### Abstract

This work is a formalization of soundness and completeness proofs for a Seligman-style tableau system for hybrid logic. The completeness result is obtained via a synthetic approach using maximally consistent sets of tableau blocks. The formalization differs from previous work in a few ways. First, to avoid the need to backtrack in the construction of a tableau, the formalized system has no unnamed initial segment, and therefore no Name rule. Second, I show that the full Bridge rule is admissible in the system. Third, I start from rules restricted to only extend the branch with new formulas, including only witnessing diamonds that are not already witnessed, and show that the unrestricted rules are admissible. Similarly, I start from simpler versions of the @-rules and show that these are sufficient. The GoTo rule is restricted using a notion of potential such that each application consumes potential and potential is earned through applications of the remaining rules. I show that if a branch can be closed then it can be closed starting from a single unit. Finally, Nom is restricted by a fixed set of allowed nominals. The resulting system should be terminating. Paper: doi.org/10.4230/LIPIcs.TYPES.2020.5.

### License

BSD License

### History

June 3, 2020
  The fully restricted system has been shown complete by updating the synthetic completeness proof.

### Topics

- Logic/General logic/Modal logic

### Session Hybrid_Logic

- Hybrid_Logic

## Cite

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Hybrid_Logic-AFP

    @article{Hybrid_Logic-AFP,
      author  = {Asta Halkjær From},
      title   = {Formalizing a Seligman-Style Tableau System for Hybrid Logic},
      journal = {Archive of Formal Proofs},
      month   = {December},
      year    = {2019},
      note    = {\url{https://isa-afp.org/entries/Hybrid_Logic.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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