The Logic of Partitions and Partition Logics: Ore's Correspondence, Contextual Pasting, and Direct-Sum Decompositions

Abstract

The term ``partition logic'' denotes two constructions at different levels. In automaton and generalized-urn models, selected partitions generate Boolean event algebras whose contextwise union forms a concrete pasted event structure; in Ellerman's framework, whole partitions are classifications governed by refinement and partition operations. For a finite set $U$, Ore's correspondence maps each generator $π$ to its Boolean algebra $\BA(π)$, but it neither identifies the pasted carrier with $\Part(U)$ nor makes pasting a partition operation. It yields $\BA(π\wedgeσ)=\BA(π)\cap\BA(σ)$ and $\BA(π\veeσ)=\langle\BA(π)\cup\BA(σ)\rangle_{\rm BA}$, where $\langle\cdot\rangle_{\rm BA}$ denotes Boolean-algebra generation. Thus meet captures the common event algebra, whereas join gives the ambient Boolean closure. Chinese-lantern, Firefly, and triangular examples distinguish shared events, atomic intertwining, and inherited concrete order. Ellerman's direct-sum decompositions (DSDs) provide a vector-space analogue: component projections of an orthogonal DSD resolve the identity and encode exclusive outcomes, but its components are not equivalence classes of vectors. Gleason and Kochen--Specker applications require globally context-consistent valuations on those projections.