 Even if it is not, \(\langle \mathcal{M}_0,H\rangle\) still provides a classical model, in which we can interpret \(\mathcal{L}^{+}_0\). With that, we can in effect apply the T-schema, relative to our hypothesis \(H\), and see what we get. More precisely, we can let \(\tau(H) = \{\left\ulcorner A\right\urcorner | A\) is true in \(\langle \mathcal{M}_0,H\rangle \}. \tau(H)\) is generally a better hypothesis about what is true in our language than \(H\) might have been. At least, clearly, if \(H\) made foolish guesses about the truth of sentence of the truth-free fragment \(\mathcal{L}_0\), they are corrected in \(\tau(H)\), which contains everything from \(\mathcal{L}_0\) true in \(\mathcal{M}_0\). Thus, \(\langle \mathcal{M}_0,\tau(H)\rangle\) is generally a better model of \(\mathcal{L}^{+}_0\) then \(\langle \mathcal{M}_0,H\rangle\).