This question is motivated by this old answer of mine. Below, by "appropriate theory" I mean any consistent finitely axiomatizable theory in the language $L_2$ of second-order arithmetic containing $RCA_0$.
Given an appropriate theory $T$, we can whip up a propositional modal logic $PL_T$ associated to $T$ by interpreting "$\Box$" as "is $T$-provable" (of course, for such $T$ we always have $PL_T=GL$). There is also a "semantic alternative" gotten by interpreting "$\Box\varphi$" as "For every structure $M$, if $M\models T$ then $M\models\varphi$."
There are a couple subtleties here. First, by "structure" I just mean "set equipped with interpretations of the symbols in $L_2$" - so, structures don't come equipped with their theories. Second, "$\models$" is expressed in any of the usual $\Sigma^1_1$ ways.
Despite the completeness theorem, for some appropriate $T$ we have $VL_T\not=PL_T$. In fact $VL_T\not\supseteq {\bf K}$ in general: for appropriate $T$ we have $\chi\in VL_T\iff ACA_0\subseteq T,$ where $$\chi=\Box(\Diamond p\rightarrow (\Diamond q\vee\Diamond\neg q)).$$ Note that this applies to $WKL_0$ despite its proving the completeness theorem - the real culprit is bivalence, which while "obvious" actually has real strength here.
My question is: what can we say about $VL_T$ for $T$ an appropriate theory strictly below $ACA_0$?
(I'm ignoring theories incomparable with $ACA_0$ for now.)
Unfortunately I suspect that this is hopelessly broad, so let me hone in on a specific sub-question:
Do we have $VL_{RCA_0}=VL_{WKL_0}$?
(I'm really interested in any progress on the broader question, but this seems like a good starting point.)
