ABSTRACT
We begin by defining multiplication of values by non-negative integers and examining its consequences, many of which are familiar. But certain principles that should be true in arithmetical value structures turn out not to be provable from our axioms. This problem can be traced to the possibility of bizarre “tubular” value structures. To eliminate these evidently useless structures from our inquiry, a new axiom (A8), provably consistent with the original seven, is introduced. It provides the deductive power to prove the heretofore missing principles and some unanticipated principles as well. To further enhance the expressive power of the formalism, we introduce quantifiers over the integers that are used as multipliers. This enables us to define the concept of delimitation, an idea essential to all forms of measurement, broadly enough to apply to nonlinear value structures. The concepts of commensurability and incommensurability are then defined in turn. In applying Cartesian models to problems in value theory, it is best to conform to certain constraints, the most important of which is that the values represented on distinct axes be incommensurable. It then becomes apparent that Cartesian models are especially well suited to representing certain objective list conceptions of welfare.
