Valuations on Lattices and their Application to Information Theory

Bi-valuations are functions that take two lattice elements and map them to a real number. The zeta function is an important example of this class of functions, since it encodes inclusion on the lattice by indicating whether one lattice element includes another. This indicator function can be generalized to quantify a degree of inclusion, which induces a measure on the lattice. In the past we have shown that for distributive lattices in general, these degrees of inclusion follow a sum rule, a product rule and a Bayes' theorem analog. When such a generalization is applied to a lattice of logical statements, we recover Bayesian probability theory. In this paper, we review our previous work in developing the lattice of questions from the lattice of logical statements which answer them. With the aid of a single postulate relating probabilities of statements to relevances of questions, we derive a natural generalization of information theory. The result is a novel and efficient derivation of Shannon's entropy based on lattice theory.