Measuring on Lattices

Abstract

Previous derivations of the sum and product rules of probability theory relied on the algebraic properties of Boolean logic. Here they are derived within a more general framework based on lattice theory. The result is a new foundation of probability theory that encompasses and generalizes both the Cox and Kolmogorov formulations. In this picture probability is a bi-valuation defined on a lattice of statements that quantifies the degree to which one statement implies another. The sum rule is a constraint equation that ensures that valuations are assigned so as to not violate associativity of the lattice join and meet. The product rule is much more interesting in that there are actually two product rules: one is a constraint equation arises from associativity of the direct products of lattices, and the other a constraint equation derived from associativity of changes of context. The generality of this formalism enables one to derive the traditionally assumed condition of additivity in measure theory, as well introduce a general notion of product. To illustrate the generic utility of this novel lattice-theoretic foundation of measure, the sum and product rules are applied to number theory. Further application of these concepts to understand the foundation of quantum mechanics is described in a joint paper in this proceedings.Comment: 13 pages, 7 figures, Presented at the 29th International Workshop on Bayesian and Maximum Entropy Methods in Science and Engineering: MaxEnt 200

Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

“…The efficacy of the methodology also raises the question of whether it can be used to derive calculi in other areas of science. This indeed appears to be the case: as mentioned in the Introduction, we have already used this methodology to derive the axioms of measure theory [7] and to develop a new calculus of questions [7,8]. However, we suspect that this work merely scratches the surface of what is possible.…”

“…The result is that the symmetry of associativity of both the logical OR and logical AND operations results in a constraint equation for the bivaluation, which is the sum rule of probability theory. More generally, the sum rule ensures the symmetry of associativity of the binary operations of the distributive algebra [7]. Another important symmetry represents the fact that one should be able to append statements that have absolutely nothing to do with the problem at hand without affecting one's inferences.…”

“…In recent years, Cox's example has been expanded into a general methodology [6] that has been used to yield insights not only into existing areas such as measure theory [7] but also to aid in the construction of new calculi, such as a calculus of questions [7,8]. It is this methodology which we employ here to derive Feynman's rules of quantum theory.…”

On the Interpretation of Probabilities in Generalized Probabilistic Models

“…But it turns out that, if the function ϕ has to be compatible with the algebraic properties of B, there are not too many options at hand: ϕ has to satisfy, up to rescaling, a set of laws which are equivalent to those given by Kolmogorov's axioms. This approach has been modified and used to derive Feynman laws of probability in the quantum setting [16][17][18][19].…”

The Origin of Complex Quantum Amplitudes

“…Associativity of OR, ensures that the values on the elementary propositions (that cannot be further decomposed) can be set arbitrarily. Ordering (that X is more precise than (X OR Y )) ensures that these values µ(X) are non-negative -this is the basis of measure theory [4,5].…”