Is it a fallacy to logically draw a conclusion from a false premise?

I read this and immediately thought of the difference between a sound argument and a valid argument. A sound argument is one where the conclusions logically derive from the premises. A valid argument is a sound argument where all the premises are true. Dr. Bennett's example of the Ohio potato flag is a sound argument. It is not a valid one.

I suppose the best way to counter that is what you suggested, reword the false premise to a true one, and see if the logic still holds.

It is a fallacy only if the falsehood of the premise is evident through poor reasoning rather than just a false statement. For example,

The capital of Ohio is Potato, therefore, it wouldn't be inappropriate to put a potato on the state flag.

The reasoning here is not bad; it's simply that the premise is false. Not knowing something is not the same as fallacious reasoning. This is why very smart and reasonable people draw some terrible conclusions—they simply are working with poor data.

To answer your other question, you wouldn't be countering their reasoning (because it would not be necessarily flawed), you would be correcting a mistake. Once the mistake is corrected (a new premise is constructed), the conclusion must be reconsidered in light of the new information. This is what is meant in science when it is said that all conclusions are provisional . Theories are constructed using solid reasoning, but sometimes the data these are based on turn out to be wrong (more often just incomplete). I think this science analogy is a good one to use (e.g., "Do you think it is wise for scientist to update their theories in light of new data? Wouldn't it therefore be wise for you to update your conclusion in light of a corrected premise?")