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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?

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  • $\begingroup$ 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 $\forall_\&$? $\endgroup$
    – Matt F.
    Dec 3, 2021 at 10:15
  • $\begingroup$ 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. $\endgroup$ Dec 6, 2021 at 2:00

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