Nested Sequents for Intermediate Logics: The Case of Gödel-Dummett Logics
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916276236976128 |
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| author | Lyon, Tim S. |
| author_facet | Lyon, Tim S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_07550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nested Sequents for Intermediate Logics: The Case of Gödel-Dummett Logics Lyon, Tim S. Logic in Computer Science Logic 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. |
| title | Nested Sequents for Intermediate Logics: The Case of Gödel-Dummett Logics |
| topic | Logic in Computer Science Logic |
| url | https://arxiv.org/abs/2306.07550 |