Sometimes we try to analyze a concept as a conjunction of two or more concepts. Thus, we might say that x knows p provided p is true and x justifiably believes p. Frequently, such proposed analyses founder on counterexamples—Gettier examples in this case.
I want to highlight one kind of failure. Sometimes analyzing x's being an F in terms of x's being a G and x's being an H, fails because to be an F, not only does x have to be a G and an H, but x's Gness and Hness have to be appropriately connected. While Gness and Hness are ingredients in Fness, their interconnection matters, just as one doesn't simply specify an organic compound by listing the number of atoms of each type in the compound, but one must also specify their interconnection.
I suspect this kind of connection-failure of conjunctive definitions is common. One way to see what is wrong with the justified true belief analysis of knowledge is to note that there has to be a connection between the justification and the truth and the belief. Specifying what the connection has to be like is hard (that is my understatement of the week).
Here's another case of the same sort. Suppose we say that an action is a murder provided it is a killing and morally wrong. Then we have a counterexample. Igor, who used to be a KGB assassin, has turned over a new leaf. As part of his turning over a new leaf, he has promised his wife that, no matter what, he will never kill again, no matter what. Maybe in ordinary cases that promise would be inappropriate. But given Igor's life history, it is quite appropriate. Now, Tatyana has just mugged Igor and is about to stab him to death so as not to leave any witnesses. Igor picks up a rock and kills her in self-defense. What he has done was a killing and it was morally wrong—it was the breaking of a promise. But it wasn't a murder because the connection between the fact that the action was a killing and the fact that the action was morally wrong wasn't of the right sort. (One might try to say that it was a killing and immoral, but wasn't immoral qua killing.)
When we hear a conjunctive analysis being given in philosophy, I think it's time to look for a connection-counterexample, a case where each conjunct is satisfied, but the satisfaction of the conjuncts lacks the right kind of interconnection. Sometimes, I think, one can intuitively tell that a proposed analysis is unsatisfactory for lack of such interconnection even without coming up with a counterexample. Here is a case in point. Consider the notion of "causal necessitation". A natural-sounding definition is this: an event E causally necessitates an event F provided that (i) it is nomically necessary that if E holds, then F holds; and (ii) E causes F. But even if it turns out that this is a correct characterization—that necessarily E causally necessitates F if and only if (i) and (ii) hold—I don't think it's a good definition. For it misses out the fact that one wants a connection between the necessitating and the causing—the co-presence of the two factors shouldn't be merely coincidental. But it's really hard to come up with an uncontroversial case where we have a difference between the two. (Interestingly, it may be possible to do so if Molinism is true.)
We are rightly suspicious of disjunctive analyses. I think we should have a similar, though weaker, suspicion of conjunctive ones.
There is a structural connection between the points in this post and Aristotle's Metaphysics H6. The point is also similar to Geach's discussion of the good. We cannot define a "good basketball player" as someone who is (i) good and (ii) a basketball player.