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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.03436 |
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| _version_ | 1866914633617506304 |
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| author | Blomet, Quentin Égré, Paul |
| author_facet | Blomet, Quentin Égré, Paul |
| contents | The set of $\mathsf{ST}$-valid inferences is neither the intersection, nor the union of the sets of $\mathsf{K}_3$- and $\mathsf{LP}$-valid inferences, but despite the proximity to both systems, an extensional characterization of $\mathsf{ST}$ in terms of a natural set-theoretic operation on the sets of $\mathsf{K}_3$- and $\mathsf{LP}$-valid inferences is still wanting. In this paper, we show that it is their relational product. Similarly, we prove that the set of $\mathsf{TS}$-valid inferences can be identified using a dual notion, namely as the relational sum of the sets of $\mathsf{LP}$- and $\mathsf{K}_3$-valid inferences. We discuss links between these results and the interpolation property of classical logic. We also use those results to revisit the duality between $\mathsf{ST}$ and $\mathsf{TS}$. We present a combined notion of duality on which $\mathsf{ST}$ and $\mathsf{TS}$ are dual in exactly the same sense in which $\mathsf{LP}$ and $\mathsf{K}_3$ are dual to each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03436 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | ST and TS as Product and Sum Blomet, Quentin Égré, Paul Logic 03B50, 03B47 The set of $\mathsf{ST}$-valid inferences is neither the intersection, nor the union of the sets of $\mathsf{K}_3$- and $\mathsf{LP}$-valid inferences, but despite the proximity to both systems, an extensional characterization of $\mathsf{ST}$ in terms of a natural set-theoretic operation on the sets of $\mathsf{K}_3$- and $\mathsf{LP}$-valid inferences is still wanting. In this paper, we show that it is their relational product. Similarly, we prove that the set of $\mathsf{TS}$-valid inferences can be identified using a dual notion, namely as the relational sum of the sets of $\mathsf{LP}$- and $\mathsf{K}_3$-valid inferences. We discuss links between these results and the interpolation property of classical logic. We also use those results to revisit the duality between $\mathsf{ST}$ and $\mathsf{TS}$. We present a combined notion of duality on which $\mathsf{ST}$ and $\mathsf{TS}$ are dual in exactly the same sense in which $\mathsf{LP}$ and $\mathsf{K}_3$ are dual to each other. |
| title | ST and TS as Product and Sum |
| topic | Logic 03B50, 03B47 |
| url | https://arxiv.org/abs/2401.03436 |