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Zusammenfassung:
In this paper we discuss two {\em hybrid languages\/}, ${\cal
L}(\forall)$ and ${\cal L}(\downarrow)$, and provide them with complete
axiomatizations. Both languages combine features of modal and
classical logic. Like modal languages, they contain modal operators
and have a Kripke semantics. Unlike modal languages, in these systems
it is possible to `label' states by using $\forall$ and $\downarrow$
to bind special {\em state variables\/}.
This paper explores the consequences of hybridization for
completeness. As we shall show, the challenge is to blend the modal
idea of {\em canonical models\/} with the classical idea of {\em
witnessed\/} maximal consistent sets. The languages ${\cal
L}(\forall)$ and ${\cal L}(\downarrow)$ provide us with two extreme examples
of the issues involved. In the case of ${\cal L}(\forall)$, we can
combine these ideas relatively straightforwardly with the aid of
analogs of the {\em Barcan\/} axioms coupled with a {\em modal theory
of labeling\/}. In the case of ${\cal L}(\downarrow)$, on the other hand,
although we can still formulate a theory of labeling, the Barcan
analogs are not valid. We show how to overcome this difficulty by
using $\mbox{{\it COV}}^{ \ \ast}$, an infinite collection of additional rules
of proof
which has been used in a number of investigations of extended modal
logic.
