The rough logic and roughness of logical theories

Tuples in an information system are taken as terms in a logical system, attributes as function symbols, a tuple taking a value at an attribute as an atomic formula. In such a way, an information system is represented by a logical theory in a logical language. The roughness of an information system is represented by the roughness of the logical theory, and the roughness of logical theories is a generalization of that of information systems. A logical theory induces an indiscernibility relation on the Herbrand universe of the logical language, the set of all the ground terms. It is imaginable that there is some connection between the logical implication of logical theories and the refinement of indiscernibility relations induced by the logical theories. It shall be proved that there is no such a connection of simple form.