Lattices and Their Consistent Quantification

TL;DR

This paper explores how lattice structures and their symmetries underpin the derivation of sum and product rules in consistent quantification, connecting order theory with logical and probabilistic reasoning.

Contribution

It introduces the concept of consistent quantification on lattices and derives fundamental rules from lattice symmetries, linking order theory to probability.

Findings

Sum rule derived from associativity in lattices

Product rule derived from distributivity in lattices

Logical statements form a Boolean lattice with these properties

Abstract

This paper introduces the order-theoretic concept of lattices along with the concept of consistent quantification where lattice elements are mapped to real numbers in 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 lattices also constrains consistent quantification and leads to a product rule. The sum and 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 requisite symmetries.