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Measurement

As Karel Berka shows very nicely in his reviewing paper about 'Scales of Measurement' (Berka, 1983) is  the usage of  the terms 'measurement' and 'scale' in the literature  very fuzzy, even contracdictory  (Berka examines explicitly Stevens, Coomb, Torgerson, Suppes and Zinnes, Pfanzagl, Carnap, Leinfellner and Bunge, as well as Ross). The only satisfying approach would be a completely worked out framework about scales and measurement which would improve all the known desiderata. Such an account is not known today (some first experimental ideas to this topic can be found in (Doeben-Henisch, 1998)). Thus we will restrict the subject here to the fundamental concepts as they are needed for the following experimental work.

As minimal conditions of measurement we will assume the following:

1. An observer (Obs) selects some situation (Sit) for measurement

2. The observer applies some measurement procedure (MEAS)

3. The measurement procedure includes the following:

   1. Some reference objects (RO) as standard units

   2. Some rules how to apply the ROs to the target objects (TOs) (this can be put into a measuring instrument)

   3. Some rules how to assign numerals to the application of ROs to TOs

   4. Some rules how to interpret numerals as numbers of a certain type

   5. Some rules how to write measurement-expressions of the format <n, unit, space, time, device, context> (where n := number of times of applications of unit; unit := the RO used; space := a description of some location; time := a description of point in timeline; device := the used instruments; context := further descriptions of properties describing the measurement event).

4. A list of measurement-expressions constitutes the set of measurent data (DATA MEAS).

1. Classification (Nominal Scale)

A special case of pre-measurement is given if the identification of an empirical object --the target object (TO)-- has to be established in the sense that a procedure shall generate the statement tha some empirical object a has the properties {P₁, ..., P_k} and therefore this object a qualifies as an instance of the class of objects which is characterized by these properties {P₁, ..., P_k}. This is called classification or categorization. The resulting statement of a classification is always C(a,{P₁, ..., P_k}) saying that the empirical object a with the properties {P₁, ..., P_k} is an instance of the class C. The class C is an abstract concept which is given by a list of characterizing empirical properties {P₁, ..., P_k}. If C, C' are different classes then it should hold 

1. a has P₁, ..., P_k ==> C(a,{P₁, ..., P_k}) /* If an empirical object has the empirical properties P₁, ..., P_k, then a belongs to the theoretical concept C, which is characterized by the properties P₁, ..., P_k */

2. a = a /* every object is identical with itself */

3. If a = b then b = a /* If an object a is identical with an object b then b is also identical with a */

4. If a = b & b = c then a = c /* Being identical is a transitiv property */

5. a = b  & C(a,{P₁, ..., P_k}) ==> C(a,{P₁, ..., P_k}) = C(b,{P₁, ..., P_k})

6. a != b & C(a,{P₁, ..., P_k})  ==> ~C(b,{P₁, ..., P_k})

If objects of a domain of investigation can be classified according to this rules then the set of these classified objects is called to represent a nominal scale. Strictly speaking this is not yet measurement; it is a pre-condition of measurement.

2. Fundamental Measurement

If it is possible to classify objects according to some previously defined properties {P₁, ..., P_k } then it is possible to apply a measurement procedure in the sense that a target object with properties {P₁, ..., P_k} is compared to a reference object defined by properties {P'₁, ..., P'_k}. If P_i equals P'_i then the target object is comparable to the reference object and it can be stated how many times the reference object can be applied to the target object. If there is at some time at some place no comparable target object then the application of the procedure would yield the result <0, unit, space, time, device, context>. A  list of measurement-expressions will then constitute the set of measured data (DATA MEAS).

Suppes and Zinnes (Suppes, &Zinnes, 1963) introduced the concept of fundmental measurement. Such a measurement is characterized by the following properties:

If one has a set A of empirical measured data as well as an empirical relation R between these empirical data and one can present a theoretical set P of numbers with a theoretical relation S such, that one can construct a homomorphic mapping f  with R(a,b) iff S(f(a), f(b)) for all elements of the set A,  then f is a fundamental measurement and the numbers of the set P according to f represent a scale.

