We present a foundation for inference that unites and significantly extends
the approaches of Kolmogorov and Cox. Our approach is based on quantifying
finite lattices of logical statements in a way that satisfies general lattice
symmetries. With other applications in mind, our derivations assume minimal
symmetries, relying on neither complementarity nor continuity or
differentiability. Each relevant symmetry corresponds to an axiom of
quantification, and these axioms are used to derive a unique set of rules
governing quantification of the lattice. These rules form the familiar
probability calculus. We also derive a unique quantification of divergence and
information. Taken together these results form a simple and clear foundation
for the quantification of inference.