Lattices and Their Consistent Quantification
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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.
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