A set of \(T\)-sentences that does not imply any false arithmetical statement may be obtained by allowing only those \(\phi\) in \(T\)-sentences \(T\ulcorner \phi \urcorner \leftrightarrow \phi\) that contain \(T\) only positively, that is, in the scope of an even number of negation symbols. Like the typed theory in Section 3.2 this theory does not prove certain generalizations but proves the same T-free sentences as the strong type-free compositional Kripke-Feferman theory below (Halbach 2009). Schindler (2015) obtained a deductively very strong truth theory based on stratified disquotational principles.