Its along the lines of the inverse square law...but, for watts, if you think about it along the lines of lux vs lumens for example, for a given source with all else being equal, twice the lumens will give twice the lux.

Lux is how bright it looks, and lumens would be the total output.

Does that help?

The 4x as bright to look 2x as bright is more along the lines of distance relationships, and is according to the inverse square law.

Example:

I have a flashlight that puts out 1,000 lumens at 100,000 cd (Lux at 1 meter equivalent).

That means at 100 meters, it will put 10 lux on my target.

If I made that 2,000 lumens with the same beam pattern, I would now have twice that, at 20 lux (Double the lumens per square meter).

If I kept the 1,000 lumens and 100,000 cd, but moved the target to 200 meters, I would drop to only 2.5 lux.

So at double the DISTANCE, the same light is going to make the target appear 1/4 as bright.

And, conversely, the light would need to be 4x brighter to make the target look the same brightness, at double the distance.