It is much more fun, and much more productive, to engage in original philosophy instead of arguing against the positions of other philosophers. But, as tedious as it is, argument is necessary in order to evaluate the strengths and weaknesses of various claims and to compare them to each other. Unfortunately this is all argument can do, it is almost unheard of for an argument against a claim to be so successful that it forces it to be given up; at most it will convince some people who are yet undecided to favor another claim, diminishing, but not eliminating, its support in the long run.
But before I can say how to best argue against a philosophical claim I should describe the general structure of philosophical claims. In general a philosophical claim consists of three parts, which may or may not receive equal attention. In order they are: the argument, the claim, and applications. Usually philosophers who wish to make a particular claim will begin with an argument from simpler or less controversial premises for that claim. The argument serves to make the claim more acceptable, more intuitive. It also serves to illuminate some of the reasoning behind the claim, and why the philosopher proposing it is motivated to make it. Next is usually the claim itself, in tedious detail proportional to the length of the paper it is in. This section serves to elaborate on exactly what is being claimed, hopefully to eliminate confusion, and to distinguish the claim from other similar claims. And finally, if we are lucky, the claim will be applied to various situations and problems. This serves as both additional clarification (sometimes more so than the statement of the claim itself), and as a way of demonstrating the value of the claim, by showing how it provides more and better explanations.
The least effective way to argue against a philosophical claim is to attack the argument, because it is so rarely successful. For one philosophers usually aren’t stupid people, and so the argument itself can be assumed in most cases to have a coherent structure. Even if we find what appears to be an unjustified move usually what we have really uncovered is a hidden assumption or a strange use of words. And that isn’t really a flaw in the argument; we might wish to point it out for reasons of clarity, but that is about the only role it can serve. A second problem with attacking the argument is that it is perfectly possible for a true claim to have a poor argument for it. For example, Galileo was motivated to say that the earth moved around the sun partly because of his observations of the moons of Jupiter (or so I am told). And that is a very weak argument by analogy for the claim that the earth moves around the sun. Nevertheless the earth does move around the sun, so it was a good thing that the scientists of the time evaluated the claim itself and not Galileo’s motivation for making it. Now this is not to say that we can never argue against the argument for another position. Attacking arguments can reveal what makes a good and bad argument, and that knowledge helps us be better at developing our own philosophy (because we are often led to new ideas by arguments for them). And an argument against a argument may be a suitable preface to a more substantial argument against a philosophical claim, as it can make the claims appear less intuitive than it is made out to be. Now, as a closing remark on this matter, let me say that attacking the argument is a natural move to make, because in other situations positions may be seen as in competition with one another, and an argument then is the primary tool with which to force people of one side to agree with the other side. But such a combative approach to philosophy is unhelpful, and no philosophical argument can be absolutely compelling anyways, as there is always some premise you could disagree with.
Similar to attacking the argument is to attack the premises. Again this at best can be used to show that the philosophical claim in question is not as intuitive as it might seem, and so, like attacking the argument, should be used only in conjunction with a more substantial argument against the claim. No more needs to be said about it then.
A better way to argue against the claim is to try and demonstrate that it implies something contradictory or absurd, a tactic often called the reductio ad absurdum. This motivates rejecting the claim, because it is hard to believe that a claim flawed in this way can really be a suitable description of the world. Unlike the previous two ways to argue against a claim this one can actually stand by itself, and thus is the first contender for a serious counter-argument. However it has several failings that make it the least preferred way to argue against a claim. Firstly it is fully possible for those who accept the claim to simply bite the bullet and accept the absurd conclusion (assuming it isn’t a flat-out contradiction). And, secondly, it is easy to weaken claims so that particular contradictions don’t follow from them. Of course the upshot from all this is that either way of reacting to such an argument against the claim can strip it of much of its intuitive appeal, and so in this way such an argument may be indirectly effective.
Arguing that the claim has unacceptable consequences can be seen as arguing against the claim directly. But it is also possible to argue against the applications of the claim. To argue against the applications is to try and demonstrate that the claim does not solve the problems it sets out to, and that its explanations are lacking. Usually this entails giving a detailed analysis of how the claim is applied to various problems, and then showing that its explanations raise as many problems as they resolve, thus not leaving us in a better position than we were before, or by showing that all the claim is doing is labeling the situations it is applied to, but that it isn’t actually saying anything substantive (this would be like noticing an unknown force acting upon an object and reacting by giving that force a name and then treating the mystery as resolved). If successful this would make the claim pointless, and thus we would be pragmatically motivated to discard it. But, although this can be convincing to those yet undecided, supporters of the claim are unlikely to pay it much heed, as it is hard to convince someone that a position they like isn’t really saying anything at all.
Thus we come to the best way to argue against a philosophical claim, which is to show that some competing and mutually exclusive claim is superior. To do that entails showing that the second claim provides better and more substantive explanations than the first. And one way to do that is by showing that the second claim makes finer distinctions than the first, that it can speak about interesting and useful features which the first is necessarily silent on. And we can also attempt to show that its explanations leave fewer and less pressing unanswered questions than those of the other claim (we can always persist in asking why until every explanation has no more to say on the matter, but explanations will differ in where they stop and in how strongly we desire to answer that unanswered “why?”). This is the best way to argue against a philosophical claim because a successful argument against a claim via some other method leaves supporters of that claim with the ability to maintain that it is still the best claim about these matters. But showing that some other claim is superior necessarily places on supporters of that claim burden of improving it, and that is about the best outcome you can hope for.