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Notes
- 1.
The language issue has deserved great attention in measurement in the last thirty years. In 1984 the International Vocabulary of basic and general terms in metrology (VIM) was published as the result of a joint effort of authoritative international scientific and technical organisations [4], which has now come to the third edition, with substantial revisions [5, 6]. Yet the proposed terminology may not be general enough for accomplish the vision of measurement in behavioural sciences [7]. For this reasons, we will sometimes depart to the VIM terminology in this book [8]. For the reader’s convenience a short collection of key terms is presented in the appendix at the end of the book.
- 2.
When modelling the measurement process, as we will do especially in Chap. 5, the terms “quantity” and measurand are often quite interchangeable. The difference concerns whether we think at the model as applied in a specific situation or intended to represent a large class of possible situations. Often both interpretation are possible and feasible.
- 3.
The distinction among these terms is not purely linguistic, yet we prefer not to dwell into this at this stage.
- 4.
In Helmholtz’s language, concrete numbers are those arising from the counting of real objects.
- 5.
Note that in this discussion we use the term “property” in three meanings: either to denote the property we want to measure or the empirical relations and operations that characterise it or even the formal properties of such empirical relations and operations. So, for example, length is a measurable property, characterised by the empirical properties of order and addition, and order, e.g., is characterised by the transitivity (formal) property. This is a drawback of using the term “property” for denoting what we want to measure, as in fact is recommendend by current international guidelines [6], that perhaps have not considered enough such theoretical aspects. This is why in other papers we have used the term “characteristic” instead [1]. Here we have preferred to follow the international recommendation, hoping that the discourse is still clear enough, although with some effort, to the reader.
- 6.
This notation is subsequent to Helmoltz and is the one we will use routinely throughout the book. The general notation principles adopted in this book and a list of some of the main symbols, grouped by subjects, are reported in the appendix at the end of the book.
- 7.
In reality, measurability under not ordered structure is also of interest, as we will see in the following of this chapter.
- 8.
Remember that we have assumed, in this example, that all the objects, including their combinations, can be put on each pan of the balance. Note also from this, the importance of properly defining the class of objects under consideration. In the notion of scale it is implicitly assumed that for each object there is an element in the scale which is equivalent to it. An element of a scale is called a standard.
- 9.
For mercury, around room temperature \(\alpha =0.00018\,^{\circ }\mathrm C ^{-1}\) [12]. In practical application, the thermal expansion of the thermometer glass should also be accounted for, but such details are ineessential here.
- 10.
Although magnitude estimation can be considered a measurement method, the other three are rather scaling procedures. The difference is subtle, often overlooked, but substantial. In a scaling method, the aim is to assigning numbers to a set of objects, in order to constitute a scale for the quantity of interest. Instead in measurement the goal is to assign numbers to one, or more objects, in respect of a previously established scale. In the case of magnitude estimation, the previously established scale is assumed to be some inner scale in the subject(s). In ratio estimation or production it is not necessary to dispose of such a scale, but only to be able to perceive ratios. This difference will be easier to understand after reading Chaps. 3–5 of this book. We will consider ratio estimation or production at a later stage in this section, when dealing with the classification of measurement scales.
- 11.
In fact this term was introduced later on—Stevens rather spoke of “mathematical group structure”—but we prefer it since it is perhaps easier to understand.
- 12.
This paved the way to the representational theory of measurement, as we will see in a moment.
- 13.
Thus, continuing the discussion in Footnote 11, whilst magnitude estimation may be regarded as a measurement method, ratio estimation and production are rather scaling procedures, which allow obtaining a ratio scale even when there is no empirical addition operation, as usually happens with perceptual quantities. We will discuss in dept this important and conceptually difficult point in Chap. 3.
- 14.
- 15.
- 16.
Note how the idea of interpreting signs produced by instruments, suggest by Rossi, comes into play.
- 17.
We will discuss loudness measurement in some detail in Chap. 8.
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Rossi, G.B. (2014). Measurability. In: Measurement and Probability. Springer Series in Measurement Science and Technology. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8825-0_1
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