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Is there a name for this manner of purposely speaking in double negatives, e.g.

I wouldn't say no to a cup of tea!

I've noticed it as a habit of some people, perhaps often going along with a particular type of sense of humour and British understatement.

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See related questions: english.stackexchange.com/search?q=litotes+ – JLG May 29 '12 at 17:12
That's not a 'double negative' anyway. 'no' is a quoted, mentioned word, not used in the sentence. It does provide some extra processing depth for interpretation, but that's all. – Mitch May 29 '12 at 17:54
@Mitch alternatively, "say no" is a phrasal verb with negative polarity. – Mark Beadles May 29 '12 at 19:54
@MarkBeadles: I hadn't even considered that (I always thought of phrasal verbs as 'V + P' where the preposition doesn't prepose anything). But still the 'double negative' only refers to the non-standard use of the adjective 'no' instead of 'a' or 'any' (as in 'I don't want no apples' for 'I don't want any apples'). There are all sorts of situations in English with multiple negative polarity words: 'I miss denying the lack in no uncertain terms the vanishing emptiness'. Just because a phrase includes two explicit 'n-' negation terms, doesn't make it the 'illegal' 'double negative'. – Mitch May 29 '12 at 20:32
@Mitch I agree, this isn't one of those evil traditional double negatives. – Mark Beadles May 29 '12 at 20:34
up vote 7 down vote accepted

This is litotes, a figure of speech using understatement - in particular, double negation - for rhetorical effect. It's not a recent habit, or even solely a British one, but rather a figure of speech used in many cultures and languages.

A prototypical English example is "not bad!"

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Indeed, "nicht schlecht!" in German. – Qube May 31 '12 at 12:32

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