There is no "human logic". Logic is logic. It encompasses thoughts both natural and foreign. Who, for example, really finds the Banach-Tarski paradox intuitive? And yet it's logical. In fact, it follows from somewhat intuitive logical rules. But, hell, we can throw those out, too. We can accept a logic that, for example, rejects the entire concept of implication and the principle of explosion. A logic which can have two contradictory things both be true, without causing the system itself to fail. The limits of logic have nothing to do with what is natural to humans.
Axioms are not things that exist just floating in space, but neither are they things that only exist to a human mind. Logical implications can be defined, and inferred, in a useful way that applies to systems that don't understand logic at all! Even if aliens thought in a way entirely, well, "alien" to us, they could still be described by logic, even if they themselves found the idea patently ridiculous, because even their formation followed physical laws, and their development followed biological laws.
Perhaps not laws that we are familiar with. Perhaps the life we hypothetically encounter is just one monolithic life-form that developed in solitude on a planet, and self-developed. Normal rules of evolution do not apply when there is only one creature. But, that's not to say that there were no rules. For to survive, it still must have done something right. And in a universe with some known laws, what is right to survive can be surmised quickly: it obviously didn't jump into its sun right away and perish. Suicide cannot coexist with life. So already we can figure out one thing: this singular life-form that "evolved" in such a strange manner does not commit suicide.
Other aspects might be harder to find. Is it peaceful? There's no way of knowing. Whether it's peaceful or not has nothing to do with whether it survived and developed to get here, because it developed entirely in solitude. But, is it very sociable? Almost certainly not. A thing that has never before seen other life won't even comprehend what it is to be social. It's just like us examining quantum physics. What on earth is going on? We can't understand things totally outside of our experience except by analogy and coincidence, and here there is almost definitely no analogy, only a freak coincidence of mental structure.
So you see, we can reason about things logically, even if they are alien. We've reasoned about quantum mechanics, despite never seeing it, and despite never even understanding it. We've reasoned about single-celled organisms, things that are incapable of even thinking. We have reasoned about automata and machines, about chemical and nuclear reactions on a scale so far beyond the realm of our comprehension that we can never hope to appreciate it on anything more than a dumb, primitive level. We do it even if we don't understand, because the tools of logic are universal. Not by virtue of some belief or god, but simply because things have patterns. Even alien things.
http://en.wikipedia.org/wiki/Banach-Tarski_paradox
http://en.wikipedia.org/wiki/Paraconsistent_logic