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

Abstract: 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.