"Everything is permitted"
"he solemnly announced in the discussion that there is decidedly nothing in the whole world that would make men love their fellow men; that there exists no law of nature that man should love mankind, and that if there is and has been any love on earth up to now, it has come not from natural law but solely from people's belief in their immortality. Ivan Fyodorovich added parenthetically that that is what all natural law consists of, so that were mankind's belief in its immortality to be destroyed, not only love but also any living power to continue the life of the world would at once dry up in it. Not only that, but then nothing would be immoral any longer, everything would be permitted, even anthropophagy. And even that is not all: he ended with the assertion that for every separate person, like ourselves for instance, who believes neither in God nor in his own immortality, the moral law of nature ought to change immediately into the exact opposite of the former religious law, and that egoism, even to the point of evildoing, should not only be permitted to man but should be acknowledged as the necessary, the most reasonable, and all but the noblest result of this situation."
--Fyodor Dostoevsky, The Brothers Karamazov (tr. Pevear and Volokhonsky)
If I used reason alone, I would reach much the same conclusion that Ivan Karamazov reaches. If the earth is livable due to a random combination of the right matter and space, if human existence occurs because of random mutations over the course of evolution's work, then reason tells me that there is no such thing as morality. If human existence has no inherent meaning, but only the meaning we apply to it, than any morality we try to affix to human existence is arbitrary, malleable, and fleeting. Random processes that allow human existence could also destroy human existence, and there is no meaning in any of it. With no reasonable grounding for any morality, we can do whatever we want without any concern for the consequences. We can act morally, but we don't have to, and if our actions undo all earthy existence even, well, it was all random and meaningless anyway.
Using reason alone, can you explain to me why anybody--humans or animals--has any inherent "right" to anything? Intellectual and cultural traditions have given us the concept of "rights," but human traditions are often irrational (in Western tradition, it was supposedly reason that brought about belief in "unalienable" rights, but I cannot see this belief as entirely rational). Institutions grant individuals rights, but you also cannot cite these: these are legal rights, granted by an authority, and subject to change.
You also cannot cite any metaphysical concept, such as "inherent value," "intrinsic value," "dignity," or "soul." These are qualities we believe individuals may hold, but they are not based on reason. Empirical study and rational argument will, I think, fail to show convincing evidence of the "inherent value" of any living human or animal.
I do not believe that the ideas of "human rights" or "animal rights" are based strictly on reason.
I don't believe that "everything is permitted." I also believe humans and animals are imbued with inherent value. But I recognize that these beliefs are not based on reason alone (in fact my beliefs have been informed by religion). Using my rational faculties, however, I could reach the conclusion that everything is permitted, and I would struggle to find a convincing rational argument for inherent rights. I do not, then, believe that any morality, including non-violence for humans and animals, is based strictly on reason. I don't think animal advocates can be confident they've reached a position based on reason alone.
I do not write this to deny the validity of animal rights (exploitation of animals is hardly based on solid reason, either). But I do deny that reason alone will guide anybody to an animal rights position. I don't think anybody can rely on the hope that eventually most people will rationally accept that eating animals is wrong. However, reason combined with other human values may lead one to accept the animal rights position.
I think animal rights may be based on "rational consequences." The rational consequence of animal sentience is that some of our existing human values (the concept of "rights," the prohibition against physically harming the innocent or vulnerable, etc.) would be applied to animals, also. When we learn more about animals' experience (their intellect, emotions, social lives, capacity to suffer, etc.), we might reach the conclusion that those same ethical considerations we offer to other humans would justly be offered to animals.
Reason alone did not lead me to an ethic of non-violence. However, the rational consequence of my knowledge of animals means that I must apply this ethic of non-violence to animals as well.