The inconsistency of the \(T\)-sentences does not rely on classical reasoning. It is also inconsistent over much weaker logics such as minimal logic and partial logic. However, classical logic does play a role in restricting the free use of principles of truth. For instance, over a classical base theory, the compositional axiom for implication \((\rightarrow)\) is equivalent to the principle of completeness, \(\forall{\scriptsize A}(T[{\scriptsize A}] \vee T[\neg{\scriptsize A}\)]). If the logic under the truth predicate is classical, completeness is equivalent to the compositional axiom for disjunction. Without the law of excluded middle, FS can be formulated as a fully compositional theory while not proving the truth-completeness principle (Leigh &amp; Rathjen 2012). In addition, classical logic has an effect on attempts to combine compositional and self-applicable axioms of truth. If, for example, one drops the axiom of truth-consistency from FS (the left-to-right direction of axiom 2 in Section 4.3) as well as the law of excluded middle for the truth predicate, it is possible to add consistently the truth-iteration axiom \(\forall{\scriptsize A}(T[{\scriptsize A}] \rightarrow T[T{\scriptsize A}])\). The resulting theory still bears a strong resemblance to FS in that the constructive version of the rule-of-revision semantics for all finite levels provides a natural model of the theory, and the two theories share the same \(\Pi^{0}_2\) consequences (Leigh &amp; Rathjen 2012; Leigh, 2013).  This result should be contrasted with KF which, if formulated without the law of excluded middle, remains maximally consistent with respect to its choice of truth axioms but is a conservative extension of Heyting arithmetic.