Fallacy of the many arguments
The fallacy of the many arguments is a formal fallacy whereby two or more arguments are given for the same conclusion, but the conclusion is considered false or ungrounded when only one of the arguments has been shown to be unsound.
This is a graver version of the ad logicam fallacy, whereby a conclusion is considered false because the argument provided has been shown to be unsound. However, in the ad logicam fallacy, it would be reasonable to infer that if the only argument for the conclusion is unsound, then the conclusion is ungrounded, so far as no other arguments are given. But on the many arguments fallacy, inferring that the conclusion is false or ungrounded would be faulty reasoning, given that other arguments have been given for said conclusion.
An all too frequent real-life example is when someone is exposed to several arguments for going vegan, finds a flaw in one of them (or believes to have found a flaw) and therefore rejects the conclusion.
After noticing this fallacy, I found that it's actually more effective to give just one argument (the strongest) rather than giving many arguments. The reason is that if many are given, the listener will probably attack the weakest, and if successful, consider the conclusion false or ungrounded. So giving only the strongest argument works best.
However, a friend later told me that the classic rhetorical strategy is to give four arguments (no more, no less). The first must be your second strongest, to stun the listener. Then, while the effect lasts, quickly slip the two weakest arguments, and finally deliver the strongest, for the final blow. That's rhetoric for you, as opposed to logic.