Reverse mathematics seeks to find subsystems of second-order logic that are equivalent to certain mathematics theorems, say over $\mathsf{RCA}_0$.

Modal logic can be regarded as a weak version of first-order logic -- with the modal necessity operator $\square$ as replacing the universal quantifier $\forall$ in first-order logic, and the possibility operator $\diamond$ as replacing the existential quantifier $\exists$. (I learned this interpretation from van Benthem's book: Modal Logic for Open Minds, p.3.) Via this replacement, a mathematics theorem written in the language of first-order logic can be converted to a sentence in modal logic. We can then ask:

Is there an axiom system in modal logic, that when regarded as a subsystem of first-order logic, is equivalent to a given mathematics theorem, say over $\mathsf{K}$? (Here $\mathsf{K}$ is a weak system of modal logic, in analogy to $\mathsf{RCA_0}$ for reverse mathematics.) Has there been work to do reverse mathematics over modal logic, in this sense?

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    $\begingroup$ In this translation, do you have a pair of modal operators $\square_x$ and $\lozenge_x$ for each variable $x$? $\endgroup$ Oct 13, 2022 at 14:16
  • $\begingroup$ Alex, thanks for pointing out my naivety. $\endgroup$
    – Colin Tan
    Oct 13, 2022 at 14:22
  • $\begingroup$ It was just a clarifying question - I didn't mean to suggest your question is naive! I think it's interesting. $\endgroup$ Oct 13, 2022 at 14:27
  • $\begingroup$ I take no offence, Alex. I am grateful for your asking a question that helps to clarify my idea. $\endgroup$
    – Colin Tan
    Oct 13, 2022 at 15:59


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