Tableaux for Quantified Hybrid Logic
Patrick Blackburn, Maarten Marx

Abstract:
We present a (sound and complete) tableau calculus for Quantified
Hybrid Logic (QHL). QHL is an extension of orthodox quantified modal
logic: as well as the usual Box and Diamond modalities it contains
names for (and variables over) states, operators @_s for asserting
that a formula holds at a named state, and a binder downarrow that
binds a variable to the current state.  The first-order component
contains equality and rigid and non-rigid designators. As far as we
are aware, ours is the first tableau system for QHL.
Completeness is established via a variant of the standard translation
to first-order logic.  More concretely, a valid QHL-sentence is
translated into a valid first-order sentence in the correspondence
language. As it is valid, there exists a first-order tableau proof for
it.  This tableau proof is then converted into a QHL tableau proof for
the original sentence. In this way we recycle a well-known result
(completeness of first-order logic) instead of a well-known proof.
The tableau calculus is highly flexible. We only present it for the
constant domain semantics, but slight changes render it complete for
varying, expanding or contracting domains. Moreover, completeness with
respect to specific frame classes can be obtained simply by adding
extra rules or axioms (this can be done for every first-order
definable class of frames which is closed under and reflects generated
subframes).

Keywords: hybrid logic, first order modal logic, tableaux