Abstract
This chapter aims to conclude the book by providing a high-level interpretation of measurement and its characterizing features. We first develop a semiotic perspective on the information gained via measurement, specifically, how, in any measurement process, syntactic information (i.e., data, in the form of indication values) grounds semantic information (in the form of measurement results), which in turn grounds pragmatic information (in the form of measurement results together with the contextual information that enables decision-making). We then briefly retread the path followed in this book, describing how we began with a minimal set of necessary conditions for measurement and then progressively explored issues critical to the development of complementary sufficient conditions, related in particular to the ontology and epistemology of measured properties; the nature of scales, measurability, and measured values; and the roles of empirical and informational processes in measurement. This culminates in a general model of a measurement process that emphasizes the importance of evaluating the quality of the information produced by measurement in terms of object relatedness (“objectivity”) and subject independence (“intersubjectivity”). We conclude with an argument that, despite differences in subject matter and application, any measurement process can be characterized as an empirical and informational process that is designed on purpose, whose input is an empirical property of an object, and that produces explicitly justifiable information in the form of values of that property.
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Notes
- 1.
This analogy between communication and measurement should be read while keeping attention also to their substantial differences, as highlighted in Sect. 4.2.1.
- 2.
The relation sign stands for entity is very general. Famously, Charles Sanders Peirce identified three ways in which it can be realized. “If we come to interpret a sign as standing for its object in virtue of some shared quality, then the sign is an icon. Peirce’s early examples of icons are portraits […]. If […] our interpretation comes in virtue of some brute, existential fact, causal connections say, then the sign is an index. Early examples include the weathercock, and the relationship between the murderer and his victim […]. And finally, if we generate an interpretant in virtue of some observed general or conventional connection between sign and object, then the sign is a symbol. Early examples include the words ‘homme’ and ‘man’ sharing a reference” (Atkin, 2013; emphasis added). In this semiotic perspective, indication values (i.e., local values) can be interpreted as indexes of measurands, and measured values (i.e., public values) as icons of measurands.
- 3.
- 4.
In the last few decades semiotics has had important developments (see, e.g., the overview by Wolf, n.d.) but—as with philosophy of language in Chap. 5—a reference to some key founders and some of their main themes is sufficient for our purposes here. Its context is provided by the “semiotic triangle” that we introduced in Box 2.1.
- 5.
Since data can be nonlinguistic, a further layer may be introduced for distinguishing between data and its representation. Depending on the context, smoke might be represented by the English word “smoke”, the Italian word “fumo”, etc. The fact that data, if it is not a linguistic entity, needs to be represented in order to be manipulated, communicated, etc. has nothing to do with the relations among syntactic, semantic, and pragmatic information. In what follows we do not deal with the problem of representation of data (an example is about the decimal representation of non-integer numbers: should the number 3/2 be represented as “1.5” or “1,5”?). The reader interested in notational issues may refer to the SI Brochure, Section 5, “Writing unit symbols and names, and expressing the values of quantities” (BIPM, 2019).
- 6.
For any given set X = {xi} equipped with a probability distribution such that p(xi) is the probability of selection of xi (so that ∑ p(xi) = 1), the quantity of information—which should then be more specifically called “quantity of syntactic information” or “quantity of data”—conveyed by the selection of xi is –log(p(xi)). Accordingly, Shannon’s entropy, −∑ p(xi) log(p(xi)), can be interpreted as the average “amount of freedom” in the selection of elements from X. The maximum freedom is when the probability distribution is uniform, and therefore it does not add any constraints to the definition of the set; as mentioned above, the minimum freedom—zero entropy, no freedom at all—is when one element is certain, and therefore all other ones are impossible. From the semiotic perspective we are discussing, this is a purely syntactic characterization: only data is involved, with no references to the property under measurement as its possible meaning.
- 7.
The foundational work made some decades ago for establishing a quantitative basis of (semantic) information—sometimes presented in terms of amount of content—did not lead to anything comparable to what Shannon’s entropy constitutes for the quantitative evaluation of the amount of (syntactic) data (see, e.g., the extensive analysis by Hintikka, 1970). From this perspective it is unfortunate that the basic mathematical entity of Shannon’s theory, −log(p(xi)), has been called “quantity of information” instead of “quantity of data”. The usual remedy is to specify “quantity of syntactic information”, or “quantity of statistical information”, or also “quantity of technical information” in the lexicon adopted by Weaver, as mentioned above.
- 8.
And from previous papers of ours, in which we have discussed the very definition of <measurement> (e.g., Mari, 2013), and its epistemology (e.g., Mari, 2003), the stereotypes that surround measurement (e.g., Mari, Carbone, Giordani, & Petri, 2017), and in particular the mistaken assumption that measurement is identical to quantification (Mari, Maul, Torres Irribarra, & Wilson, 2017).
- 9.
“According to this account, the three conditions—truth, belief, and justification—are individually necessary and jointly sufficient for knowledge of facts” (Steup & Ram, 2020: 2.3).
- 10.
A clear and simple example of this fundamental characterization of science is given, by difference, by Daniel Dennett: “There are many strategies, some good, some bad. Here is a strategy, for instance, for predicting the future behavior of a person: determine the date and hour of the person’s birth and then feed this modest datum into one or another astrological algorithm for generating predictions of the person’s prospects. This strategy is deplorably popular. Its popularity is deplorable only because we have such good reasons for believing that it does not work. When astrological predictions come true this is sheer luck, or the result of such vagueness or ambiguity in the prophecy that almost any eventuality can be construed to confirm it. But suppose the astrological strategy did in fact work well on some people. We could call those people astrological systems—systems whose behavior was, as a matter of fact, predictable by the astrological strategy. If there were such people, such astrological systems, we would be more interested than most of us in fact are in how the astrological strategy works—that is, we would be interested in the rules, principles, or methods of astrology. We could find out how the strategy works by asking astrologers, reading their books, and observing them in action. But we would also be curious about why it worked. We might find that astrologers had no useful opinions about this latter question—they either had no theory of why it worked or their theories were pure hokum. Having a good strategy is one thing; knowing why it works is another” (1987: p.16). We claim exactly the same of measurement: that its results work (in some sense) is not enough; we want to know why they work. And this requires “opening the box” of the process and examining its structure and functioning.
- 11.
A more specific condition is then that measurement results need to be reproducible by all relevant social stakeholders. Even neglecting the practical constraints related to the fact that setting up a measurement system may have costs which are not affordable for all interested parties, we must acknowledge that some measurements are not repeatable, for example when they alter the state of the object under measurement in an irreversible way (see, e.g., destructive testing, www.electropedia.org/iev/iev.nsf/display?openform&ievref=151-16-29). Hence reproducibility cannot be taken as a characterizing condition.
- 12.
We are referring here to the epistemic justification of measurement results, and therefore to the principled possibility of interpreting the information produced by measurement in a social context where it becomes shared knowledge. Higher level forms of justification are not only possible but usually also desirable for measurement, and in particular pragmatic justification, aimed at showing that the measurement results deserve the resources used for obtaining them.
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Mari, L., Wilson, M., Maul, A. (2021). Conclusion. In: Measurement across the Sciences. Springer Series in Measurement Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-65558-7_8
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