Answer

Hello,
There is a practice in yeshivot to think that every verse teaches the opposite of what it says. After all, if what appears in the verse is the rule, then why do we need the verse? So clearly the rule is the opposite, and the verse comes to teach that there is an exceptional case to the rule. The joke about this is: “Whatever Sarah tells you, listen to her voice”—from here, one might conclude that a person should listen to his wife. But in yeshivot, of course, they learn the opposite: from here we see that one does not need to listen to his wife, and therefore we needed the verse to teach Abraham that he does.
Similarly, the Talmud at the beginning of Yevamot learns that a positive commandment overrides a prohibition from the case of wool and linen in tzitzit. There too one could have said that a positive commandment does not override a prohibition, and tzitzit involving wool and linen is an exception, but the Talmud derives from there a rule for the whole Torah.
 
As for your question, I am not familiar with any principled treatment of this problem, and my theory is as follows:
 
First, one must distinguish between extending from the case in the verse to the whole Torah—that is, creating a general rule—and an analogy from one specific case to another specific case. A binyan av from one verse is usually an analogy, and from two verses it is sometimes that and sometimes a generalization. To be sure, the boundary is not sharp, since an analogy to several specific cases is also an analogy. It seems to me that if we are dealing with several cases and not a principled rule, then it is still an analogy. So, for example, extending to all forbidden relatives seems to me to be a collection of analogies and not a generalization (each of the forbidden relatives is a case similar to the forbidden relative in the verse). By contrast, creating a rule about positive commandments that are time-bound is a clear generalization (because the similarity to the case before us lies in a very non-trivial parameter—dependence on time. In the language of logicians, this is what is called abduction. I discussed this a bit at the beginning of my fourth booklet on the site).
 
An analogy from the detail in the verse to another detail (or several details), that is, a binyan av from one verse, in my opinion is always made so long as there is similarity. The question is when one makes a generalization from the case in the verse into a broad rule. And here I have always thought that it probably depends on the question of what the starting point is: is this rule reasonable or not? To be sure, even here one could put it in two ways:
A.
1. If the rule is reasonable, then the verse is unnecessary, which implies that it specifically comes to limit it to this case alone and say that the rule is not correct. 2. If the rule is not reasonable, then the verse is not needed in order to single out the case it deals with, and the conclusion is that it comes to teach the rule.
But then something strange comes out: specifically illogical rules are learned from verses, while what is logical is rejected.
B.
1. If the rule is reasonable, then it makes sense to expand the verse and create from it a sweeping rule. 2. If the rule is not reasonable, then the verse is probably introducing a special law, and there is no reason to expand it beyond that (there is no proof that there is a general rule here. You have only its own novelty and nothing more).
 
So which of these approaches is correct? It seems to me that one must distinguish between different degrees of reasonableness. If the rule is not unreasonable, but we still would not have said it without the verse, then one should go with approach B. The verse comes to authorize us to formulate the general rule (because we would not have done so on the basis of reasoning alone). But if the rule is entirely reasonable, then if the verse comes to tell us to expand it, it is unnecessary, and therefore the required conclusion is that it comes specifically to limit it, in the manner of approach A.
 
All this obviously requires broader and deeper study. These are only preliminary lines of thought.
 
Goodbye,