Suppes and Zinnes - basic measurement theory.pdf - Ted Sider

22 BASIC MEASUREMENT THEORY
The answer to the second problem, the uniqueness problem, for pointer
measurement has been the source of some confusion. The uniqueness
of the readings is determined by the uniqueness of the corresponding
fundamental or derived numerical assignment, not, as might appear, by
the method of calibrating the pointer instrwnent. The fact that the pointer
instrument for measuring mass gives a ratio scale is not because of the
equal spacing of the divisions on the dial; it would be quite possible to
use a nonlinear spring and have the divisions of the dial unequally spaced
without altering the scale type, On the other hand, neither is it because
of the fact that two points (e.g., 32 kg and 212 kg) are generally fixed or
determined by the caiibration procedure. Suppose, for example, the
fundamental measurement of mass yielded only an ordinal scale and the
extension of the spring were monotonically related to mass. Then,
although precisely the same calibration procedure could be carried out,
that is, two points could be fixed and the dial divided into equally spaced
divisions, the resulting readings of the instrument wonld nevertheless be
only on an ordinal scale. Thus the scale type of a pointer instrument
derives directly from the scale type of the corresponding fundamental or
derived numerical assignment. As a corollary to this, it follows that
merely inspecting the divisions on the dial of a pointer instrument, or even
observing the calibration pfocedures of the instrument, does not enable one
to infer the scale type of the measurements obtained from the instrument.
3. EXAMPLES OF FUNDAMENTAL MEASUREMENT
In this section some concrete examples of different empirical systems
are given, and in each case solutions to the representation and uniqueness
problems are exhibited. The empirical systems themselves are not
necessarily of interest. Many are too restrictive in one way or another to
have direct application to significant psychological situations. These
systems, however, permit bringing into focus some of the essential aspects
of measurement.
The proofs establishing representation and uniqueness theorems are
generally long and involved. For most empirical systems, therefore, we
shall have to content ourselves with simply stating the results. This means
that for each empirical system described a full numerical system is stated,
and then, more particularly, at least one subsystem which is a homomorphic
image of the empirical system is given. The uniqueness problem is
answered by giving the relationship between any two subsystems (of the full
system) that are homomorphic images of the empirical system, Two
proofs are given in Sec. 3.1 and 3.2. The first proof (Sec. 3.1) is relatively
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EXAMPLES OF FUNDAMENTAL MEASUREMENT
simple and it will serve to introduce some of the necessary terms and ideas
that are generally encountered. The second proof (Sec. 3.2) is considerably
more difficult so that it will give some notion of the complexity of the
representation problem.
There is, as was indicated previously (Sec. 1.1), some degree of arbitrariness
in the selection of a full numerical system for'a given empirical
system. In each case other full numerical systems could have been chosen
and a different representation theorem established. Thus it should be
emphasized that, although the representation problem is solved for each
empirical system by exhibiting one homomorphic numerical system having
certain reasonably natural and simple properties, the existence of other,
perhaps equally desirable, nwnerical systems is not ruled out.
3.1 Quasi-Series
Nominal and classificatory scales are somewhat trivial examples of
measurements so that we shall consider first an empirical system leading
to an ordinal scale. The empirical system to be described is one that might,
for example, be applicable to a lifted weight experiment in which subjects
are given the weights in pairs and are instructed to indicate whether one
weight of each pair seems heavier or whether they seem equally heavy.
Some useful definitions are as follows. A relational system 'lf = (A, R)
consisting of a set A and a single binary relation R is called a binary
system, A binary system (A, I) is called a classificatory system if and only
if I is an equivalence relation on A, that is, if and only if I has the following
properties:
1. Reflexive: if a E A, then ala;
2, Symmetric: if a, bE A and alb, then bla;
3. Transitive: if a, b, c E A and if alb, blc, then ale,
where a E A means that a is a member of the set A. A common example
of an equivalence relation is the identity relation =. In psychological
contexts it is convenient to think of the indifference relation or the "seems
alike" relation as an equivalence relation, although there are many cases
in which the transitivity property fails to hold,
We can now define a quasi-series (Hempel, 1952).
Definition L The relational system 'lf = (A,/, P) is a quasi-series if \
and only if
L (A, I) is a classificatory system;
2. Pis a binary, transitive relation.
3, If a, bE A, then exactly oneofthefollowingholds: aPb, bPa, alb,