It's not exactly a lie, just a standard. Nominal board sizes were based on the unfinished lumber size. Another 1/4 inch is taken off each side to get a smooth surface that makes it easier to work with.
that shows the rough cuts of boards from a log. When they look at a log, they determine how many of each size they can get from it, and at that point, a 2x4 is 2 inches by 4 inches.