  Cie\'sli\'nski asked whether compositional truth theory with the additional
axiom that all propositional tautologies are true is conservative over Peano
Arithmetic. We provide a partial answer to this question, showing that if we
additionally assume that truth predicate agrees with arithmetical truth on
quantifier-free sentences, the resulting theory is as strong as
$\Delta_0$-induction for the compositional truth predicate, hence
non-conservative. On the other hand, it can be shown with a routine argument
that the principle of quantifier-free correctness is itself conservative.
