Entropic geometry from logic - Coecke, B (Elsevier, 2003)
We establish the following equation: Quantitative Probability = Logic + Partiality of Knowledge + Entropy I.e.: 1. A finitary probability space Δn (= all probability measures on {1,…, n}) can be fully and faithfully represented by the pair consisting of the abstraction Dn (= the object up to isomorphism) of the partially ordered set (Δn , ⊑) introduced in [3], and, Shannon entropy; 2. Dn 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; DA ≅ (Δn ,⊑) when A is the n-element powerset and DA ≅ (Ωn, ⊑), the domain of mixed quantum states also introduced in [3], when A is the lattice of subspaces of a Hilbert space.