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 of morality of course, but you should at the very least try and distill it into an elegant system that is a) consistent and b) has sufficient predictive power to be useful.
An example of a system that fulfills a) but not b) would be something like: There's no morality anyone can do anything they want. That's only one axiom and pretty elegant as systems go. The problem with it is that it's so trivial that it can't assign value judgements to anything at all, rendering it all but worthless.
The system that I tend to follow has three primary axioms:
1) There are two kinds of things in the world: people and property.
2) People own property and they can do anything they want with said property as long as it doesn't violate the rights of others to do the same. Moreover, the person himself is property and thus is solely owned by himself.
3) By violating the axioms above, one forfeits his right to the above axioms.
While not perfect, this system is internally consistent and also has sufficient predictive power:
If someone punches me, he's violating my property of myself (2), and therefore can be sent to jail (3). If I break into someones house, same thing: violates (2) and therefore can be punished by (3).
There are some things that the system can't assign judgements to, however: What are children? What rights do they have? What about abortion? What exactly violates someone else's property? Does air pollution can't as violation? What about the environment? What rights to animals have (the system says they're property, but some people might have a problem with others torturing animals, for example)? Where does the government get it's legitimacy? Can the government levy taxes? etc etc etc
In truth, the things it can't make judgements about are probably than the things it can. But I personally haven't found a better or more elegant system to frame the world in, so I generally stick with it while being aware of it's limitations.