# AXIOMATIC APPROACHES TO STEVENSâ€™ MAGNITUDE SCALING: RECENT DEVELOPMENTS

### Abstract

In 1996, Narens showed that Stevensâ€™ methods of magnitude scaling are based on but a few qualitative assumptions that are straightforward to evaluate empirically. Two crucial assumptions are commutativity (the outcome of a sequence of two assessments does not depend on their order) and multiplicativity (the outcome of a sequence of two assessments equals a single assessment if the number associated with the single assessment equals the product of the numbers associated with the sequenced assessments). In an initial test of these axioms in the loudness production of 1 kHz-tones (Ellermeier & Faulhammer, 2000), the authors found commutativity to hold and multiplicativity to fail in the majority of lis- teners, leading to the conclusion that, while respondents seem to be able to base their judgments on a ratio-scale of sensation strength, the numerals used in the assessments do not correspond to the scale values proper. This situation inspired research into the gener- alizability of both the empirical findings and Narensâ€™ (1996) theory. This paper will give an overview of these recent developments, focussing on empirical evaluations of commu- tativity and multiplicativity in different sensory modalities and experimental tasks on the one hand, and on theories that are relaxing these axioms which are inherent in Stevensâ€™ approach, on the other.