Tableaux for Quantified Hybrid Logic

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 □ and ◊ modalities it contains names for (and variables over) states, operators @_s for asserting that a formula holds at a named state, and a binder ↓ 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).

This research was supported by the Netherlands Organization for Scientific Research (NWO, grants# 612.000.106 and 612.062.001). This work was carried out as part of the INRIA funded partnership between LIT (Language and Inference Technology, ILLC, University of Amsterdam) and LED (Langue et Dialogue, LORIA, Nancy).