We believe that there's always something gained and something lost in an arguement. However, when the time comes that we think our gf decides she's not going to let something go or try to help us work things out, that is the time we shut down and become silent and sometimes.

There are always 2 sides to and arguement or disagreement. It is beneficial for both parties to talk things out when they can in an orderly fashion. Sometimes even if you feel you aren't in the wrong, you need to swallow your pride for a moment and see where things are going. Sometimes the other party feels more deeply hurt that what once was thought. Something like this may never surface if one part of the relationship won't bow out for a minute or so.
It takes patience, yes, but any relationship worth fighting for deserves some type of patience on both sides, even internet relationships. Sometimes that patience comes from letting something go till things calm or both sides agree on something.