idiot-matic expressions

So it occurred to me today that the expression “at arm’s length” could indicate close just as easily, if not more so, than the far it’s usually taken to mean. Obviously convention dictates far, but if something is one arm’s length from you it’s really not distant at all. “At arm’s length” would by definition also be “within reach.” And is it supposed to be my arm? How does that bare ‘arm’ achieve NP status without a determiner or quantifier?

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