Split Interpolation: Refining Craig's Theorem via Three-Valued Logics
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913760851001344 |
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| author | Blomet, Quentin |
| author_facet | Blomet, Quentin |
| contents | Which choices of truth tables and consequence relations for two logics $\mathsf{L}_1$ and $\mathsf{L}_2$ ensure the satisfaction of the following split interpolation property: If two formulas $ϕ$ and $ψ$ share at least one propositional atom and $ϕ$ classically entails $ψ$, then there is a formula $χ$ that shares all its propositional atoms with both $ϕ$ and $ψ$, such that $ϕ$ entails $χ$ in $\mathsf{L}_1$ and $χ$ entails $ψ$ in $\mathsf{L}_2$? We identify the cases in which this property holds for any pair of propositional logics based on the same three-valued Boolean normal monotonic scheme for connectives and two monotonic consequence relations. Since the resulting logics are subclassical, every instance of this property constitutes a particular refinement of Craig's deductive interpolation theorem, as it entails the latter and further restricts the range of possible interpolants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Split Interpolation: Refining Craig's Theorem via Three-Valued Logics Blomet, Quentin Logic Which choices of truth tables and consequence relations for two logics $\mathsf{L}_1$ and $\mathsf{L}_2$ ensure the satisfaction of the following split interpolation property: If two formulas $ϕ$ and $ψ$ share at least one propositional atom and $ϕ$ classically entails $ψ$, then there is a formula $χ$ that shares all its propositional atoms with both $ϕ$ and $ψ$, such that $ϕ$ entails $χ$ in $\mathsf{L}_1$ and $χ$ entails $ψ$ in $\mathsf{L}_2$? We identify the cases in which this property holds for any pair of propositional logics based on the same three-valued Boolean normal monotonic scheme for connectives and two monotonic consequence relations. Since the resulting logics are subclassical, every instance of this property constitutes a particular refinement of Craig's deductive interpolation theorem, as it entails the latter and further restricts the range of possible interpolants. |
| title | Split Interpolation: Refining Craig's Theorem via Three-Valued Logics |
| topic | Logic |
| url | https://arxiv.org/abs/2503.20924 |