  Three classes of models of QHC, the joint logic of problems and propositions,
are constructed, including a class of subset/sheaf-valued models that is
related to solutions of some actual problems (such as solutions of algebraic
equations).
  To test the models, we consider a number of principles and rules, which
empirically appear to cover all "sufficiently simple" natural conjectures about
the behaviour of the operators ! and ?, and include two hypotheses put forward
by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these
turns out to be either derivable in QHC or equivalent to one of 14 principles
and rules, of which 11 are conservative over classical and intuitionistic
logics. The three classes of models together suffice to confirm the
independence of these 11 principles and rules, and to determine all
implications between them, apart from one potential implication.
