ABSTRACT
This chapter generalizes the ordinary mathematical concept of an absolute value to partially ordered value structures with least upper bounds. To do so, it posits a new axiom (A9) asserting that the least upper bound of any finite, nonempty set of values is itself a value. The absolute value of a given value is then defined as the least upper bound of the set containing just that value and its negative. Many familiar theorems about absolute values then become provable, as do some new ones involving incomparable values. The absolute value operator is used to extend the definitions of delimitation, commensurability, and incommensurability, which were previously applicable only to positive values, to all values—including those that are negative, zero, or incomparable with zero. Some new concepts are introduced as well: absolute difference, an analog of distance that provides a measure of the difference between any two values; the modes of orthanticity that result from the repeatedly bifurcating structure of absolute value models; and the notion of a value’s reflections, which also result from that bifurcating structure.
