The two halves of disjunctive correctness

Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic (PA). In this paper, we show that principle “every true disjunction has a disjunct” is equivalent to bounded induction and thus it not conservative. On other hand, converse implication “any with disjunct true” can be conservatively added PA. The methods introduced here allow us give direct nonconservativeness proof DC.