This is almost exactly what happened to Ivan around when he turned 2 weeks old, however there were a few differences, the most important being that he recovered in one week. From what you have described, this baby hasn't recovered since birth, let alone improved in his condition, therefore I doubt he will at this point.
What happened with Ivan, was that he started off by being unable to move his front 2 paws, but pushed himself around with his back feet. I took him to the vet who flipped him over and noticed (like what caiti pointed out,) that he would attempt to flip himself over, but that it took multiple tries to get there. Later that day, his back legs stopped working as well. By the next day, his tail no longer moved.
However, as he didn't seem to be in any pain, because his eyes were totally alert and open, and also because he had a voracious appetite, I hand fed him for a week and put 'the decision' on hold. Had he stayed the same or not gotten better after the first 3 days, I would have put him to sleep. However, a day after his tail stopped was paralysed, it started again. The next day, his hind legs worked, and he learned how to get around on just hind legs very well. 2 days later, all but his front paws themselves were working--his arms and little elbows were moving, but his paws were not, therefore he couldn't grasp food, so I was still hand feeding him.
But 1 week after I spotted the problem in the first place and took him to the vet, he was back to normal.
I believe what happened was that he fell from a high ledge in his cage, or he got out of the cage and whilst running around, chewed on something he wasn't supposed to --a wire or some paint. Obviously I made many mistakes when I (okay, Sonya) had Ivan, because I didn't have this forum, so he was in a non-baby safe cage with ledges that were way too high, and Rodya and Sonya were together for a few months until I found Chins-n-hedgies, and fixed all my mistakes.
In my opinion, the problem with Logan seems much more severe than Ivan's case, and I would suggest you do right by him and let him go.
Then again, I also suggest that you do right by all your animals and cease breeding/collecting chinchillas, as you have stated multiple times on many different threads that you do not have the money to vet your animals properly, nor do you seem to have transportation much of the time. It would be in their best interest if they were with someone who had a steady source of income and a serious amount of cash tucked away for emergencies like this one.
