Rationality

Is there any way to prove that you are telling the truth when it seems false to others?

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My answer is bound to disappoint, but here goes anyway.

The obvious options for proving that I'm telling the truth are 1) to give reasons for thinking what I say is actually true, 2) to give reasons for thinking that I'm honest and 3) to give people a basis for doubting their own reasons for doubting me.

1) The best way to prove that you're telling the truth is to give people good reasons to believe that what you're saying is actually true. Unfortunately, in some cases this is really hard. Suppose I really did hear John tell Mary that he planned to break into Sam's computer. That might really have happened, and I might have heard it. But I might not have any independent way of showing that John and Mary really had this conversation, and if it's my word against theirs, there's not a lot that I can do.

2) I might be able to provide evidence that I'm generally honest, and that I don't have any special motive for lying about John. That would help my case indirectly. It would tend to show that I'm not deliberately lying. But even if I convince people that I'm honest and that I'm trying to report things as they happened, there would still be room for doubt. Maybe I misheard what John said. Or, if the issue is whether John was planning to do something illegal, maybe I missed some important bit of context that would give John's words a different meaning.

3) I might be able to give reasons for doubting John's honesty. But while that clears some of the obstacles to believing me, it's not the same as showing that what I've said is actually true.

So there are various things you could do that might help you make your case, but they'll depend on the circumstances. There's no one way that fits all cases, and there's very often no foolproof way in any case. If there were, legal trials would be a lot easier.

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