Entropic Geometry from Logic

Abstract

We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic + Partiality of Knowledge + Entropy. That is: 1. A finitary probability space \Delta^n (=all probability measures on {1,...,n}) can be fully and faithfully represented by the pair consisting of the abstraction D^n (=the object up to isomorphism) of a partially ordered set (\Delta^n,\sqsubseteq), and, Shannon entropy; 2. D^n itself can be obtained via a systematic purely order-theoretic procedure (which embodies introduction of partiality of knowledge) on an (algebraic) logic. This procedure applies to any poset A; D_A\cong(\Delta^n,\sqsubseteq) when A is the n-element powerset and D_A\cong(\Omega^n,\sqsubseteq), the domain of mixed quantum states, when A is the lattice of subspaces of a Hilbert space. (We refer to http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.html for a domain-theoretic context providing the notions of approximation and content.)