While I have never seen this notion before (it may be common or this may be the first paper that uses those terms), Comment 1 basically explains the idea. A relation $r$ in $\mathcal{R}_{n,m}$ is an $n+m$-arity relation (on propositional formulas I believe), where the first $n$ inputs are considered "positive", and the last $m$ are "negative".

To see what these mean, consider the following case. Say $r$ is a $\mathcal{R}_{1,0}$ relation, i.e. the only input is positive. Further, say $B$ is deducible from $A$. (You will have to look in the paper to see what deducible exactly means here.) Then since the arity is positive, $r(B)$ is deducible from $r(A)$. An example would be the trivial relation $r(A) := A$. Another example with a positive arity of $2$ is conjunction in classical logic. If $B_0$ and $B_1$ are deductible from $A_0$ and $A_1$ respectively, then $B_0 \wedge B_1$ is deducible from $A_0 \wedge A_1$.

In the negative case, it is the opposite. Say $r$ is a $\mathcal{R}_{0,1}$ relation, i.e. the only input is negative. Again, say $B$ is deducible from $A$. Then since the arity is negative, $r(A)$ is deducible from $r(B)$---the opposite direction as before. An example would be the negation relation in classical logic; if $B$ is deducible from $A$, then $\neg A$ is deducible from $\neg B$. (The contrapositive.)

These can be combined so that a relation has both positive and negative arities. An example is implication in classical logic, which is a $\mathcal{R}_{1,1}$ relation, i.e. it has one positive arity and one negative arity. If $B_0$ and $B_1$ are deducible from $A_0$ and $A_1$ respectively, then $A_0 \rightarrow B_1$ is deductible from $B_0 \rightarrow A_1$. (This is basically $A_0$ implies $B_0$ implies $A_1$ implies $B_1$.) The antecedent is the negative input, while the consequent is the positive one.

I hope this helps.