 Third, he considered dropping the assumption that w is closer to itself than any other world, or even the assumption that w is among the worlds that are closest to it. When we think in terms of similarity (or indeed of metrics) these assumptions seem perfectly natural, but some philosophers have held that they have bad proof theoretic consequences. Given the truth conditions Lewis adopts, the assumption that w is closer to itself than any other world is equivalent to the claim that p ∧ q entails p □→ q, and the assumption that w is among the worlds that are closest to it is equivalent to the claim that p □→ q and p entail q. The first of these entailments in particular has been thought to be implausible. But Lewis ultimately decided to endorse it, in large part because of the semantic model he was using. When we don’t think about entailments, and instead simply ask ourselves whether any other world could be as similar to w as w is to itself, the answer seems clearly to be no.