I believe I now understand the basic point of the paper. Stevens' experiments asked subjects to make ratio judgements, i.e. given a stimulus x, to be deemed to have one unit of intensity, say for various stimuli y how many times y is greater than x. Narens' notation for this is (x,p,y), where p is the number named by the subject.

Stevens assumed (according to Narens) that the words uttered by the subject could unproblematically be understood as numerical estimates of the actual ratio of subjective intensities. Narens points out that this is an assumption that might not hold. In general, there will be some function f mapping a verbally expressed ratio p to the actual (but unobserved) number f(p) relating the subjective intensities.

Narens calls the scale implied by a subject's judgements multiplicative if (x,p,y) and (y,q,z) imply (x,pq,z). This is a property of the subject's verbal responses, and therefore can be observed to hold or fail to hold in the experimental data. But the unknown function f implies that even if the experimental data are multiplicative, that may not be true of the underlying objective relationships between intensities f(p), f(q), and f(pq). If f(p) = log(p), then instead we would have f(pq) = f(p) + f(q), giving an additive objective scale underlying the verbal scale. Narens only mentions multiplicative and additive scales (and unless I've overlooked it, manages not to mention the log function at all, although it must be lurking offstage). However, there are an unlimited number of other possibilities. Provided only that f is monotonic, we could define an operation * on real numbers by f(p)*f(q) = f(pq). When f is the identity, * is multiplication. When f is log, * is addition. If f(p) = tanh(p), then * would be the formula for relativistic addition of velocities.

Narens explores various assumptions that might be made to ensure that f(pq) = f(p)f(q), although the derivation is not deep. There seems to be an assumption that the objective relationships ought to form a multiplicative scale. That seems to me to be rather undercut by the paper of Torgensen that Narens cites. It reports that when subjects are asked to give differences instead of ratios, they generate additive scales, i.e. (x,p,y) and (y,q,z) imply (x,p+q,z).

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Richard Kennaway, jrk@cmp.uea.ac.uk, http://www.cmp.uea.ac.uk/~jrk/

School of Computing Sciences,

University of East Anglia, Norwich NR4 7TJ, U.K.