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For example, usually up and down refer to opposite directions. However, in American English, we could say we are "down" to do something or we are "up" to do it and mean the same thing -- that we want to do it.

Similarly, hot and cool are opposite temperatures, but can also both be used to mean fashionably attractive/impressive.

Totally open to the possibility that the fact that I can only think of very colloquial examples of this phenomenon means I'm overthinking this, but if there's already a term for this out there I'd love to hear it.

(Also, please let me know if there are other more appropriate tags for this question -- on review, these seemed to be the best)

Edit: From thinking about this more, a more precise statement might be: There is a set of four words which are two pairs of homographs. One word in each pair is the antonym of one in the other pair, while the other in each pair is a synonym for the other word in the pair.

For example, call hot1 the word which means "a high temperature" and hot2 the word which means "fashionable/trendy". Then call cool1 the word which means "a cool temperature" and cool2 the word which means "fashionable/trendy". hot1 and hot2 and homographs, as are cool1 and cool2. Further, hot1 is an antonym of cold1, and hot2 is a synonym of cold2.

Still not sure if a general term for the situation has sprung up.

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    Related: english.stackexchange.com/questions/334096/… – MetaEd Jun 28 '16 at 16:18
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    Much like "shit". Good: "I loved that movie! It was the shit." Bad: "Why do you listen to that shit music?" Online Slang Dictionary – NVZ Jun 28 '16 at 19:03
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    @EdwinAshworth -- Auto-antonyms came to mind for me as well! I guess more precisely, one could say that the phenomenon I'm thinking of is when you have a set of four words which are two pairs of homographs. One word in each pair is the antonym of one in the other pair, while the other in each pair is a synonym for the other word in the pair. – verybadatthis Jun 28 '16 at 20:02
  • @MετάEd -- Interesting. I'm not sure that this case is quite the same -- it seems like hot and cold, or up and down, are pretty classic examples of antonyms, not just collocated words. I might be missing something in the response though -- really interesting! – verybadatthis Jun 28 '16 at 20:07
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I think a better way to describe the relationship you're looking for is: Term for two synonym phrases that use antonyms; ie: I'm up for that / I'm down for that. In this example, "up" and "down" are antonyms, but they both contextually mean "willing/able."

I was unable to find a discrete term for the phenomenon, considering I was unable to come up with any examples that aren't slang, I doubt a simple term for the specific case exists. Off the top of my head: "paradoxical synonym phrases?"

Edit: You're not alone Opposonym (Suggested on wikipedia but not anything formal) and Paradoxical Heteronym

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