Can Claims Without Rational Evidence Be Disproven?

Depends on your worldview. In pure rationalism—based on the principle of sufficient reason—everything can be proven or disproven. It is just a question of how clever you are by inventing premises in support for your conclusion. But this is a dead horse, although many people still try to ride it.

Logical Disproof and the Münchhausentrilemma

The only way to disprove a claim is to show that there is an internal logical contradiction in that claim. If you can’t do that, because there is no logical contradiction involved, basically you're stuck. Pure rationalism, ironically, has to violate its own principle. If you try to give a reason for something, you either end up with an infinite regress, which is impossible, or start with a vicious logical circle, or you have to arbitrarily break up the reasoning and start with some dogmas. This trilemma is called the Münchhausentrilemma.

Trying to provide a reason for stopping reasoning will lead to another start of a further trilemma, which goes on and on. You can judge the quality of a worldview by observing how well it can handle the trilemma. Most worldviews are subpar.

Examples and Illustrations

To illustrate, the theist worldview often uses God as a way to stop reasoning, resorting to a dogma. It cannot provide a reason for stopping at that point. Evidence points to facts or uses facts about the world to draw conclusions. If there are no facts, you might think about falling back into pure rationalism to create new conclusions that are disconnected from all facts. This means you can use the same reasoning to come to the opposite conclusion. Thus, neither proving nor disproving it is conclusive. So, yes, you can disprove it seemingly but at the same time you can't. Which means you cannot disprove it. It is fruitless to disprove it with a way of thinking that leads to the problem.

Principles and Solutions

The simple solution is: If you can neither prove nor disprove a claim with known facts, you cannot know if it is true or not. You might show that it is nonsensical, which is far more valuable than being "just false." If you can show that the claim cannot collide with any facts, known or unknown, you know that the claim does not have any meaning or any logical connection to the world. Your only way out is to use claims that can be disproven by evidence. If you don't, the claim can be rejected on the basis that it does not have a logical connection to facts about the world, and on equal grounds, the claim can be true, untrue, or meaningless. Because most claims are meaningless anyway, they are probably nonsense.

Hitchens' Razor and Volker's Razor

Hitchens' razor states that what is asserted without proof can be rejected without proof. This is a principle and it is a logical fallacy category error notorious hard to find to ask for proof for that and reject it based on the fact that it cannot be proven. Principles have to stand the test of time, and that means you can't trace them directly to facts about the world.

I would use a sharper version of this and introduce you to Volker's razor: If you cannot provide possible knowable facts that would disprove your claim, your claim might be worse than just being false—it might be nonsense. If your claim is to be taken seriously, provide the facts that support it. Provide some possible knowable facts that might contradict your claim. Because claims cannot be proven, they can be falsified (disproven). Only claims that make it harder to falsify something should raise red flags. Looking only for facts that support your claim leads to confirmation bias, a diabolical trap for thinking. Treat all evidence as preliminary. If someone comes up with a claim that cannot be refuted, treat it as it deserves to be treated—as nonsense. These are principles that only science adheres to if not, it is bad science. Any assertion that is not based on facts is likely to be unrelated to the world of facts.

Conclusion

Your approach to claims and evidence determines how you engage with the world. By adhering to principles like Hitchens' razor and Volker's razor, you can ensure that your reasoning is sound and your claims are meaningful. The next time you encounter a claim without proper evidence, remember to approach it with the mindset of a skeptic, always questioning and seeking verifiable facts.