  Taha and Nielsen have developed a multi-stage calculus {\lambda}{\alpha} with
a sound type system using the notion of environment classifiers. They are
special identifiers, with which code fragments and variable declarations are
annotated, and their scoping mechanism is used to ensure statically that
certain code fragments are closed and safely runnable. In this paper, we
investigate the Curry-Howard isomorphism for environment classifiers by
developing a typed {\lambda}-calculus {\lambda}|>. It corresponds to
multi-modal logic that allows quantification by transition variables---a
counterpart of classifiers---which range over (possibly empty) sequences of
labeled transitions between possible worlds. This interpretation will reduce
the "run" construct---which has a special typing rule in
{\lambda}{\alpha}---and embedding of closed code into other code fragments of
different stages---which would be only realized by the cross-stage persistence
operator in {\lambda}{\alpha}---to merely a special case of classifier
application. {\lambda}|> enjoys not only basic properties including subject
reduction, confluence, and strong normalization but also an important property
as a multi-stage calculus: time-ordered normalization of full reduction. Then,
we develop a big-step evaluation semantics for an ML-like language based on
{\lambda}|> with its type system and prove that the evaluation of a well-typed
{\lambda}|> program is properly staged. We also identify a fragment of the
language, where erasure evaluation is possible. Finally, we show that the proof
system augmented with a classical axiom is sound and complete with respect to a
Kripke semantics of the logic.
