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  "body": "\n## Summary\n\nQuantum logic is the study of the propositions about a quantum system and of the logical operations on them.\nIn the approach of Birkhoff and von Neumann, a proposition is a closed subspace of the [[Hilbert space|HM2HHE3J51]] of the system.\nThese propositions form an [[orthomodular lattice|6FPKEFX0X4]].\nWhen the Hilbert space has dimension at least $2$, this lattice is not distributive, so the distributive law of classical logic fails.\nAbstract quantum logics are studied as orthomodular lattices or orthomodular partially ordered sets.\n\n## Explanations\n\n### L1 human\n\nIn classical physics, a statement such as \"the position is in this interval\" is true or false of each state.\nSuch statements combine like subsets, with \"and\", \"or\" and \"not\".\nIn quantum mechanics, a yes-no question corresponds to a closed subspace of a Hilbert space, the states for which the answer is surely yes.\n\"And\" is intersection, \"not\" is the orthogonal complement, and \"or\" is the smallest closed subspace containing both.\nWith these operations \"$p$ and ($q$ or $r$)\" can differ from \"($p$ and $q$) or ($p$ and $r$)\".\nThis reflects the fact that some quantum questions cannot be answered at the same time.\n",
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