Lattices and Their Consistent Quantification

This paper introduces the order-theoretic concept of lattices along with the concept of consistent quantification where lattice elements are mapped to real numbersin such a way that preserves some aspect of the order-theoretic structure. Symmetries, such as associativity, constrain consistent quantification, and lead to a constraint equation known as the sum rule. Distributivity in distributive latticesalso constrains consistent quantification and leads to a product rule. The sumand product rules, which are familiar from, but not unique to, probability theory, arise from the fact that logical statements form a distributive (Boolean) lattice, which exhibits the requisitesymmetries.