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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.14776 |
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| _version_ | 1866909233303257088 |
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| author | Bezhanishvili, N. Martins, M. Moraschini, T. |
| author_facet | Bezhanishvili, N. Martins, M. Moraschini, T. |
| 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 $\mathsf{bi}$-$\mathsf{LC}$ of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice $Λ(\mathsf{bi}$-$\mathsf{LC})$ of extensions of $\mathsf{bi}$-$\mathsf{LC}$.
We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of $\mathsf{bi}$-$\mathsf{LC}$. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of $\mathsf{bi}$-$\mathsf{LC}$. We introduce a sequence of co-trees, called the finite combs, and show that a logic in $\mathsf{bi}$-$\mathsf{LC}$ is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest non-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$ and consequently, a unique pre-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_14776 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bi-intermediate logics of trees and co-trees Bezhanishvili, N. Martins, M. Moraschini, T. 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 $\mathsf{bi}$-$\mathsf{LC}$ of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice $Λ(\mathsf{bi}$-$\mathsf{LC})$ of extensions of $\mathsf{bi}$-$\mathsf{LC}$. We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of $\mathsf{bi}$-$\mathsf{LC}$. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of $\mathsf{bi}$-$\mathsf{LC}$. We introduce a sequence of co-trees, called the finite combs, and show that a logic in $\mathsf{bi}$-$\mathsf{LC}$ is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest non-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$ and consequently, a unique pre-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular. |
| title | Bi-intermediate logics of trees and co-trees |
| topic | Logic |
| url | https://arxiv.org/abs/2211.14776 |