Image size / Resolution

In your example, the "true" quality of the image is defined by the pixel dimensions of the image, which stays at 8000 x 8000 pixels (64 megapixels and 183.1 megabytes) both before and after your size change, because Resample is off. The number of pixels did not change, so the inherent quality of image detail did not change.

Because the amount of detail did not change, what now controls how detailed the image looks is the print size combined with the viewing distance. 72 ppi might seem like a very low resolution for the 111.111 inch version. But since 111.111 inches is almost 14 times larger than 8 inches, if a person was to stand 14 times farther away from the 111-inch print than from the 8-inch print (like when you see a large ad from a car), the quality of detail would appear to be the same for both prints, because they have the same number of pixels.

If you select the Resample option, then the number of pixels will change as Rob Day showed, and that will actually change the quality of the image. But in your example where Resample is off, there is no difference in quality; the two examples spread out the same 8000 x 8000 pixels over different distances.

Here are my two cents:

It is better to print with higher resolution if possible and it is also better to view image on higher monitor resolution if possible, image will get on quality in viewers eye but that does not mean image itself is better in quality. Changing resolution number ppi inside Photoshop will not change image quality nor monitor resolution, it can only affect quality when printing same image - higher resolution means better quality of printed version which is exactly the same in data as one viewed on screen. With higher monitor resolution or higher print resolution the only difference is in more dots-pixels or smaller dots-pixels which is harder to see what counts as better quality.

The Image dialog lets you either resize or resample an image. If you uncheck Resample the size of the pixels are constrained with the print output dimensions—if you make the output dimensions smaller the pixels become smaller.

If you check Resample pixels are added or subtracted as needed depending on the chosen resolution. In your case if you check Resample with the dimensions at 8" x 8" and Resolution at 72ppi, pixels will have to be removed (down sampled). The image pixel dimensions will drop from 8000x8000 to 576x576.

The image has pixels - a fixed amount. In your example 8000px wide  x 8000 pixels high.

The ppi number just tells the printer whether to spread those pixels over a larger or smaller area - no pixels are added or taken away by changing the ppi number so the amount of data contained in the image is unchanged.

Dave

Ppi is strictly a print parameter, stored as metadata alongside the file. It's not, repeat not, a property of the file. The file is just pixels, with no size and no resolution.

In fact, you have the answer right in front of your nose. Yes, ppi is a direct function of physical print size.

Ppi = pixels per inch. Read it literally and think about what it means.You have so many pixels in your file, and you spread them out over a certain area. The bigger the area, the lower the density and the lower the ppi figure.