Quantificationalists should theorize in a free logic

Argument

On the intended reading of domain specifiers, the classical principle of Universal Instantiation

$$\forall xP\to P[x/a]\text{\hspace{1em}(UI)}$$

fails within the scope of domain specifiers.

For example, it is surely true that, among the French, everything is French:

$$\mathsf{At}(F)(\forall xFx)\text{\hspace{1em}(1)}$$

But Barack Obama is not French. Nor is he French among the French:

$$\mathsf{At}(F)(\neg Fb).\text{\hspace{1em}(2)}$$

Thus, assuming that domain specifiers commute with application and that Boolean operators are stable, we get

$$\mathsf{At}(F)(\forall xFx\wedge \neg Fb),\text{\hspace{1em}(3)}$$

which is a counter-example to the $\mathsf{At}(F)$-necessitation of (UI).

So, if Quantificationalists wish to hold that logical truths remain true under arbitrary domain specifiers, they should reject (UI) and theorize in a free logic, for example this one.

Caveats on stability

I take (1) to be intuitive, especially if we take the domain specifier $\mathsf{At}(F)$ to formalize the English exceptive “with the exception of the non-French.” But (1) can also be derived from the assumption that $F$ is stable. For if it is, then $\mathsf{At}(F)(F)=F$. Given this claim, $\mathsf{At}(F)(\forall xFx)$ can be proved in a background logic where domain specifiers commute with application and the laws of free logic hold within the scope of domain specifiers.¹

Likewise, (2) is derivable from $Fb$ given the assumption that both $F$ and $b$ are stable. While one may object about the stability assumptions in the specific example given, so long as one admits a non-trivial stable property $G$ that fails to apply to at least one stable individual $a$, a counter-example to the $\mathsf{At}(G)$-necessitation of (UI) can be constructed.

Footnotes

1. Specifically, we use the $\mathsf{At}(F)$-necessitation of the FH axiom FrUI and the rule (At-UG) from AtQ. ↩