Valuations on Lattices: Fuzzification and its Implications

The notion of ordering is perhaps one of the most fundamental of abstract concepts. The zeta function is used to algebraically describe the ordering of elements in a lattice. An appropriate generalization of the zeta function generalizes the notion of inclusion to degrees of inclusion. However, the lattice structure imposes strong constraints on the values that these degrees can take. Here we review our previous work [1] 1 in studying these degrees of inclusion and relate these notions to the fuzzification of the lattice (independently introduced by Vassilis Kaburlasos). We show that an inclusion measure on the Boolean lattice of logical statements leads to Bayesian probability theory, which suggests a fundamental relationship between fuzzification of a Boolean lattice and Bayesian probability theory. Last, we explore the challenges of generalizing inclusion on the lattice of questions.