Earlier quoted context omitted.
Human rights are not a mathematical or computer science concept, they are in fact a law concept. I know it's tempting for computer people on hackernews to want to solve problems with the tools of their own domain - but law is exactly the place where rights need to be protected, not code.
If you don't have axiomatic beliefs, then where do your beliefs come from? Some abstract feeling? Whatever "feels" right? Clearly (to me at least), you need a system of axiomatic beliefs. If your axioms don't form a coherent system (i.e you can prove a contradiction), then you really don't believe anything at all. It's surprising to me that very few people think about morality this way. Everyone has a different view…
The answer to your first question is "oftentimes, yes" to the second two.
There is this pervasive myth held by people who tend to gravitate towards "functional" fields like ours--that is, technology, math, the logical sciences, and so on--that the human existence can be distilled into a series of logical statements that can be evaluated at different times and, given the same inputs, will always produce the same outputs. (It's the same origin as the myth of the "perfectly rational actor" in economic systems that makes a lot of assumptions that are invalid in the context of how people truly function.) This myth misses a key thing about humans: thinking animals are not computers.
What feels right or what beliefs someone holds can and usually do change over time as someone's existence progresses and new experiences are had. This is where some of our greatest benefits, like the ability to experience art in new and different ways depending on the context or situation or even just our mood, and some of our greatest tragedies, like holding one set of people to be negatively different based on an arbitrary criteria, originate. Saying that if someone's beliefs can't be reduced to "code" causes them to be invalid is against the human condition.