The general logical structure of extensive quantitative concepts has been worked out as early as the beginning of this century (Hšlder, 1901). Later in this century general measurement theory attracted a considerable amount of attention, mostly in connection with quantitative methods in psychology. The short summary of general measurement theory presented here is based on the reviews by Suppes and Zinnes (1963) and Luce and Krumhansl (Luce and Krumhansl, 1988).
The basic idea of any measurement theory is that a quantitative scale
is a map between some empirical objects and associated numerical values.
The prototype of a scale is the mapping of physical bodies to a measure of
their physical mass. This mapping, however, is not arbitrary but is supposed
to meet some requirements. The most important of them is that the mapping
includes also a map between empirical relations between the objects to
algebraic relations between the numerical values. Again the simplest example
is that of physical mass. In this case the quantitative measurements are
constructed in such a way that, for instance, the operation of combining
objects
corresponds to the addition of the masses of objects 
with the mass of
to
obtain the mass of 
:
The physical operation of combining objects corresponds
to the mathematical operation of summation. This is the most important aspect
of defining a scale, since it implicitly defines the scientific meaning of the
concept and determines how to use the measured values for predictions; for
instance predicting the mass of a filled container from the masses of the
container and that of the cargo.
There are other, more technical aspects of general measurement theory that we will not review here. They concern the types of scales and the uniqueness of scales. Scales are for instance classified as fundamental or derived, depending on whether they are based on existing scales or not. For those interested in these aspects of measurement theory we recommend the excellent summary by Suppes and Zinnes (1963).