It is a known result by Scott and Myhill that the second-order version of $L$ yields $\mathrm{HOD}$. 
Recently, Kennedy, Magidor, and Väänänen ([Inner models from extended logics: Part I][1]) investigated inner models given by logics with generalized quantifiers, which yields a logic intermediate between first-order and second-order logic.

> Is there a logic that produces the mantle?

(Here the choice of the mantle is somewhat arbitrary; we may replace it by 'generic mantle', 'symmetric mantle' or whatever. I will focus on the mantle in this question, but I welcome discussing other cases.)

Of course, the answer is trivial if we assume, like $V=L$ or $V=L[G]$ for some $L$-generic $G$. I want to ask the existence of logic which defines the mantle *uniform to* models of ZFC. 

> Is there a (ZFC-definable) abstract logic $\mathcal{L}$ such that the inner model given by $\mathcal{L}$ is (ZFC-provably) the mantle?
>
>(Under model-theoretic terms, is there $\mathcal{L}$ such that for any model $M$ of $\mathsf{ZFC}$, the inner model given by $\mathcal{L}$ is the mantle of $M$?)

Here are some of my rough thoughts:

- Sublogics of higher-order logics are not the candidate for $\mathcal{L}$: the corresponding inner models of higher-order logics are $\mathrm{HOD}$ (if my reasoning is correct), so the sublogics yield a submodel of $\mathrm{HOD}$. However, $\mathrm{HOD}$ need not be the mantle. (Theorem 70 of Fuchs, Hamkins, and Reitz ([Set-theoretic geology](https://arxiv.org/abs/1107.4776)).)

- We can rule out $\mathcal{L}_{\kappa\kappa}$, which yields Chang model. The inner model given by $\mathcal{L}_{\kappa\kappa}$ is the least transitive model of ZF that contains all ordinals and is closed under $<\kappa$-sequences (Theorem II of Chang's [*Sets constructible using $L_{\kappa\kappa}$*](http://www.ams.org.proxy.lib.umich.edu/books/pspum/013.1).) However, the mantle need not be closed under $<\kappa$-sequences. (A generic extension of $L$ would be an example.)

I would appreciate any comments or answers.


  [1]: https://arxiv.org/abs/2007.10764
  [2]: https://arxiv.org/pdf/1107.4776.pdf