However, the axioms \(\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})\) and \(\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})\) are inconsistent over weak theories of syntax, so one of them has to be given up. If \(\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})\) is retained, one will have to find weaker axioms or rules for truth iteration, but truth remains a classical concept in the sense that \(\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})\) implies the law of excluded middle (for any sentence either the sentence itself or its negation is true) and the law of noncontradiction (for no sentence the sentence itself and its negation are true). If, in contrast, \(\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})\) is rejected and \(\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})\) retained, then it will become provable that either some sentences are true together with their negations or that for some sentences neither they nor their negations are true, and thus systems of non-classical truth are obtained, although the systems themselves are still formulated in classical logic. In the next two sections we overview the most prominent system of each kind.