> The classical logical operators form a neat topology. Should we expect there to be such symmetry and structure amongst the quantum operators as well?
From Quantum Logic https://en.wikipedia.org/wiki/Quantum_logic :
> Quantum logic can be formulated either as a modified version of propositional logic or as a noncommutative and non-associative many-valued (MV) logic.[2][3][4][5][6]
> Quantum logic has been proposed as the correct logic for propositional inference generally, [...] group representations and symmetry.
> The more common view regarding quantum logic, however, is that it provides a formalism for relating observables, system preparation filters and states.[citation needed] In this view, the quantum logic approach resembles more closely the C*-algebraic approach to quantum mechanics. The similarities of the quantum logic formalism to a system of deductive logic may then be regarded more as a curiosity than as a fact of fundamental philosophical importance. A more modern approach to the structure of quantum logic is to assume that it is a diagram—in the sense of category theory—of classical logics
Quantum_logic#Differences_with_classical_logic: https://en.wikipedia.org/wiki/Quantum_logic#Differences_with...