# Negation-quantifier-negation blocks in nonclassical logic: reference request

I'm looking for references to discussion of a certain question in the literature on non-classical first-order logics. I suspect it must have been investigated thoroughly, but I can't seem to find anything on it.

Let's say that a paraquantifier $$\mathsf{Q}$$ is a triple consisting of a sequence $$s_1$$ of zero or more negations, a quantifier $$\mathcal{Q}$$ (either $$\forall$$ or $$\exists$$), and a sequence $$s_2$$ of zero or more negations. Where $$\mathbf{v}$$ is a variable, we write $$\mathsf{Q}\mathbf{v} \, \phi$$ for $$s_1\mathcal{Q}\mathbf{v}s_2 \, \phi$$. Let's say that two paraquantifiers $$\mathsf{Q}_1$$ and $$\mathsf{Q}_2$$ are equivalent in a logic $$L$$ just in case, for any $$\mathbf{v}$$ and $$\phi$$, $$\vdash_L \mathsf{Q}_1 \mathbf{v} \, \phi \leftrightarrow \mathsf{Q}_2 \mathbf{v} \, \phi$$. We define the paraquantifier number of a logic $$L$$ to be the number of equivalence classes of paraquantifiers under this relation.

Of course, the paraquantifier number of classical first-order logic is four: every paraquantifier is equivalent to one of $$\{\forall, \exists, \forall \neg, \exists \neg \}$$. The paraquantifier number of intuitionistic first-order logic is ten: every paraquantifier is equivalent to one of $$\{ \forall, \neg \neg \forall, \forall \neg \neg, \exists, \neg \neg \exists, \exists \neg \neg, \forall \neg, \exists \neg, \neg \neg \exists \neg, \neg \forall\}$$ (see Dummett, Elements of Intuitionism [Oxford, 1977], 29).

I'm curious about the paraquantifier numbers of other non-classical first-order logics. Does anyone know of work on this question?

• What other logics are you interested in? It would help to specify some examples and what counts as a quantifier in them. E.g.: For modal logic, is $\square\forall$ a different quantifier from $\forall$? For linear logic, is $\forall_\otimes$ a quantifier or is there only $\forall_\&$? Dec 3, 2021 at 10:15
• I was only thinking about logics with the usual classical/intuitionistic syntax, rather than cases such as modal or linear logic, but of course the question generalizes interestingly. Dec 6, 2021 at 2:00