Example: a subject has to compare a set of n different cigaret machines; for every  two cigaret machines a,b  the subject shall decide which one is 'easier to use'. Then there will arise some ordering which can be related to numerals which in turn can be mapped into numbers of the set P.

The intuition behind this concept is clear, but the problem with this concept is that the theoretical set P is infinite, the set A of measured data is always finite. There can be no complete mapping!. The other point is that the relation S is always defined with regard to P, but the empirical relation R is only valid with regard to the comparisons which really have been done, and this means  that the domain of this empirical relation R is also always finite.  Furthermore it cannot be garanteed that the relation R is really transitiv (if R(a,b) & R(b,c) then R(a,c)).  This has to be shown in every individual case. Thus the idea of a fundamental measurement can oly be used as  a 'regulativ idea' a la Kant, but not as an established concept.  The only thing one can do in this situation is a partial mapping of proved empirical data in a formal structure.

3.Partial Mapping and Weak Ordering

Knowing about the inherent limits of a mapping between finite sets of empirical objects and infinite sets of theoretical objects one can construct partial mappings with some kind of ordering assuming the  following conditions (a,b are empirical objects with a,b in A; P_i are empirical properties, <_e is an empirical comparison operation, < is a theoretical  ordering relation, P'_i are theoretical properties,  a',b' in E  are theoretical objects,  f: A --> E):

1. a has P₁, ..., P_k ==> C(a',{P'₁, ..., P'_k}) /* If an empirical object has the empirical properties P₁, ..., P_k, then exists there a theoretical property a' with the theoretical properties  P'₁, ..., P'_k  and a' belongs to the theoretical concept C, which is characterized by the properties P_'1, ..., P'_k */

2. a = a /* every object is identical with itself */

3. If a = b then b = a /* If an object a is identical with an object b then b is also identical with a */

4. If a = b & b = c then a = c /* Being identical is a transitiv property */

5. a = b  & a has {P₁, ..., P_k} & b has {P_i, ..., P_m} ==> C(a',{P'₁, ..., P'_k}) = C(b',{P_'i, ..., P_'m})

6. a != b & a has {P₁, ..., P_k}  ==> ~C(b',{P'₁, ..., P'_k})

7. For all a,b,c in A: a <_e b   & b <_e c  ==> a <_e c    /* Transitivity of ordering */

8. a <_e b ==> a' < b'

9. For all a,b in A: a = b    ==> ~ a <_e b /* Irreflexivity of Orderiung */

10. For all a,b in A: a != b    ==> a <_e b or b <_e a /* The ordering is connex */

With these postulates it is assumed that  a,b in A and the empirical data constitute a finite structure X = <A,<_e> and  a,b' in E and the theoretical entities constitute a theoretical structure Y = <E,<>  with f: X --> Y. This is weaker then a homomorphism. And for all those elements of P which are not in the range of f it holds that they have a purely theoretical status and have to be validated by explicit measurements if necessary. An ordering of data according to such an empirical  relation <_e (or theoretical relation <) is often called  an ordinal scale. Because we have here no full homomorphic mapping f we call the numbers of P with regard to f a limited ordinal cale.

4. Existence and Uniqueness; other scales

If one wants  to establish a  mapping f:X --> Y  one has to clarify at least the following conditions for this mapping:

(i) does there exist such a mapping f?

(ii) if there exists more than one    mapping f, under which conditions are all these mappings f₁,..., f_k  unique with regard to the (limited) homomorphism?

There are much more scales conceivable (and indeed in the literature used, e.g. like interval scale, ratio scale, logarithmic interval scale etc.). But we will restrict our considerations to these two simple types of scales and we will make some experiments with these.

Experiment 1: Students run a first experiment

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