 Recall that Transititivity  and Contraposition  entail antecedent monotonicity, so it remains to show that both hold for the strict conditional. To see why Contraposition  holds for the strict conditional, note again that if \(\phi\strictif\psi\) is true in w, then all \(\phi\)-worlds in \(R(w)\) are \(\psi\)-worlds, as depicted in the left Venn diagram in Figure  7. Now suppose w is a \(\neg\psi\)-world in \(R(w)\). As the diagram makes clear, w has to be a \(\neg\phi\)-world, and so \(\neg\psi\strictif\neg\phi\) must be true in w. Similarly, if \(\neg\psi\strictif\neg\phi\) is true in w, then all \(\neg\psi\)-worlds in \(R(w)\) are \(\neg\phi\)-worlds, as depicted in the right Venn diagram in Figure  7. Now suppose w is a \(\phi\)-world in \(R(w)\). As depicted, w has to be a \(\psi\)-world, and so \(\phi\strictif\psi\) must be true in w.