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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.20512 |
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| _version_ | 1866908675704094720 |
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| author | Anishchenko, Dmitry M. |
| author_facet | Anishchenko, Dmitry M. |
| contents | BS4 is a natural Belnapian conservative extension of Lewis modal system S4 via strong negation. In [24] it was proved that the translation TB that naturally generalises the Godel-Tarski translation T embeds faithfully Nelsons logic N4 into BS4. So it is natural to define a modal companion of a logic extending N4 as an extension of BS4. In this paper we construct a representation of an N4-lattice similar to the representation of a Heyting algebra as an open elements algebra for a suitable topoboolean algebra. Using this algebraic result we construct a wide class of N4- extensions, elements of which have modal companions. In particular, all N3- extensions have modal companions. Also we prove that there are a continuum of N4- extensions that have no modal companions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Modal Companions of Logics with Strong Negation Anishchenko, Dmitry M. Logic BS4 is a natural Belnapian conservative extension of Lewis modal system S4 via strong negation. In [24] it was proved that the translation TB that naturally generalises the Godel-Tarski translation T embeds faithfully Nelsons logic N4 into BS4. So it is natural to define a modal companion of a logic extending N4 as an extension of BS4. In this paper we construct a representation of an N4-lattice similar to the representation of a Heyting algebra as an open elements algebra for a suitable topoboolean algebra. Using this algebraic result we construct a wide class of N4- extensions, elements of which have modal companions. In particular, all N3- extensions have modal companions. Also we prove that there are a continuum of N4- extensions that have no modal companions. |
| title | On Modal Companions of Logics with Strong Negation |
| topic | Logic |
| url | https://arxiv.org/abs/2511.20512 |