On three-valued presentations of classical logic
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915025594089472 |
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| author | da Ré, Bruno Szmuc, Damian Chemla, Emmanuel Égré, Paul |
| author_facet | da Ré, Bruno Szmuc, Damian Chemla, Emmanuel Égré, Paul |
| contents | Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the collapsible schemes, in which the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16035 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On three-valued presentations of classical logic da Ré, Bruno Szmuc, Damian Chemla, Emmanuel Égré, Paul Logic 03B05, 03B47, 03B50 Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the collapsible schemes, in which the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible. |
| title | On three-valued presentations of classical logic |
| topic | Logic 03B05, 03B47, 03B50 |
| url | https://arxiv.org/abs/2312.16035 |