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Three classes of models of QHC, the joint logic of problems and propositions,are constructed, including a class of subset/sheaf-valued models that isrelated to solutions of some actual problems (such as solutions of algebraicequations).  To test the models, we consider a number of principles and rules, whichempirically appear to cover all "sufficiently simple" natural conjectures aboutthe behaviour of the operators ! and ?, and include two hypotheses put forwardby Hilbert and Kolmogorov, as formalized in the language of QHC. Each of theseturns out to be either derivable in QHC or equivalent to one of 14 principlesand rules, of which 11 are conservative over classical and intuitionisticlogics. The three classes of models together suffice to confirm theindependence of these 11 principles and rules, and to determine allimplications between them, apart from one potential implication.

A Galois connection between classical and intuitionistic logics. II: Semantics