Theories of truth based on the \(T\)-sentences, and their formal properties, have also recently been a focus of interest in the context of so-called deflationary theories of truth. The \(T\)-sentences \(T\ulcorner \phi \urcorner \leftrightarrow \phi\) (where \(\phi\) does not contain \(T)\) are not conservative over first-order logic with identity, that is, they prove a sentence not containing \(T\) that is not logically valid. For the \(T\)-sentences prove that the sentences \(0=0\) and \(\neg 0=0\) are different and that therefore at least two objects exist. In other words, the \(T\)-sentences are not conservative over the empty base theory. If the \(T\)-sentences are added to PA, the resulting theory is conservative over PA. This means that the theory does not prove \(T\)-free sentences that are not already provable in PA. This result even holds if in addition to the \(T\)-sentences also all induction axioms containing the truth predicate are added. This may be shown by appealing to the Compactness Theorem.