An extended argument is one that proves as set of propositions from certain premises, and them from those propositions it proves further propositions, and so on. In essence an extended argument is a long argument that is constructed like a tower; the argument consists of multiple levels, each of which rests on the previous one. The extended argument seems to have fallen out of favor in recent times, perhaps because of the modern emphasis on analysis, although it does still make the occasional appearance.
However there are more than stylistic reasons to avoid the extended argument. Given any derivation of a conclusion from premises there is a certain percentage of people who will disagree with it. Even if you cast it in completely formal terms some will dispute your formalization. Often this percentage is small enough, usually, that it need not trouble us. After all, we can evaluate our philosophical theories by seeing how well they aid us in understanding their subject matter. If our philosophical theory is a good vehicle for understanding, meaning that it always provides a correct and clear explanation, as far as we can tell, then we can be fairly confident of its validity, even if the connection between the theory and simpler principles may be questioned by some. However, the extended argument is a sequence of these derivations, such that to doubt one step is to doubt the entire derivation. By the reasoning above we can conclude that there will always be at least some small percentage of people who will doubt each step of the derivation. And because to doubt the validity of one step is to doubt the validity of the entire thing there will thus be a substantial percentage who will doubt the entire thing, if the argument contains enough stages. And that is a problem because the sheer number of objections that will be raised by different people will make the theory seem suspicious, no matter how good of an explanation it is.
But we can’t simply abandon extended arguments, because sometimes what we want to demonstrate is complex enough that there is no straightforward demonstration of it from simple principles. In such a situation there are measures we can take to strengthen the argument. The most desirable course of action is to derive our desired conclusion from several independent extended arguments. Although many will doubt at least one step in one of the extended arguments for that conclusion fewer will doubt one step in each of the arguments for it. Thus the conclusion will seem more robust, and thus be more acceptable. But developing multiple extended arguments for the same conclusions may not be possible, possibly for reasons of space, or possibly because other arguments just don’t seem to present themselves. In that case the best tactic to take is to add extra support to each step of the extended argument. In addition to deriving each step from the previous ones an intuitive or commonsensical argument should be provided. Such additions attempt to show that the propositions being argued for in that step are a good fit for certain of our intuitions, or that denying them is in some way deeply unintuitive. Obviously such support is not a conclusive argument from them. However, it makes it more likely that someone who doubts the derivation of a single step will be inclined to think that the conclusions of that step are true anyways, or that they have some other derivation. Again, this makes the argument seem more robust.
Unfortunately the vulnerability of the extended argument to a single weak link in its chain of derivations is not its only problem. Extended arguments also run the risk of becoming so abstract that their conclusions are meaningless or useless. Consider an argument that starts with an analysis of some relatively intuitive and ordinary topics. It then goes on to develop certain technical terms to explain certain arrangements and properties of those ordinary topics. And then in the next stage it develops technical terms to talk about those technical terms, and so on. In the end we have claims about terms that are very close to being meaningless. In theory they have meaning, if we could track the argument backwards to understand what facts about ordinary objects they are really about. However few bother with such a study, including the author, because these technical terms are often deceptively named. For example, the author may define a technical term such as pleasure to describe an increase in a being’s perfection (actual Spinoza definition), and then go on to make claims about pleasure, and claims on the basis of those claims, and so on. And some point almost everyone loses track of what pleasure really means in this context and simply thinks of the later claims about pleasure as being about the experience that we usually think of as pleasure (which has little relation to this technical definition of pleasure).
Of course the solution to this problem is actually fairly simple. First refrain from using terms that already have a well-understood meaning as technical terms, or redefining them in any way, without at least adding a subscript of some other notation. And, in addition to this, it is helpful to apply the conclusions of the extended argument to certain problems or situations, in order to guarantee that they haven’t become abstracted away from any possible usefulness.
I myself am somewhat sympathetic to the extended argument. It seems reasonable to suppose that the more complicated the conclusion the longer the argument to establish it will have to be. And there is no reason, in principle, to restrict ourselves only to simple matters (assuming we understand the simple matters, and aren’t ignoring them). Of course I rarely get the opportunity to develop such an extended argument myself, given my space constraints.