now, assuming the 4K has pixel doubling, the image should now be stretched by a factor of 2 both height and width wise (but the image retains same aspect ratio since the 4K has same aspect ratio as 1080p). however, because of this stretching and the fact that you are still sitting THE SAME X distance away from the 4K, the image on the 4K should look not as sharp. However, if you move farther away from the 4K, eventually you will perceive the sharpness of the image on the 4K to be similar as the 1080p display, right?
To me, the above description would be the same as simply comparing two 1080p displays, where one display is twice as big as the other, and they were arranged at the same viewing distance.
The bigger display will look not as sharp if at the same viewing distance as the smaller display. But if the bigger display is at a further viewing distance, you can achieve the same experience. I think your constraint in your example is you describe the 4K display as being twice as big as the 1080p display, so the person will be much "closer" to the 4K according to the size and the fact that you are using 1080p content.
But, here are two picture examples that I think sum up the issues discussed.
The first example is showing "dumb" pixel doubling. Notice how the two left examples appear to be identical to each other, you are just using pixel doubling (please ignore the two images on the right). So imagine the leftmost image is a 1080p display, and the image to its right is a 4K display of exactly the same size. It perfectly reproduces the 1080p content, using 4x the pixels:
Now consider the second example, shown below, where you have a 1080p content on the left with visible pixels. But the right image would be the 4K display "interpolating" and using its extra pixels to try to guess at additional information and insert "curves" into the image. Different algorithms can produce different results, but notice how the two images look different from each other? Which is better? Manufacturers trying to sell 4K displays will really try to convince you the image on the right is better, so you should get a 4K display to perform this kind of interpolation to "improve" your paltry 1080p content:
Just as a comparison, there are some really slick interpolation algorithms, e.g., this one is tuned to optimize icons or other "pixel art" type cartoons, and it's pretty amazing:
Here's another example, where seemingly the algorithm can intelligently fill in the blanks between the pixels of a (presumably) 1080p image:
Read more here:
http://www.extremetech.com/extreme/132950-csi-style-super-resolution-image-enlargment-yeeaaaah
That would be pretty cool to have a 4K display that could do this sort of thing without lots of distracting artifacting or messing up, but we'll just have to see. Perhaps the drawback would be that you'd need the equivalent of a $500 GPU built into the TV. Hmm I think I'd rather just install a software driver on my existing computer that already has a fancy GPU to handle this, and then buy a really dumb and cheap Korean 4K display that will just display the content that has been interpolated by the GPU that I already own. I don't want to be forced to buy a TV that has its own GPU if it will just be redundant, or worse, if it cannot perform as well as the GPU I already have where I could just use GPU scaling instead of TV scaling.