Previous derivations of the sum and product rules of probability theory
relied on the algebraic properties of Boolean logic. Here they are derived
within a more general framework based on lattice theory. The result is a new
foundation of probability theory that encompasses and generalizes both the Cox
and Kolmogorov formulations. In this picture probability is a bi-valuation
defined on a lattice of statements that quantifies the degree to which one
statement implies another. The sum rule is a constraint equation that ensures
that valuations are assigned so as to not violate associativity of the lattice
join and meet. The product rule is much more interesting in that there are
actually two product rules: one is a constraint equation arises from
associativity of the direct products of lattices, and the other a constraint
equation derived from associativity of changes of context. The generality of
this formalism enables one to derive the traditionally assumed condition of
additivity in measure theory, as well introduce a general notion of product. To
illustrate the generic utility of this novel lattice-theoretic foundation of
measure, the sum and product rules are applied to number theory. Further
application of these concepts to understand the foundation of quantum mechanics
is described in a joint paper in this proceedings.