The Argument from Fallacy is a common logical fallacy. It is the error of believing that because an argument contains a fallacy, the conclusion of the argument is therefore false.
A Fallacious Argument:
1. God is immortal
2. I am not God.
3. Therefore, I am not immortal.
Argument from Fallacy:
4. The previous argument committed the fallacy of denying the antecedent
5. Therefore the argument is invalid
6. Therefore the previous conclusion is false; you are immortal.
Obviously, this final conclusion, #6, does not follow from the premises. Moreover, the final conclusion is false. However, the fact that the conclusion does not follow has no bearing on the fact that it is false. It does, of course, indicate that we should disregard the argument and look for other evidence as to the truth value of statements #3 and #6.
This fallacy is very common in politics where very complex issues are debated as if there are only one or two important premises under consideration. Generally people who make this fallacy will accept that you are wrong if you own up to it, and allow you to create a valid argument (if you can). It is usually useless to point out that they have made a fallacy; they are only interested in attacking your argument. However, they will probably enjoy having another chance to argue with you, so keep forging ahead! It is important to be clear that you are dumping the initial argument and replacing it with another; otherwise you may open up the possibility of your conversation partner making the much more socially acceptable tu quoque fallacy, something to be avoided if at all possible.
It is important to note that calling out a fallacious argument is not in any way a fallacy; the fallacy only arises when you assume that because one argument proved invalid, you no longer have any reason to consider better arguments.
If you think that you have found an example of the argument from fallacy, be careful. The conclusion "this guy is not a reliable source; therefore I should not waste my time listening to him" is perfectly acceptable. The fallacy arises when you confuse "I have no evidence for X" and "X is not true".
This fallacy is also called the Argument to Logic, argumentum ad logicam, the Fallacy Fallacy, and the Fallacist's Fallacy.