I've never found out the name of this fallacy, or even if there is a name for it, but it seems to me it's the fallacy that occurs by far the most often. You are having a debate with someone and then you make 6 crucial points...then the other person responds by picking only point #5 and trying to refute it.
More often than not their attempt to refute the single point they singled out was invalid anyway, but as it appears as if they "hit the ball back" to you to continue the rally, they actually didn't since they ignored almost all the points made. But still since it appears that way, at least to them, it comes back to you to somehow respond.
So to be clear it looks like this:
Person A's argument:
And then person B responds with:
5. argument point...
That's not true because....etc
Then it's back to person A who has to do what, repeat all his points again??
In my experience it's almost always the points they ignore that are most damaging to their whole argument, and the single point they cherry picked out to refute is the weakest and more simple to refute.
Add to that the points aren't normally numbered like they are here, that's just for clarity in this example. So it's particularly difficult to respond to.
It just gives them this opportunity to say in some form or another "no, you're wrong" in response to what you said. Which is powerful in itself but seems like a fallacy to me considering the way it's achieved.
EDIT: Since many people seem to misunderstand what I mean by argument point I'll try to explain a bit more. Imagine you're reviewing a product or service and you have to make a list of pros and cons or advantage and disadvantages. So you could claim a product is not great due to the cons.
- very expensive
- not waterproof
- takes too long to recharge
- customer support rude
So my fallacy would be that someone would claim this product is in fact good and would argue against these five disadvantages by stating "no you're wrong, it's not a bad product because it actually doesn't take that long to recharge". So regardless of whether what he said is true or not, he simply has ignored all the other cons that add up to a bad product. So even if true it's still a bad product based on the other cons.