You can't divine rights out of some fundamental logic. The only ones who can decide what rights humans have are the humans themselves, and then we structure our society to make sure those rights aren't violated.

Didn't you say this earlier, though?

The purpose of government is to protect our rights.

It sounds like you also think "you logically have a right to the services of someone else", otherwise who's doing the protection?

"You can't divine rights out of some fundamental logic" - of course you can. This is the role of philosophy.