Non-distributive relatives of ETL and NFL
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913264582000640 |
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| author | Kozhemiachenko, Daniil |
| author_facet | Kozhemiachenko, Daniil |
| contents | In this paper, we devise non-distributive relatives of Exactly True Logic (ETL) by Pietz and Riveccio and its dual (NFL) Non-Falsity Logic by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL's and NFL's entailment relations on the De Morgan lattice $\mathbf{4}$. We generalise these pre-orders and determine which distributive properties that hold on $\mathbf{4}$ are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics, we also devise a truth table semantics which uses non-distributive lattice $\mathbf{M3}$ as their lattice of truth values. We also provide analytic tableaux systems that work with sequents of the form $ϕ\vdashχ$. We also prove the correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-distributive relatives of ETL and NFL Kozhemiachenko, Daniil Logic In this paper, we devise non-distributive relatives of Exactly True Logic (ETL) by Pietz and Riveccio and its dual (NFL) Non-Falsity Logic by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL's and NFL's entailment relations on the De Morgan lattice $\mathbf{4}$. We generalise these pre-orders and determine which distributive properties that hold on $\mathbf{4}$ are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics, we also devise a truth table semantics which uses non-distributive lattice $\mathbf{M3}$ as their lattice of truth values. We also provide analytic tableaux systems that work with sequents of the form $ϕ\vdashχ$. We also prove the correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices. |
| title | Non-distributive relatives of ETL and NFL |
| topic | Logic |
| url | https://arxiv.org/abs/2403.09137 |