  Since \(p\) does imply \(p\), \(p \rightarrow p\), i.e., at every world in which \(p\), also \(p\), then to show that arbitrary \(q\) is not sufficient for this self-implication, there needs to be a counterexample: one where \(q\) holds without \(p \rightarrow p\). Generally, to preserve sufficiency, the same strategy goes for any logical truth, \(B\): if we are to avoid \(A \rightarrow B\) when \(B\) is logically necessary but irrelevant to \(A\), there must be points at which even logical truths fail. For example, there would be a world at which the law of non-contradiction \(\twiddle (A \amp \twiddle A)\) fails. These points are known as non-normal or impossible worlds, about which there is a growing literature (e.g., Nolan 2013; Berto and Jago 2019).