When your 'arguments' consist of making fun of the opposition and relying on logically flawed quips then you become biased, even if you reached your original opinion based on objective analysis.

TBH, I feel like this is a logical impossibility. If your "argument" is making fun of people or making logically flawed quips, then you didn't "reach your original opinion based on objective analysis." If you reached the opinion based on objective analysis, then how could your arguments be either of these?

You arent biased if you came to a conclusion justly, but then present your point in unjust ways, you're just kind of being a douche. Your "view" is still the same, regardless of what you say.

For example, lets say I reason the following: If P then Q, P is true, therefore Q is true. But I go and talk to someone who says that not-Q is true. I say to him, smugly, "Heh, anyone who believes not-Q is just a [bad thing]." Here, the I am not being biased in my opinion, I am just being a douche.

I feel like I'm not being clear enough so here is what I mean in short: the validity of an argument and the way you present it are two seperate issues. I suppose you could use the term "bias" to indicate variance in either. PunkRockMakesMeSmile is talking about "bias" in his position; he is saying there is no bias in the validity of his argument. You are talking about something elese, "bias" in his presentation, which is a seperate issue which, finally, I think is more rightly attributed to "douchebaginess" than "bias."