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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2409.14998 |
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| _version_ | 1866916407041589248 |
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| author | Martins, Miguel Moraschini, Tommaso |
| author_facet | Martins, Miguel Moraschini, Tommaso |
| contents | A bi-Heyting algebra validates the Gödel-Dummett axiom $(p\to q)\vee (q\to p)$ iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension $\operatorname{\mathsf{bi-GD}}$ of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of $\operatorname{\mathsf{bi-GD}}$ is locally tabular.
Notably, if $L$ is an extension of $\operatorname{\mathsf{bi-GD}}$, then $L$ is locally tabular iff $L$ is not contained in $Log(FC)$, the logic of a particular family of finite co-trees, called the finite combs. We prove that $Log(FC)$ is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Tabularity is Decidable for Bi-Intermediate Logics of Trees and of Co-Trees Martins, Miguel Moraschini, Tommaso Logic A bi-Heyting algebra validates the Gödel-Dummett axiom $(p\to q)\vee (q\to p)$ iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension $\operatorname{\mathsf{bi-GD}}$ of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of $\operatorname{\mathsf{bi-GD}}$ is locally tabular. Notably, if $L$ is an extension of $\operatorname{\mathsf{bi-GD}}$, then $L$ is locally tabular iff $L$ is not contained in $Log(FC)$, the logic of a particular family of finite co-trees, called the finite combs. We prove that $Log(FC)$ is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem. |
| title | Local Tabularity is Decidable for Bi-Intermediate Logics of Trees and of Co-Trees |
| topic | Logic |
| url | https://arxiv.org/abs/2409.14998 |