Editing Quantum logic (section)

== Introduction ==
The most notable difference between quantum logic and [[classical logic]] is the failure of the [[propositional logic|propositional]] [[distributive law]]:<ref>Peter Forrest, "Quantum logic" in ''[[Routledge Encyclopedia of Philosophy]]'', vol.&nbsp;7, 1998. p.&nbsp;882ff: "[Quantum logic] differs from the standard sentential calculus....The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity."</ref>	
: ''p'' and (''q'' or ''r'') = (''p'' and ''q'') or (''p'' and ''r''),
where the symbols ''p'', ''q'' and ''r'' are propositional variables.

To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the [[reduced Planck constant]] is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as [[Operator (quantum mechanics)|quantum-mechanical operators]].  They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that ''are'' feasible.  See {{Slink||Quantum logic as the logic of observables}} ''et seq.'' for details.</ref>
: ''p'' = "the particle has [[momentum]] in the interval {{closed-closed|0, +{{frac|1|6}}}}"	
: ''q'' = "the particle is in the interval {{closed-closed|−1, 1}}"
: ''r'' = "the particle is in the interval {{closed-closed|1, 3}}"
We might observe that:
: ''p'' and (''q'' or ''r'') = ''true''	
in other words, that the state of the particle is a weighted [[quantum superposition|superposition]] of momenta between 0 and +1/6 and positions between −1 and +3.

On the other hand, the propositions "''p'' and ''q''" and "''p'' and ''r''" each assert tighter restrictions on simultaneous values of position and momentum than are allowed by the [[uncertainty principle]] (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So there are no states that can support either proposition, and
: (''p'' and ''q'') or (''p'' and ''r'') = ''false''