Freely adding one layer of quantifiers to a Boolean doctrine
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918177972158464 |
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| author | Abbadini, Marco Guffanti, Francesca |
| author_facet | Abbadini, Marco Guffanti, Francesca |
| contents | We describe the layer of quantifier alternation depth at most one of the quantifier completion of a Boolean doctrine over a small category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. The resulting construction satisfies a universal property that makes it the free QA-one-step Boolean doctrine.
To achieve this version of Herbrand's theorem, we characterize, within the doctrinal setting, the classes $A$ of quantifier-free formulas for which there is a model $M$ such that $A$ is precisely the class of formulas whose universal closure is valid in $M$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16328 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Freely adding one layer of quantifiers to a Boolean doctrine Abbadini, Marco Guffanti, Francesca Logic Category Theory Primary: 03G30. Secondary: 03B10, 18C10, 08B20 We describe the layer of quantifier alternation depth at most one of the quantifier completion of a Boolean doctrine over a small category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. The resulting construction satisfies a universal property that makes it the free QA-one-step Boolean doctrine. To achieve this version of Herbrand's theorem, we characterize, within the doctrinal setting, the classes $A$ of quantifier-free formulas for which there is a model $M$ such that $A$ is precisely the class of formulas whose universal closure is valid in $M$. |
| title | Freely adding one layer of quantifiers to a Boolean doctrine |
| topic | Logic Category Theory Primary: 03G30. Secondary: 03B10, 18C10, 08B20 |
| url | https://arxiv.org/abs/2410.16328 |