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Cie\'sli\'nski asked whether compositional truth theory with the additionalaxiom that all propositional tautologies are true is conservative over PeanoArithmetic. We provide a partial answer to this question, showing that if weadditionally assume that truth predicate agrees with arithmetical truth onquantifier-free sentences, the resulting theory is as strong as$\Delta_0$-induction for the compositional truth predicate, hencenon-conservative. On the other hand, it can be shown with a routine argumentthat the principle of quantifier-free correctness is itself conservative.

Compositional truth with propositional tautologies and quantifier-free correctness