Please correct me if I err - it's been some time since my college statistics class.
It looks like the author is talking about setting a threshold, as in the context of creating a test. Please have a look at [1].
In this setting, it is not possible to decrease the false-negative rate without increasing the false-positive rate.
I think the author is phrasing things this way simply as a matter of rhetoric, not to express the point you are suggesting.