When I do presentations to groups of parents, I commonly ask "How many of you have kids that you would describe as argumentative?" Typically, more than two-thirds of the audience raise their hand (and I assume that there are others who are reluctant to acknowledge the problem publicly). Parents arguing with their children appears to be Occuring at epidemic levels.
The good news about the problem of argumentative children is that it is the easiest of all the child behavior problems to solve. It is easy for two reasons: 1) The child's cooperation is not required and 2) The parent doesn't have to do anything to solve the problem. Let's look at the dynamics of parent/child arguments:
Parents argue with their children because they labor under the delusion that they will convince them of the wisdom of their point of view, despite the fact that this has never happened. During an argument, children never say, "Gosh, Mom, you know I've never seen it from that perspective before and, having heard you, I will now abandon my point of view on this subject and adopt yours." Yet parents believe that the application of logic and rationality will sway their children.
For an argument to occur, three elements are required: a topic (of which there is never a shortage): an arguer (a role your child is extremely ready to play); and an arguee (the person--you--willing to engage the arguer in the nonproductive dispute). Take away any of these elements and there is, by definition, no argument.
So, the solution to the problem of argumentative children is very simple: simply refuse to be the arguee. If you remain silent, there cannot--it is impossible--for an argument to ensue. Problem solved. Keep your focus on your child's compliance while saying nothing.
Now, this is not to say that your children will be silent--they won't. But a child spewing unilaterally is a very different circumstance than a toe-to-toe donnybrook. And once you assume this stance repeatedly in the face of their attempts at arguing, your children will come to realize the futility of trying to hook their parents into a dispute.
So, the solution to argumentative children is exceedingly simple--place your bottom lip against your top lip and hold it there.