on the context: therefore the reference to time and context is usually omitted.
- 31.
This is the paradox known as sorites, a term which derives from the Greek word “soros”, meaning <heap> (see Hyde & Raffman, 2018). The classical way to present it is in terms of logical properties, for example as follows. Let an be a set of n grains of wheat. Of course a0 is not a heap, i.e., is_heap(a0) = false. Moreover, an and an+1 are indistinguishable in their being heaps, in the sense that if a set is not a heap, adding a grain to it does not make it a heap, i.e., if is_heap(an) = false then is_heap(an+1) = false. Then starting from the first clause and by the repeated application of the second clause the conclusion is reached that is_heap(an) = false no matter how large n is.
- 32.
Depending on how radical (or consistent) this position is, one could say the same about objects: the world is in principle an undifferentiated blob, and objects are only conceptual constructions we provide for understanding the world.
- 33.
Note that this alternative is presented with no reference to values of properties, which are instead discussed in Chap. 6.
- 34.
With or possibly even without objects: in an extreme position, objects could be just considered as the bundles of their properties—see Maurin (2018) (and since properties change as time passes, this also means that objects have no persistence: I am not the same person as I was one second ago, but we are only equivalent, according to a more or less complex criterion of equivalence; this might be an interpretation of the radical impermanence mentioned in Footnote 26 about Buddhism).
- 35.
The alternative between realism and nominalism relates not only to properties, as realism and nominalism clash about the existence of universals as such. Take these two examples: “the steam engine was invented at the end of the seventeenth century” and “the tiger is an endangered species”. For a nominalist they cannot be literally true, because “the steam engine” and “the tiger” do not refer to anything in the world, but only to concepts we adopt to organize our knowledge. He or she would explain the meaning of the two sentences by considering them to be shorthands for something like “the first object that we presently conceptualize as <steam engine> was invented at the end of seventeenth century” and “the current number of objects that we conceptualize as <tiger> is less than a given threshold”, or even more explicitly “the extension of <steam engine> was empty before the end of seventeenth century” and “the cardinality of the current extension of <tiger> is less than a given threshold”. This reduction strategy may become cumbersome. For example, in the case of “Shakespeare’s works include 39 plays”, the nominalist would claim that the quantification is on concepts, so that, e.g., A Midsummer Night’s Dream does not exist as such but it is only a concept by means of which we identify a subset of the objects (paper volumes, digital files, theater performances, etc.) with the property of having Shakespeare as their author. Assessing the truth of the sentence would then require first of all assessing an equivalence criterion between such disparate entities, and then counting the number of the so obtained equivalence classes.
- 36.
There is one more reason supporting realism about properties, related to the status of values of properties, a subject that we explore in Chap. 6. Just as a mention here, though obtained through the conventional definition of a unit, an entity such as 1.2345 m seems to have an existence independent of the knowledge that we have of it. In other words, values are not concepts.
- 37.
An analogous distinction is put between the intension and the extension of a concept (see Sect. 2.1). Hence, the intensions of the concepts <∑ 1/(i 2i)> and <ln(2)> are different, while their extension is the same, i.e., the number 0.693147 …. Note that intensions and extensions are sometimes attributed to terms too; see, e.g., Chalmers, 2002.
- 38.
This is about individual properties. There is also an extensionalist interpretation of general properties. 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).
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Mari, L., Wilson, M., Maul, A. (2021). What is measured?. In: Measurement across the Sciences. Springer Series in Measurement Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-65558-7_5
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