Minimal Conditions for Additive Conjoint Measurement and Qualitative Probability1

At the present time, there is one set of techniques for proving representation theorems for finite measurement structures an another set for infinite structures. Techniques for finite structures were developed in Scott (1964) and basically consist of solving finite sets of inequalities; techniques for infinite structures in one way or another resemble those used in Holder (1901) an d consist of the construction of fundamental sequences. Although finite structures often admit good axiomatizations in the sense that necessary and sufficient conditions for their representations can be given, they do not admit good uniqueness results. Infinite structures, however, often have uniqueness results for their representations but assume structural (nonnecessary) conditions in their axiomatizations. In this paper, new techniques are developed which allow infinite structures to be represented in terms of their finite substructures and thus simultaneously achieve good axiomatizations and representation theorems. These new techniques use the compactness theorem of mathematical logic in a way similar to Abraham Robinson’s use in his Nomtandurd Analysis (Robinson, 1966). However, to avoid the introduction of a large amount of mathematical logic into this paper, algebraic constructions are given for the various uses of the compactness theorem. This makes the paper relatively self-contained. These new techniques also allow a bridge to be built from finite to infinite structures. Thus, in Section 7 it is shown that certain infinite structures with unique representations are limits of sequences of finite structures. In terms of representations this means that as more elements are included into the qualitative structure the more “unique” the representation becomes. These new techniques also avoid the use of Archimedean axioms.