  The BHK interpretation interprets propositional statements as descriptions of
the world of proofs; a world which is hierarchical in nature. It consists of
different layers of the concept of proof; the proofs, the proofs about proofs
and so on. To describe this hierarchical world, one approach is the Russellian
approach in which we use a typed language to reflect this hierarchical nature
in the syntax level. In this case, since the connective responsible for this
hierarchical behavior is implication, we will use a typed language equipped
with a hierarchy of implications, $\{\rightarrow_n\}_{n=0}^{\infty}$. In fact,
using this typed propositional language, we will introduce the hierarchical
counterparts of the logics $\mathbf{BPC}$, $\mathbf{EBPC}$, $\mathbf{IPC}$ and
$\mathbf{FPL}$ and then by proving their corresponding soundness-completeness
theorems with respect to their natural BHK interpretations, we will show how
these different logics describe different worlds of proofs embodying different
hierarchical behaviors.
