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