How come does
no hace falta mean "there's no need/necessity" and the like?
What's the logic behind the meaning?
no hace falta (si no haces nada)/(si no pasa nada)? == nothing wrong happens (if you refrain from doing that).
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No hace falta is just a form of the phrase hacer falta "to be necessary". The logic behind it is that falta means "lack". When something is needed, you can say that something makes or causes a lack, or something is an instance of lack. If you say
no hace falta
then that literally means (something) does not make or cause a lack; it's not lacking, and therefore not needed.
There's no tone of judgement in this phrase; no hace falta can be used to tell someone off, or to spare them unneeded work.