Quantifier-free formulas and quantifier alternation depth in doctrines
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2024
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| author | Abbadini, Marco Guffanti, Francesca |
| author_facet | Abbadini, Marco Guffanti, Francesca |
| contents | This paper aims to incorporate the notion of quantifier-free formulas modulo a first-order theory and the stratification of formulas by quantifier alternation depth modulo a first-order theory into the algebraic treatment of classical first-order logic.
The set of quantifier-free formulas modulo a theory is axiomatized by what we call a quantifier-free fragment of a Boolean doctrine with quantifiers. Rather than being an intrinsic notion, a quantifier-free fragment is an additional structure on a Boolean doctrine with quantifiers. Under a smallness assumption, the structures occurring as quantifier-free fragments of some Boolean doctrine with quantifiers are precisely the Boolean doctrines (without quantifiers). In particular, every Boolean doctrine over a small category is a quantifier-free fragment of its quantifier completion.
Furthermore, the sequences obtained by stratifying an algebra of formulas by quantifier alternation depth modulo a theory are axiomatized by what we call QA-stratified Boolean doctrines. While quantifier-free fragments are defined in relation to an "ambient" Boolean doctrine with quantifiers, a QA-stratified Boolean doctrine requires no such ambient doctrine, and it consists of a sequence of Boolean doctrines (without quantifiers) with connecting axioms. QA-stratified Boolean doctrines are in one-to-one correspondence with pairs consisting of a Boolean doctrine with quantifiers and a quantifier-free fragment of it. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_08551 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantifier-free formulas and quantifier alternation depth in doctrines Abbadini, Marco Guffanti, Francesca Logic Category Theory Primary: 03G30. Secondary: 03B10, 18C10, 08C15, 08B20 This paper aims to incorporate the notion of quantifier-free formulas modulo a first-order theory and the stratification of formulas by quantifier alternation depth modulo a first-order theory into the algebraic treatment of classical first-order logic. The set of quantifier-free formulas modulo a theory is axiomatized by what we call a quantifier-free fragment of a Boolean doctrine with quantifiers. Rather than being an intrinsic notion, a quantifier-free fragment is an additional structure on a Boolean doctrine with quantifiers. Under a smallness assumption, the structures occurring as quantifier-free fragments of some Boolean doctrine with quantifiers are precisely the Boolean doctrines (without quantifiers). In particular, every Boolean doctrine over a small category is a quantifier-free fragment of its quantifier completion. Furthermore, the sequences obtained by stratifying an algebra of formulas by quantifier alternation depth modulo a theory are axiomatized by what we call QA-stratified Boolean doctrines. While quantifier-free fragments are defined in relation to an "ambient" Boolean doctrine with quantifiers, a QA-stratified Boolean doctrine requires no such ambient doctrine, and it consists of a sequence of Boolean doctrines (without quantifiers) with connecting axioms. QA-stratified Boolean doctrines are in one-to-one correspondence with pairs consisting of a Boolean doctrine with quantifiers and a quantifier-free fragment of it. |
| title | Quantifier-free formulas and quantifier alternation depth in doctrines |
| topic | Logic Category Theory Primary: 03G30. Secondary: 03B10, 18C10, 08C15, 08B20 |
| url | https://arxiv.org/abs/2404.08551 |