Relational logic is, in all likelihood, a subset of predicate logic and has to do with, as the name implies, relations.
Jones (j) is Smith's (s) brother. Bxy = x is brother to y. So Bjs. This relation is symmetric i.e. Bjs implies and is implied by Bsj
Brown (b) is as fat as Smith. Fxy = x is as fat as y. So Fbs and also Fbb (the relation is reflexive)
Smith is taller than Jones. Txy = x is taller than y. So Tsj. Now for some relational logic: (Tsj & Tjb) implies Tsb (the relation is transitive).
The above are dyadic relations.
An example of a triadic relation is Smith (s) asked Jones (j) to call Brown (b) which in symbolic form would be Csjb; the general expression is Cxyz which translates as x asked y to call z.