  The effect of Richard T. Cox's contribution to probability theory was to
generalize Boolean implication among logical statements to degrees of
implication, which are manipulated using rules derived from consistency with
Boolean algebra. These rules are known as the sum rule, the product rule and
Bayes' Theorem, and the measure resulting from this generalization is
probability. In this paper, I will describe how Cox's technique can be further
generalized to include other algebras and hence other problems in science and
mathematics. The result is a methodology that can be used to generalize an
algebra to a calculus by relying on consistency with order theory to derive the
laws of the calculus. My goals are to clear up the mysteries as to why the same
basic structure found in probability theory appears in other contexts, to
better understand the foundations of probability theory, and to extend these
ideas to other areas by developing new mathematics and new physics. The
relevance of this methodology will be demonstrated using examples from
probability theory, number theory, geometry, information theory, and quantum
mechanics.
