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

BASIC MEASUREMENT THEORY
One might be tempted to conjecture that the first four axioms of Def. I 6
would characterize all finite difference systems for which a numerical
representation could be found (the representations of a given system
would not necessarily be related by a linear transformation). The resulting
theory would then represent one forma1ization of Coombs' ordered metric
scale. However, Scott and Suppes (1958) have proved that the theory of
all representable finite difference systems is not characterized by these
four axioms and, worse still, cannot be characterized by any simple finite
list of axioms.
The f. d. systems are not as artificial or as impractical as they may seem.
One theory for approximating these systems is to be found in Davidson,
Suppes, & Siegel (1957, Chapter 2). However, these systems can have
more general usefulness if they are used to establish a "standard set" of
stimuli. In the case of tones, for example, a set of tones may be selected in a
successive manner so that the set satisfies Axiom 5. If this standard set of
tones also satisfies the remaining four axioms, then we know from Theorem
IS that the tones may be assigned numbers that are on an interval scale.
Arbitrary tones that are not in the standard set but that satisfy the first
four axioms may then be located within intervals bounded by adjacent
tones in the standard set. This means that by decreasing the spacing
between the standard tones any arbitrary tone may be measured within
any desired degree of accuracy. This is in fact what a chemist does in
using a standard set of weights and an equal arm balance to determine
the weight of an unknown object. His accuracy of measurement is limited
by the size of the smallest interval between the standard weights or, if he
also uses a rider, by the gradations on the rider.
Other relational systems closely related to f.d. systems may appropriately
be mentioned at this point. Among the simplest and most appealing are
the bisection systems "21 = (A, B), where B is a ternary relation on the set
A with the interpretation that B(a, b, c) if and only if b is the midpoint of
the interval between a and c. The method of bisection, which consists in
finding the midpoint b, has a long history in psychophysics. The formal
criticism of many experiments in which it has been used is that the variety
of checks necessary to guarantee isomorphism with an appropriate
numerical system is not usually performed. For example, if B(a, b, c)
implies that aPb and bPc, where P is the usual ordering relation, then
from the fact that B(a, b, c), B(b, c, d), and B(c, d, e) we should be able to
infer B(a, c, e). But the explicit test of this inference is too seldom made.
Without it there is no real guarantee that a subjective scale for a stimulus
dimension has been constructed by the method of bisection.
Because of the large number of axiomatic analyses already given in this
section, we shall not give axioms for bisection systems. The axioms in
1.
.(.
.•
'EXAMPLES OF FUNDAMENTAL MEASUREMENT
:any case are rather similar to those of Def. 16, and the formal connection
between the difference relation D and the ternary bisection relation B
should be obvious:
B(a, b, c) if and only if abDbc and be Dab.
As an alternative to giving general axioms for bisection systems, it may
be of some interest to lqok at the problem of characterizing these systems
in a somewhat different manner, namely, by simply listing for a given
number n of stimuli the relations that must hold. In perusing this list it
should be kept in mind that we assume that bisection systems have the
same property of equal spacing possessed by f. d. systems. As examples,
let us consider the cases of n = 5 and n = 6.
For n = 5; let A= {a, b, c, d, e) with the ordering aPbPcPdPe. We
then have exactly four instances of the bisection relation, namely, B(a, b, c),
B(b, c, d), B(c, d, e), and B(a, c, e).
For n = 6, we may add the element f to A with the ordering
aPbPcPdPePf To the four instances of the bisection relation for n = 5,
we now add two more, namely, B(d, e,f) and B(b, d,f). We may proceed
in this manner for any n to characterize completely the bisection system
with n stimuli, none of which is equivalent with respect to the property
being studied. Establishing the representation and uniqueness theorems
is then a trivial task. The disadvantages of this approach to characterizing
those relational systems for which numerical representation theorems
exist are twofold. In the first place, in contrast to the statement of general
:axioms, the listing of instances does not give us general insight into the
structure of the systems. Second, for systems of measurement that have a
more complicated or less sharply defined structure than bisection systems,
the listing of instances can become tedious and awkward-semiorders
:provide a good example.
3.5 Extensive Systems
We consider next a relational system leading to a ratio scale. Since this
relational system contains an operation a that corresponds to an addition
operation, we may justifiably call this system an extensh'e system (see
Sec. 1.4). The axioms that we shall use to define an extensive system
(Suppes, 1951) are similar to those first developed by Holder (1901).
HOlder's axioms, however,. are more restrictive than necessary in that they
require the homomorphic numerical relational systems to be nondenumerable
(and nonfinite). The present set of axioms applies both to
denumerable and nondenumerable but infinite relational systems.