As for non-scientific theories, for example moral ideas, we cannot use the scientific method to prove them wrong (since they don't make testable predictions). Instead, we look for contradictions to prove them wrong.
And we can also justify them to a satisfactory degree through sound reasoning and supporting evidence.
And those are mistakes. If a person has a justification for a theory, and since all ideas are fallible, that means that the justification could be wrong.
So what? All theories could be wrong. That's already the way of things. Nobody ought to be subscribing to a theory as a "true" thing in the first place. Theories don't really operate on truth, they operate on utility and explanatory power. Our models of understanding can only ever be approximations and should always be subject to revision. We don't have to second guess every move we make in the world, but certainly the more far reaching ideas should not be declared as true in anything other than a colloquial sense of the word truth. "Justification" isn't an absolute, perfect, unconditional concept. It's subjective, variable, often accepted impulsively.
And they are wrong. Whether or not they know it's wrong doesn't change that.
You don't get to set the standard as to what other people will accept as evidence or insist they must never be persuaded by evidence as a rule. You can't stop people being persuaded proactively by evidence and you don't get to dictate what other people deem justified.
You're welcome to look for contradictions as your sole method of navigating the world. I'm happy to accept or subscribe to certain things based on the compelling nature of them. Again, this is not matter for debate.
If you think it can work in the positive direction, then please explain a hypothetical example of how it could work.
For example, you could claim that you've said (typed) a thing, and if I doubted you, you could point to the comment in question as evidence (assuming you were correct and that the words actually existed).