# Formalizing a Seligman-Style Tableau System for Hybrid Logic

 Title: Formalizing a Seligman-Style Tableau System for Hybrid Logic Author: Asta Halkjær From Submission date: 2019-12-20 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. Change history: [2020-06-03]: The fully restricted system has been shown complete by updating the synthetic completeness proof. BibTeX: @article{Hybrid_Logic-AFP, author = {Asta Halkjær From}, title = {Formalizing a Seligman-Style Tableau System for Hybrid Logic}, journal = {Archive of Formal Proofs}, month = dec, year = 2019, note = {\url{http://isa-afp.org/entries/Hybrid_Logic.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.