When someone fails to be convinced by our arguments one is faced with a small dilemma. There is something irksome about such a response. Does that person mean that we have not excercised enough reason or feeling in pursing the point? Have we failed to live up to the other person's standards? And why should we feel the need to live up to such standards?
The language of persuasion is a difficult one to master. One cannot help but be moved by the religious or political martyrs of the past. It is difficult to doubt their convictions but that did not make them right.
Likewise, to have millions of people be persuaded by an argument does not make it right either. As the friends of Socrates (the philosophical martyr) were aware, democracy is not co-existent with 'Truth.'
What is irksome, is that in our everyday lives, we have to convince and persuade people. In job interviews, meetings with manangers, when pursing love interests..., we need to know how to persuade. The Ancient Greeks called this art of persuasion sophistry. How to speak in public, how to compose a convincing letter, how to get things done all rely upon this art.
However, there is something demeaning with this pursuit. It is a mark of slavery. To live a life were one does not need to persuade... now that is a life worth living!