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In the first paragraph, what does $r^{-1}(B)$ mean? Does it have something to do with relations? e.g if $A$ is a relation then $A^{-1}$ is the inverse of that relation.

Its from a paper about the hilbert system: https://www3.cs.stonybrook.edu/~cse541/chapter8.pdf

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    $\begingroup$ If we imagine a rule as a function that, given the premise(s) as input produces the conclusion B, then $r^{-1}$ will be a sort of inverse. $\endgroup$
    – Mauro ALLEGRANZA
    Jun 5 at 15:21

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