A two full F/stop change to either side of the starting F/stop will either double or half the amount of light from the original F/stop.
False. Going from f/2 to f/2.8 will halve the light; going from f/2 to f/4 will quarter it.
A one full f-stop change to either side of the starting f-stop will either double or halve the amount of light.
Why? That's just the way f-stops were designed. One f-stop means a doubling or halving of light.
Note that when talking about aperture, that is stated relative to its diameter. Each f-stop increases or decreases the diameter of the aperture by a factor of about 1.4 (which is root 2). However, that causes a change in area of the aperture opening, hence the amount of light being let through, to be by a factor of 2, since the area of a circle is relative to the square of its radius.
So a sequence of full f-stops in f-numbers (for aperture) go 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11.2, 16, etc with each stop letting in double the amount light of light as the previous stop, even though the diameter only needs to increase by about 1.4 each time.