The problem with the situation you describe, Knigel, is not that saying, “If you believe that 2 + 2 = 5, then you are an idiot” is a logical fallacy. Clearly it’s not. The problem is that it’s a bad rhetorical strategy. Calling someone an idiot is generally likely to make it harder to persuade them that they are incorrect in their arguments.

Somebody is going to say, “But what if you’re not trying to persuade that person? What if you’re trying to persuade some other group of people that your argument is correct? Is it then O.K. to call the person claiming that 2 + 2 = 5 an idiot?”. First, surely there’s no real situation in which you would be debating something as trivial as what 2 + 2 equals. I’ve never seen a public debate organized in which one sides defends the idea that 2 + 2 = 4 and the other side defends the idea that 2 + 2 = 5. There don’t even seem to be any private discussions in which people would argue about this. The only case in which someone would discuss what the result of a mathematical calculation is would be when the answer is truly hard to understand. For instance, on the SDMB we have occasionally discussed why it’s true that .9999999… equals 1. This is actually rather hard to understand without the proper mathematical background. You have to understand what that mathematical notation means, and that takes some careful exposition.

Suppose then that you are trying to explain a rather tricky mathematical statement to someone else. If they didn’t understand it, you would go back and try to give them the mathematical background to understand it. If everything you tried failed to make them understand it, you would just say, “I’m sorry. I think you don’t understand this mathematical statement, but I can’t think of any further way to explain it.” Telling them that they are an idiot is giving up and insulting them. At that point they will cease to listen to any further arguments that you make. Furthermore, anyone listening into your discussion might say to you, “Look, I know you understand math better than the other person does, but that’s no reason to insult them.”

This is why I think it’s a bad rhetorical strategy to call someone an idiot, since it justs causes them to cease to listen to you.