  This paper contains initial work on attempting to bring recent developments
in the foundations of quantum mechanics concerning the nature of the
wavefunction within the scope of more logical and structural methods. A first
step involves dualising a criterion for the reality of the wavefunction
proposed by Harrigan & Spekkens, which was central to the Pusey-Barrett-Rudolph
theorem. The resulting criterion has several advantages, including the
avoidance of certain technical difficulties relating to sets of measure zero.
By considering the 'reality' not of the wavefunction but of the observable
properties of any ontological physical theory a new characterisation of
non-locality and contextuality is found. Secondly, a careful analysis of
preparation independence, one of the key assumptions of the PBR theorem, leads
to a precise analogy with the kind of locality prohibited by Bell's theorem.
Motivated by this, we propose a weakening of the assumption to something
analogous to no-signalling. This amounts to allowing global or non-local
correlations in the joint ontic state, which nevertheless do not allow for
superluminal signalling. This is, at least, consistent with the Bell and
Kochen-Specker theorems. We find a counter-example to the PBR argument, which
violates preparation independence, but does satisfy this physically motivated
assumption. The question of whether the PBR result can be strengthened to hold
under the relaxed assumption is therefore posed.
