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"I am sick". How many interpretations do you think this sentence has?
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1. ill with any range of illnesses
2. ill with nausea (a.k.a. being sick)
3. having a macabre sense of humour (sick jokes)
4 "I am sick of it" just means being fed up about some issue or other ..

why do you ask?
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Because I tought it meant mentally ill, too.
It does.

It is one of those words with a multiplicity of meanings and connotations.