Deaf Ears

"He who believes a lie will deny the truth even if it is screamed into his ear"

While that may sound a little melodramatic, it's not far from the truth in reality.  If someone believes something, it can be very hard to invalidate that belief.  Putting aside the debate of whether or not it is right to try in the first place let's just consider the scenario where someone holds a belief about you that is evidently untrue.  You might argue that you wouldn't care what they think, that what other people think of you is none of your business, and while that in itself is another debate we can at least agree we care what the people we love think about us.  For that reason sometimes you do care what someone else thinks of you, and a false belief can make you uncomfortable, or even cause you great upset.

How do you convince someone you are telling the truth when they believe a lie?  That's not an easy question to answer because the answer relies on a number of variables that aren't easy to quantify, not least of all the level of trust the other person has in you.  You can ask someone what they are thinking and they can tell you, but that relation is only ever a second hand experience.  You can't experience their thoughts and feelings for yourself, you are entirely reliant on their ability to accurately communicate their mental state.  Communication of things which are often mired in illogical conclusions such as the nature of emotion, can be incredibly hard to do.  When you have no other reason for feeling something other than "I just do" - the inability to express a reason for that feeling does not magically negate it.  You still feel that emotion even when you have absolutely no idea why, and even more so, when you know it's entirely illogical, or the opposite of what you should be feeling.

"How do you convince someone you are telling the truth when they believe a lie" - needs an addendum - "and how can you be sure you've actually convinced them when they say you have?" - as this flips the issue of trust back on ourselves.  Do they trust us, becomes, do I trust them?

Fundamentally a relationship without trust will never build upon itself.  It will crumble when pressure is applied.  A strong foundation is needed for a relationship to last, and trust is the cornerstone of a strong foundation.  Communication can build a relationship up, but if there is no trust, then what is communicated will be lost as those words fall on deaf ears.  That does lead to the rather interesting conclusion that perhaps trust can actually be quantified by looking at exactly how much someone listens to you.  If they take everything in that you say then they trust you completely.  If they don't listen to a word you say and everything goes in one ear and out the other then arguably they don't trust you at all.