On the Circular Logic in Arguments

So here is the scenario you are sitting at the dinner table. You are both quietly eating your meal while discussing what happened over the course of the day. When you slowly realize the statement just made is incorrect. So you decide to correct the individual when they fire back that in fact they didnt really say what you just corrected. Then they feel the need to tell you "exactly" what they did say, but what that is clearly not the truth.

The above scenario is wrong in two ways
1. It is clearly from the point of view of only one person meaning that it has several words and key phrases that are biased toward the party that corrected the original comment.
2. The other reason is this conversation is well on the way to a circular argument where both people are just gonna try and tell the other what they said when in reality there is no way to prove any of this.

The way to rectify the condition is to just take a step back and realize that this argument is going absolutely nowhere and if the conversation continues on this course both people will eventually end up pissed.
So if you find your self in, or creating, this situation, just be easy and let it go.