Adverbial at any rate has 2 senses
  1. anyhow, anyway, in any case, at any rate, in any event - used to indicate that a statement explains or supports a previous statement; "Anyhow, he is dead now"; "I think they're asleep; anyhow, they're quiet"; "I don't know what happened to it; anyway, it's gone"; "anyway, there is another factor to consider"; "I don't know how it started; in any case, there was a brief scuffle"; "in any event, the government faced a serious protest"; "but at any rate he got a knighthood for it"
  2. at least, leastways, leastwise, at any rate - if nothing else (`leastwise' is informal and `leastways' is colloquial); "at least he survived"; "they felt--at any rate Jim felt--relieved though still wary"; "the influence of economists--or at any rate of economics--is far-reaching"
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