The predicate version of the joint logic of problems and propositions

We consider the joint logic of problems and propositions $\mathrm{QHC}$ introduced by Melikhov. We construct Kripke models with audit worlds for this logic and prove the soundness and completeness of $\mathrm{QHC}$ with respect to this type of model. The conservativity of the logic $\mathrm{QHC}$ over the intuitionistic modal logic $\mathrm{QH4}$, which coincides with the ‘lax logic’ $\mathrm{QLL}^+$, is established. We construct Kripke models with audit worlds for the logic $\mathrm{QH4}$ and prove the corresponding soundness and completeness theorems. We also prove that the logics $\mathrm{QHC}$ and $\mathrm{QH4}$ have the disjunction and existence properties.
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