The argument for animal rights and veganism that convinced me was the unfortunately named "Argument from Marginal Cases." The argument essentially follows from the recognition of speciesism, which I discussed in another post. Basically, the argument goes as follows:
There are many basic rights that we believe are owed to all human beings, such as the right not to be killed, tortured, or used as property. If we're not going to be speciesist, we have to consider other animals as potential holders of these rights. Some people suggest that humans possess certain mental capacities that justify the recognition of these rights. Non-human animals are supposed to lack these mental capacities and hence lack the correlated rights. However, for any set of capacities that someone can deem important, there will be some "marginal" group of humans who lack these capacities (the cognitively impaired) to whom we still grant basic rights. Therefore, we have no rational basis for denying basic rights to non-humans.
I believe this argument is sound (though perhaps could be put more clearly.) There are a few possible responses to it, all of which I find rather ad hoc and hardly plausible. However, one might have observed that I do not in general use this form of argument to advocate for animal rights veganism, despite its incredible impact on my own thinking. It might even seem disingenuous of me that I offer other arguments instead of the one argument that actually convinced me of the existence of animal rights. Despite this worry, there are several reasons I do not rely on this form of argument.
The first reason is that some people might find it offensive towards the disabled, in comparing them to non-humans. I do not, in fact, think it is an offensive argument, because it intends to raise the status of non-humans rather than denigrate the disabled. It even focuses on the strengths of our convictions about the rights of the disabled to derive the force of the argument. Nevertheless, advocacy should avoid appearing offensive if possible, as offense polarizes otherwise helpful discussions.
A second reason for avoiding this argument is that it opens up the possibility for people to deny that we have obligations to the mentally disabled. I find such a suggestion indefensible, but I've heard versions of it articulated. For me personally, it's not a profitable line of dialogue to pursue, because I usually get too upset and defensive at the suggestion to carry on the conversation. I know some people who do think this way, and I'd prefer to not encourage this line of thought.
Relatedly, without relying on the intuitions about our obligations to the cognitively impaired, I am able to bolster support for such obligations while also arguing in favor of animal rights. This makes my advocacy and writing (hopefully)doubly effective, especially because I don't think our society focuses enough on our obligations to the disabled, and as I said, not everyone thinks these obligations are particularly strong. If I can advocate for animal rights and disability rights simultaneously, all the better.
Of course, I don't avoid the basic features of the argument altogether. Because even if I can convince you that animals have rights regardless of their capabilities, that obviously relates to the rights of humans whose capabilities vary from the "norm." There's no way of discussing animal rights and getting away from discussion of "marginal cases," which is probably how it should be. Which leads to a philosophical benefit of discussing animal rights without an explicit appeal to this argument, which is that if other forms of argument can lead to the same conclusion, that conclusion is further supported(and better understood.) As a side effect, these discussions can be more philosophically interesting, as well.