Quantifier-free formulas and quantifier alternation depth in doctrines

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.