Schopenhauer strikingly noted that the causal law was not a hired cab. You can't accept it and take it only as far as you want; once you've accepted it, you are committed to it all the way down the line. The same is true of rejecting a claim as accepting it; if you reject a claim by rejecting premise A, you have, as it were, committed yourself to never appealing to premise A, and have to accept whatever consequences there might be to it. This is one reason why it is simply an error for a philosopher to focus only on single arguments at a time. (It's fine to focus on a single argument at a time; it's an error only to do this.) Your rejection of a causal principle in natural theology may cause extraordinary complications in philosophy of science; your rejection of a premise in aesthetics might throw into confusion entire realms of political philosophy; your acceptance of an argument in ethics might commit you to rejecting any number of political views.
So how does this work? There seem to be two basic routes, that work rather differently:
(1) Logical Implication: I've been in three discussions in the past year, with nonphilosophers, in which I argued that their claim A logically implied claim B, and their only response to this was that they didn't say B, only A. If one claims that A implies B, one should certainly be willing to back this up with argument; but it's quite clear that you can't avoid logical implications simply by not stating them. But, of course, there is also the fact that our knowledge of the implications of our claims does not have closure: we don't ever know all the logical implications of what we say, and may be logically committed to something that we would consider absolutely unacceptable.
(2) Parallel: There's a somewhat trickier route, arising because of the relations between different classes of arguments. We might describe it as one argument raising the question of another argument's viability. If one accepts a particular other-minds argument for the human body, this raises the question of whether one should accept some analogue for the whole world. If you accept a particular position on what makes literature worthwhile, this might raise the question of whether you should incorporate an analogous analogous position into your account of what makes music worthwhile. Unlike logical implication, it provides no guarantee; you can take one without the other. But because the two cases resemble each other in important points, you would, as we say, need to have a principled reason for not treating them similarly. This is, indeed, precisely the principle that seems operative here: We should treat similar things similarly unless there is a difference relevant to the treatment itself.
This would seem to be exhaustive -- if A leads to B, it seems to do so by either requiring it or by suggesting it -- but I have no definite proof of that; possibly there are more complicated ways positions and arguments can be linked.