 We can think of \(\mathcal{L}_0\) as interpreted by a classical model \(\mathcal{M}_0\). Kripke shows us how to build an interpretation \(\mathcal{M}^{+}_0\) for the expanded language. The main innovation is to see the truth predicate as partial. Rather than simply having an extension, it has an extension (set of things of which it is true), and an anti-extension (set of things of which it is false). The extension and anti-extension are mutually exclusive, but they need not jointly exhaust the domain of \(\mathcal{M}_0\). Pathological sentences like \(L\) fall in neither in the extension or the anti-extension of \(\Tr\). (Actually, we could have interpreted the base language \(\mathcal{L}_0\) by a partial model as well, but the intended application sees partiality as only arising with semantic predicates like \(\Tr\).)