“Propositions” as Projectors

The difference between experimental yes-or-no questions and logical propositions is that there may not be experimental setups that correspond to all the possible logical combinations of experimental “propositions”.

The operations of negation (by switching 0&1 values) and implication (when one question is a refinement of another) seem to have natural experimental interpretations but it is not obvious that there must be experiments which implement the operations of logical conjunction and disjunction of any two given experimental propositions or that the relation between logical implication and negation also applies to experimentally testable questions. The assumption that these things are actually possible leads us to classical theories of physics, but it appears that in the actual world they are not. A slightly weaker assumption is that the lattice of experimentally testable propositions has the structure as the lattice of projectors in a C* algebra – which can be represented as operators on a Hilbert space. It would be nice if we could identify more fundamental and physically natural conditions (ie “axioms”) which would imply this structure, but although some have been proposed (and shown to work) their physical naturalness is not obvious to me.