Editing Quantum logic (section)

=== Standard semantics ===
The standard semantics of quantum logic is that quantum logic is the logic of [[projection operator]]s in a [[separable space|separable]] [[Hilbert space|Hilbert]] or [[pre-Hilbert space]], where an observable ''p'' is associated with the [[eigenspace|set of quantum states]] for which ''p'' (when measured) has [[eigenvalue]] 1.  From there,
* ''¬p'' is the [[orthogonal complement]] of ''p'' (since for those states, the probability of observing ''p'', P(''p'') = 0),
* ''p''∧''q'' is the intersection of ''p'' and ''q'', and
* ''p''∨''q'' = ¬(¬''p''∧¬''q'') refers to states that [[quantum superposition|superpose]] ''p'' and ''q''.

This semantics has the nice property that the pre-Hilbert space is complete (i.e., Hilbert) if and only if the propositions satisfy the orthomodular law, a result known as the [[Solèr theorem]].<ref>{{harvnb|Dalla Chiara|Giuntini|2002}} and {{harvnb|de&nbsp;Ronde|Domenech|Freytes}}.  Despite suggestions otherwise in Josef Jauch, ''Foundations of Quantum Mechanics'', Addison-Wesley Series in Advanced Physics; Addison-Wesley, 1968, this property cannot be used to deduce a vector space structure, because it is not peculiar to (pre-)Hilbert spaces.  An analogous claim holds in most [[Category (math)|categories]]; see John Harding, "[https://www.ams.org/journals/tran/1996-348-05/S0002-9947-96-01548-6/S0002-9947-96-01548-6.pdf Decompositions in Quantum Logic]," ''Transactions of the AMS'', vol.&nbsp;348, no.&nbsp;5, 1996.  pp.&nbsp;1839-1862.</ref> Although much of the development of quantum logic has been motivated by the standard semantics, it is not characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic.{{sfn|Dalla&nbsp;Chiara|Giuntini|2002}}