Editing Quantum logic (section)

=== Logic of classical mechanics ===
The [[Hamiltonian mechanics|Hamiltonian]] formulations of [[classical mechanics]] have three ingredients: [[Classical mechanics|states]], [[observable]]s and [[Dynamics (mechanics)|dynamics]].  In the simplest case of a single particle moving in '''R'''<sup>3</sup>, the state space is the position–momentum space '''R'''<sup>6</sup>. An observable is some [[real-valued function]] ''f'' on the state space.  Examples of observables are position, momentum or energy of a particle.   For  classical systems, the value ''f''(''x''), that is the value of ''f'' for some particular system state ''x'', is obtained by a process of measurement of ''f''.

The [[proposition]]s concerning a classical system are generated from basic statements of the form
: "Measurement of ''f'' yields a value in the interval [''a'', ''b''] for some real numbers ''a'', ''b''."

through the conventional arithmetic operations and [[Limit (mathematics)|pointwise limits]].  It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to the [[Boolean algebra (structure)|Boolean algebra]] of [[Borel subset]]s of the state space.  They thus obey the laws of [[Classical logic|classical]] [[propositional logic]] (such as [[de&nbsp;Morgan's laws]]) with the set operations of union and intersection corresponding to the [[Boolean operator (Boolean algebra)|Boolean conjunctives]] and subset inclusion corresponding to [[Material implication (rule of inference)|material implication]].

In fact, a stronger claim is true: they must obey the [[infinitary logic]] {{Math|''L''<sub>&omega;<sub>1</sub>,&omega;</sub>}}.

We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguished ''orthocomplementation'' operation:  The lattice operations of ''meet'' and ''join'' are respectively set intersection and set union.  The orthocomplementation  operation is set complement.  Moreover, this lattice is ''sequentially complete'', in the sense that any sequence {''E''<sub>''i''</sub>}<sub>''i''∈'''N'''</sub> of elements of the lattice has a [[least upper bound]], specifically the set-theoretic union: <math display="block"> \operatorname{LUB}(\{E_i\}) = \bigcup_{i=1}^\infty E_i\text{.} </math>