 The central opponent of Campbell in this debate was Stevens, whose distinction between types of measurement scale was discussed above. Stevens defined measurement as the “assignment of numerals to objects or events according to rules” (1951: 1) and claimed that any consistent and non-random assignment counts as measurement in the broad sense (1975: 47). In useful cases of scientific inquiry, Stevens claimed, measurement can be construed somewhat more narrowly as a numerical assignment that is based on the results of matching operations, such as the coupling of temperature to mercury volume or the matching of sensations to each other. Stevens argued against the view that relations among numbers need to mirror qualitative empirical structures, claiming instead that measurement scales should be regarded as arbitrary formal schemas and adopted in accordance with their usefulness for describing empirical data. For example, adopting a ratio scale for measuring the sensations of loudness, volume and density of sounds leads to the formulation of a simple linear relation among the reports of experimental subjects: loudness = volume × density (1975: 57–8). Such assignment of numbers to sensations counts as measurement because it is consistent and non-random, because it is based on the matching operations performed by experimental subjects, and because it captures regularities in the experimental results. According to Stevens, these conditions are together sufficient to justify the use of a ratio scale for measuring sensations, despite the fact that “sensations cannot be separated into component parts, or laid end to end like measuring sticks” (1975: 38; see also Hempel 1952: 68–9).