From the article, "In order to yield twice larger prints at the same PPI, you would need to multiply sensor resolution by 4."
Sigh!!!
To get twice larger prints you need twice the sensor resolution or 4 times the number of pixels.
The word resolution has its origins in the number of lines you can resolve. Since then marketers and journalists realized that they could play the numbers games by equating number of pixels with resolution.
The item you quoted from the article seems to define by length at a constant PPI. In other words, double linear resolution, for a twice larger print. The quote may be worded funny, but it seems quite clear. Pixels go up by the square of the linear resolution.
My point is that he is playing with words to create a sort of confusion that he then tries to explain. He is a journalist making a living by writing "profound" thoughts in a sensational way (and perpetuating the misuse of unqualified "resolution" in the process).
I am not confused, nor was I ever, and neither are you confused.
On another topic, that I don't think he covered, is that sharpness of the final image is also dependent on the processing that normally takes place within the camera. I normally shoot raw, and therefore have to explicitly do this processing by hand. The main ones are chroma and luminence noise reduction, sharpening and chromatic aberration correction.