One of the reasons anthropologists rarely present a united front and often direct their ire at good academics - like, say, Steven Pinker - is their inability to agree about the most basic facts concerning Homo sapiens. They discuss long-term debates like 'structure' versus 'agency', many of which become highly politicised, and work from philosophers (often very bad ones) and concepts (often very vague ones) instead of first principles.
Cognitivism is a set of first principles that should be agreed on. The fundamental principle of cognitivism is that all human behaviour results from the brain, or the mind (assuming that the 'mind' reduces to the functions of the brain). Except for bodily movements generated entirely by external forces, like being knocked down by a fist or a gust of wind, all of your actions (which are all, of course, body movements of some kind) are generated by your nervous system. Everything you've ever done has resulted from a calculation of some kind in your brain.
That's one principle. Another is that these calculations follow a set of principles that we share as a result of our phylogenetic inheritance. We're all a little different, and our brains are all different, succumbing as they do to a variety of influences, but there are commonalities. These commonalities are sufficient to allow us to generalise about how humans think, and as humans thinking is the very basis of humans acting, we can generalise about how humans act. We're able to use accounts of how humans act to understand how the brain works, as well as accounts of how the brain works to understand how humans act.
A third general principle, one that might be a little more controversial, is that humans look for patterns in what they observe and extract generalisable rules, usually unconsciously. They learn language by inducing rules from the spoken and written data that they come into contact with, and there might even be specialised parts of the brain for extracting this kind of information. They learn how to differentiate rocks from sticks by learning complexes of rules like "rocks are cooler to the touch than sticks, all things being equal", without even realising it.
Humans learn not by being perfect empiricists - sticking only to the data they have, which would stunt learning or render it impossible - but by being rationalists, deriving general, testable, changeable principles from their experiences. Or, to put it more technically, the beliefs that humans have are always underdetermined by their perceptions, meaning that some cognitive gear must be working to allow them to learn anything at all. They must have some kind of "human nature", a set of innate biases that cause them to induce roughly the same things from a similar set of experiences.
Noam Chomsky calls this the "poverty of stimulus" argument, which is a good term for it. We can never have enough experiences to absolutely determine our language ability (for instance), and so there must be something going on in the brain deriving the principles of syntax and vocabulary from the limited set of experiences we have. Chomsky has called this language-learning ability various things over the decades, but the general way in which people derive such principles from their experiences could be called "human nature". "Human nature" is not a popular term in anthropology departments, but I see no reason not to use it.*
Those seem to me to be reasonable principles. People act on the basis of mental states; in order for anything else (like falling rain, or the imposition of a law) to cause them to act, it has to affect their mental states. A blind-deaf person won't be swayed by an eloquent speech, and a person on drugs believing himself to be in the Sahara desert without water or shelter won't pick up an umbrella to shield himself from the rain, even if it is obvious to everyone else that it is falling on his head. Things can only motivate us if they affect our mental states. This applies to all action. People react in similar ways to similar stimuli because of the way their brains work, but the fact that people act in consistent ways, allowing us to generalise about 'social structure' (amongst other things), is not evidence against the idea that all human action has a mental cause.
All theorising in social science should flow from this set of principles, or at least be consistent with them. And if you believe in another kind of causation, such as "social" causation that is different from the mental causation of action, then it is up to you to provide evidence that this unnaturalistic thing can occur.
* "Human nature" is also very different from a "human essence"; human nature is encoded
in genes, which vary in how they are realised, so that some people are
incapable of learning language like other people can. A "human essence"
would imply an absolute lack of variation. Some people think "human nature" implies an essence, too, but whatevs.