Abstract
Suppose that we want to compare the working environments of two offices. We would probably measure the width of the working space allocated to each person, and the temperature, humidity and noise level, in typical working conditions.
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Notes
- 1.
The meaningfulness of statement concerning measurement on a given scale depends upon the uniqueness conditions for that scale, as briefly mentioned in Sect. 1.7 and as will be discussed in detail, in the following of this chapter.
- 2.
The resolution, for a reference scale, is the difference between two adjacent standards. For a measuring system, it is the capability of detecting and measuring small differences between different objects in respect of the quantity under consideration. In both cases, such differences can be very small but they must be finite. The infinitely small, as well the infinitely large, are not attainable in actual experimentation.
- 3.
Then, for any measurement problem, we can think at it as belonging to a class of similar problems, for which an appropriate finite reference scale can be established.
- 4.
The attentive reader could object that when we consider a finite structure, we mean that the total number of measurable objects is finite. So, for example, if we have, for each element of the reference scale, \(m\) objects that have the same length of it, the total number of object is \(N=nm\). Thus, our assumption actually implies that \(N\) is finite. But, after some reflection, it is easy to get persuaded that what really matters is the number \(n\) of elements of the reference scale, which is also the number of distinct ways (or states) in which the quantity manifests itself. We will see in the following of the chapter that this is the number of the equivalence classes for the quantity. It is indeed immaterial how many objects, equivalent to each elements of the scale, are there. So, we can safely assume that they are in a finite number, and even that that number is the same, \(m\), in all cases.
- 5.
In fact, \(a\succ b\) means “\(a\) scratches \(b\)”, \(a\sim b\) means “\(a\) neither scratches nor is scratched by \(b\)”, then \(a\succcurlyeq b\) means “\(a\) is not scratched by \(b\)”.
- 6.
In practice, we will consider binary, ternary or quaternary relations.
- 7.
I do not want to comment here this, somewhat questionable, definition, but just note that ordinal quantities have been defined and consequently accepted, in this environment.
- 8.
The cardinality of a (finite) set is the number of its elements.
- 9.
We could introduce a special symbol for this relation in \(S\), but we avoid to do that, since we think that a proliferation of symbols is rather confusing than clarifying. In fact, an alternative way of formulating the representation theorem is to consider an isomorphism between \(A^{*}\) and \(I\), as Narens, e.g., does, [5], rather than an homomorphism between \(A\) and \(I\), as we, and many others [3, 6], do.
- 10.
The symbol \(\Box \) denotes the end of a proof.
- 11.
The notion of interval is one of the most fundamentals in measurement. This is why we dwell somewhat on it. The concept of interval, together with that—at another level of discourse—of probability, constitutes the pillars on which most of the theory presented in this book is constructed.
- 12.
Note the origin is an arbitrary point since the line extends to infinity both to the left and to the right, and there is no special point on it that merits to be origin. Different will be the case for ratio scales, to be treated at a later stage.
- 13.
Note that if we fix the origin in any other point, the procedures still works.
- 14.
The term “solvability” suggests that, for \(\varDelta _{bc}\succ \varDelta _{ab}\), the equations
-
\(\varDelta _{bd^{\prime \prime }}\sim \varDelta _{ab}\) and
-
\(\varDelta _{d^{\prime }c}\sim \varDelta _{ab}\),
where \(d^{\prime }\) and \(d^{\prime \prime }\) are the unknowns, always have a solution.
-
- 15.
In fact, the term “monotonicity” suggests that adding equivalent intervals to two intervals does not change the order that exists between them. A monotonic transformation, in general, is one that does not alter order.
- 16.
Remember the discussion about the sign of empirical differences, at the end of the previous section.
- 17.
Proofs in this section, as well as in following of this chapter, are somewhat technical and can be omitted in a first reading, without substantial loss of continuity.
- 18.
The induction principle is a well-established argument in mathematics [7]. It states that if \(\phi \) is some proposition predicated of \(i\) where \(i\) is a natural number, and if:
-
\(\phi (0)\) holds true and
-
whenever \(\phi (j)\) is true, then \(\phi (j+1)\) is also true
then \(\phi (i)\) hold true for all \(i\). Bi-argumental induction is the same as simple induction, but concerns statements predicated for a couple of argument, \(\phi (i,k)\).
-
- 19.
Remember that a generic function \(f{:}\,X\rightarrow Y\), where \(X\) and \(Y\) are finite sets, can be defined by listing, in a set, all the pairs \((x,y)\) that satisfy it, that is, \(f=\{(x,y)|x\in X, y\in Y, y=f(x)\}\).
- 20.
The term “octave” (Latin: octavus, eighth) comes from musical acoustics and denotes the interval between two sounds whose pitch is one the double of the other. This happens when the second sound is the eighth in a musical (heptatonic) scale starting from the first.
- 21.
Remember the way we have defined positive differences in Sect. 3.4.1.
- 22.
On finiteness, remember discussion in Sect. 3.1.
- 23.
Remember Sect. 1.2.
- 24.
Instead of actually placing them on the reference line, we have drawn them parallel to it, to make the figure more readable.
- 25.
Remember that the symbol “\(\circ \)” denotes addition of entities other than numbers.
- 26.
The notion of “perfect copy” can be stated formally, but this results in a rather cumbersome mathematical framework [3]. We prefer to simply assume, when needed, that perfect copies, or replicas, of objects are available and we will often denote them with the same symbol of the original element.
- 27.
We have encountered a few monotonicity conditions so far. In fact, monotonicity concerns order conservation, and since order is a key property in measurement, monotonicity is also important.
- 28.
Additional details on loudness are provided in Chap. 8, Sect. 8.2.
- 29.
This notion of “cross-order” seems to be not standard in set theory, although it sounds quite “natural” from our standpoint. So, we provide a formal definition for it.
- 30.
There are now 53 members of the BIPM, including all the major industrialised countries.
- 31.
See footnote 1, in Chap. 1.
- 32.
The GUM was presented in Sect. 2.2.6.
- 33.
In fact key comparisons are just inter-comparisons performed at the highest level.
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Rossi, G.B. (2014). The Measurement Scale: Deterministic Framework. In: Measurement and Probability. Springer Series in Measurement Science and Technology. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8825-0_3
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