Nested Sequents for Intermediate Logics: The Case of Gödel-Dummett Logics

Tim LyonTim Lyon

DOI Link

Tim Lyon
Nested Sequents for Intermediate Logics: The Case of Gödel-Dummett Logics
Journal of Applied Non-Classical Logics, 33(2):121-164, 2023

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KurzfassungAbstract
We present nested sequent systems for propositional Gödel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these Gödel-Dummett logics, we introduce a new structural rule, called the "linearity rule," which (bottom-up) operates by linearizing branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking if data exists along certain paths within a nested sequent. Such rules require us to generalize our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalization of the usual nested sequent formalism. Our calculi exhibit favorable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems.
Projekt:Project: DeciGUT
Forschungsgruppe:Research Group: Computational LogicComputational Logic

@article{L2023,
  author  = {Tim Lyon},
  title   = {Nested Sequents for Intermediate Logics: The Case of
             G{\"{o}}del-Dummett Logics},
  journal = {Journal of Applied Non-Classical Logics},
  volume  = {33},
  number  = {2},
  year    = {2023},
  pages   = {121-164},
  doi     = {10.1080/11663081.2023.2233346}
}