In order to obtain systems that also prove universally quantified truth-theoretic principles, one can turn the inductive clauses of Tarski’s definition of truth into axioms. In the following axioms, \(AtomSent_{PA}(\ulcorner{\scriptsize A}\urcorner)\) expresses that \({\scriptsize A}\) is an atomic sentence of the language of arithmetic, \(Sent_{PA}(\ulcorner{\scriptsize A}\urcorner)\) expresses that \({\scriptsize A}\) is a sentence of the language of arithmetic.