General Measurement Model

Everyone is familiar with everyday situations in which one measures the weight of an object in kilograms [kg], the length in meters [m], the temperature in degrees Celsius [C$^{0}$], etc. In this particular measurement processes you can detect a general structure of of the kind

$$measuring: \cal{TO} \longrightarrow \cal{R} \times \cal{RO}$$ (4.1)

Measuring is understood as a mapping from target objects $TO$ into reference objects $RO$ associated with real numbers $\cal{R}$. Assume e.g. the target object $t$ is a piece of paper in DIN A4 format, then a comparison of the length of the paper with the reference object $m$ would result in the values $(0.296, m)$.

The theory of measurement as such is extensive and complex. It is not the place here to be able to discuss all the necessary topics. Basic is that you usually try to anchor specific measurement scales on a fundamental scale on which one seeks to build then abstract units as derived scales or as associated scales. However, Karel Berka in a review article 'Scales of Measurement' (Berka, 1983[14]) shows that the use of the terms 'Measure' and 'scale' in the literature is quite inaccurate and vague. He analyzed explicitly authors such as Stevens, Coomb, Torgerson, Suppes and Zinnes, Pfanzagl, Carnap, Leinfellner, Bunge, and Ross). So far, no publication is known which resolves this problem completely from the bottom up (examples of well known textbooks for measuring are: Gigerenzer (1981)[49], Plutchik (1983)[116], Shultz et.al (2005)[122], Sixtl (1982)[124], Stegmueller (1970)[130], Suppes et.al. (1963)[147], Sydow et.al. (1981)[148])