Restricted truth predicates in first-order logic

It is well-known that there exist consistent first-order theories that become inconsistent when we add Tarski’s schema T. This is Tarski’s Theorem. To avoid the inconsistency result, one can restrict Tarski’s schema in different ways. In our paper we restrict Tarski’s schema T by only instantiating the schema with a proper subset of the set of all sentences. We prove several results concerning the sets of sentences M for which Tarski’s schema T instantiated with the sentences of M is relatively consistent with any first-order theory. Let L be any first-order language containing the one-place predicate symbol T (intended to denote truth). Let M be a subset of the set of sentences of L. By the truth predicate over M we understand the instances over M of Tarski’s schema T, that is, the theory