  In previous work [Lewitzka, Log. J. IGPL 2017], we presented a hierarchy of
classical modal systems, along with algebraic semantics, for the reasoning
about intuitionistic truth, belief and knowledge. Deviating from G\"odel's
interpretation of IPC in S4, our modal systems contain IPC in the way
established in [Lewitzka, J. Log. Comp. 2015]. The modal operator can be viewed
as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are
partially adopted from Intuitionistic Epistemic Logic IEL [Artemov and
Protopopescu, Rev. Symb. Log. 2016]. In the present paper, we show that the
S5-style systems of our hierarchy correspond to an extended
Brouwer-Heyting-Kolmogorov interpretation and are complete w.r.t. a relational
semantics based on intuitionistic general frames. In this sense, our S5-style
logics are adequate and complete systems for the reasoning about proof combined
with belief or knowledge. The proposed relational semantics is a uniform
framework in which also IEL can be modeled. Verification-based intuitionistic
knowledge formalized in IEL turns out to be a special case of the kind of
knowledge described by our S5-style systems.
