Real normativity is a relatively simple beast, what someone should do is what best accomplishes their goals. About the goals themselves not much can be said; although they change over time it is only in corner cases, where the goals of a single individual become diametrically opposed, or where an additional goal is a necessary compliment of the individual’s existing goals, that we have grounds to say that someone should or shouldn’t have a particular goal.
There are, of course, other understandings of normativity, which usually doesn’t make any difference. On occasion though they can be anti-normative, which is a problem. But before I discuss anti-normativity let me first explain why deviating from the understanding of normativity outlined above is often unproblematic. You see, most people are raised so that they develop a desire to act ethically. Acting ethically is one of their goals, but they only have an intuitive sense of what is and isn’t ethical. So if they are presented with a system, based on some other understanding of normativity, that tells them what they should and shouldn’t do, on the grounds that it reveals the rules of ethics, and it agrees with their intuitive preconceptions of what ethics is, then most people have no problems living by it. They already had the goal of living ethically, and so accepting this system as a description of ethics modifies that goal to be living by this system. Thus the system becomes really normative for them, and the conflict between the conception of normativity that it is based on and real normativity is swept under the rug.
Of course this results in some amusing situations when people who accept different systems as describing what is ethical disagree. Since neither system is founded on anything better than a set of axioms such disputes dissolve to assertions that one axiom set is better than another. This means that one system can be normative for some people and non-normative, or even anti-normative, for others, but this is not the kind of anti-normativity I had in mind. Earlier I mentioned that there are times when we can say that a person should or shouldn’t have certain goals, namely when they are in conflict. Now many conflicts between goals are insignificant, and can be resolved simply by dividing resources between them. But some conflicts are deeper, sometimes two goals pull in opposite directions, such that to satisfy one of them to any extent is to frustrate the other. And sometimes one goal pulls in a direction opposite from every other goal. Such goals, and things that create and sustain them, are anti-normative.
Following a set of rules that are believed to be normative, and which aren’t designed to pursue some or all of our goals, have the potential to be anti-normative (more precisely, our goal of following them has the potential to be anti-normative). Because of this we are in a position to say “you shouldn’t follow that set of rules”. This “shouldn’t” may be weak or strong, depending on how often the set of rules is anti-normative, and maybe it is weak enough to be overcome by the goal of following them. But, regardless of whether it will trump other considerations, there is still reason to oppose then. And this is not an opposition founded only in theory. Practically anti-normativity has some downsides. Naturally being in situation where one of your goals is anti-normative is unpleasant. And the mind tends to react to this situation by automatically adjusting its priorities, either by decreasing the perceived importance of being ethical, undesirable, or decreasing the importance of the other goals that happen to oppose it, also undesirable. And from that standpoint we might say: if you care about the wellbeing of other people don’t encourage them to adopt a set of rules that can be anti-normative, it might be bad for them.
This covers human normativity, which is defined almost exclusively by our goals, since as intelligent beings those are the most important. In other situations we may run across what can be called “emergent” normativity. Emergent normativity describes a situation in which certain features tend to dominate a group because of selection pressures. A feature that increases evolutionary competitiveness might be described as emergently normative. In such situations criterions that value traits other then those that promote competitiveness can be described as anti-normative. In many cases of emergent normativity anti-normativity is a bit of a strange beast anyways. Say that we considered passivity in a predator, a lion, to be better, normatively better. Obviously as things currently are passivity for them is anti-normative, and so passivity will tend to be selected out of lions, no matter how desirable we may see it as. But, on the other hand, as soon as we try to enforce normativity as we see it, say by killing the most aggressive lions, then it stops being anti-normative because our actions have introduced a new selection pressure into the environment.
So normally talk of anti-normativity in the context of emergent normativity serves no real point, since to act on what we think should be normative is to make it normative, in the emergent sense. But an important exception to this is the emergent normativity present in societies. Each society (a group of people living under a shared social framework) can be considered subject to selection pressures (or at least the social framework can be). These selection pressures are more complicated than evolutionary selection pressures because they stem from both external (other societies competing for resources) and internal (people acting to change society, or leaving it, because it is frustrating their goals) sources. Here anti-normativity can be actually harmful. An anti-normative position would be one that would reduce the competitiveness of the society were those ideas to be adopted. And that is bad for society, and bad for the people living in it; it leads to greater dissatisfaction or reduced resources for the people composing it. Anti-normativity in the context of society then is self-defeating; anti-normative ideas are put into place because they are thought to make society better on the basis of some analysis based on first principles, but in the end they make society worse off, eventually leading to a rejection of those very principles (example: communism).