 Here the notation \(S \vdash \alpha\) means that \(\alpha\) is provable in the theory S, and \(\phi \langle \psi \rangle\) is short for \(\phi(\langle \psi \rangle)\). Assume a formula \(\phi(x)\) is given that is intended to express some property of sentences – truth, for instance. Then the diagonal lemma gives the existence of a sentence \(\psi\) satisfying the biimplication \(\psi \leftrightarrow \phi \langle \psi \rangle\). The sentence \(\phi \langle \psi \rangle\) can be thought of as expressing that the sentence \(\psi\) has the property expressed by \(\phi(x)\). The biimplication thus expresses that \(\psi\) is equivalent to the sentence expressing that \(\psi\) has property \(\phi\). One can therefore think of \(\psi\) as a sentence expressing of itself that it has property \(\phi\). In the case of truth, it would be a sentence expressing of itself that it is true. The sentence \(\psi\) is of course not self-referential in a strict sense, but mathematically it behaves like one. It is therefore possible to use sentences generated by the diagonal lemma to formalise paradoxes based on self-referential sentences, like the liar. The diagonal lemma is sometimes called the fixed-point lemma, since the equivalence \(\psi \leftrightarrow \phi \langle \psi \rangle\) can be seen as expressing that \(\psi\) is a fixed-point of \(\phi(x)\).