A lot of people I've met in my lifetime often will espouse a belief without giving me reason why they believe it. These conversations always revolve around to the concept of proof.
What is proof?
Understand that when you want to convince someone of your position you must be able to provide physical examples, repeatable experiments, or in the absence of such evidence provide a reasonable well thought out and logical conclusion as to the nature of your statement, this is proof.
Statements that begin with "I feel..." or "In my opinion..." are not proof and never will be.
If you have to look at your "proof" through a certain lens, i.e. the Christian perspective, then that is not proof and never will be.
A lot of people believing in something is also not proof and never will be.
In a world where misinformation is spread on a daily basis we must strive for truth. We must not accept truth without proof. To do otherwise is to demean yourself and waste the brain you were born with.
Don't be an idiot, think things through guys.
It would be nice if we lived in a world where every belief was arrived at through a bayesian-like probability assessment, but that's not the way the human brain works. The human brain does not need proof to form a belief, it needs justification. Justification is much easier, because proof is only one kind of justification. The human brain is hard wired to be inclined to accept all sorts of other justifications for its beliefs, and it's very well equipped to hold on to them. Perhaps one of the most spectacular things about the higher consciousness which arises from our brains is its capacity to bullshit itself. The number and nature of the psychological tools the brain has available to fool itself is staggering. Some of them can be turned off by being careful, thorough and vigilant about our thinking, but most of them can only be mitigated to some degree.
Research backs up something that magicians have been saying for long, long time; the smarter an audience is, the easier they are to fool. This also holds true for tricks of cognition. The smarter and more capable the human brain is, the greater its capacity to lie to itself. That's because smart people can come up with really smart sounding justifications for their beliefs. And, because these justification seem superior to the justifications they hear dumb people offer, they assume they are right.
So, when we are trying to get people to rely more on intellectual rigor rather than gut feelings and prejudices, we have to be careful that we are not simply spurring them to come up with better lies to tell themselves. That is, of course, easier said than done. Cognitive dissonance is always there, and it's not easy (perhaps impossible) to construct a message that will not set it off to some degree. But we know the big road blocks to avoid. Indignation, disgust and fear are the three biggest buttons that can be pushed to suppress intellectual thought. (that's why these are favorite tools of politicians, anti-vaxxers, creationists and the like. Because they can have the effect of convincing someone of something while side-stepping logic and rationality)
Obviously when we are confronting someone's beliefs we probably aren't going to use themes of disgust or fear, but it's hard to avoid spurring indignation. One sure way to make someone indignant is to insult them. For example, if we call someone an idiot for holding a belief, we present them with a choice. Either I can give up my belief and accept I am an idiot, or I can hold on to my belief and keep thinking I'm smart. It's not hard to see which choice our subconscious is likely to make. It's best not to frame things in this way.
Beliefs are principles of action. We have beliefs systems because we are capable of movement. Plants and trees have no belief system because they cannot move. Without movement, the decisions we need to make are simple. They can be guided by nothing more than instinct, which is really just a result of past trial and error. There isn't much involved when a plant decides to switch from veg to flower other than the fact that its genes come from ancestors that did it, and it worked.
But, once something is capable of movement, the decisions get more complicated. While instinct can still go a long way in informing those decisions, we needed something more. We needed a way to base decisions on many factors. Those factors include things like past experiences, feelings, intuition, and knowledge. Because reality is the thing we are moving through, those belief systems which most accurately reflect reality will be the most effective at helping us navigate. For a long time our knowledge was very limited, and so we evolved to rely on the other tools we have. Of those tools, personal experience was the most capable of giving us an accurate picture of reality, and so we evolved to rely on it most of all. At some point, our ability to gather knowledge led to breakthroughs in thinking which allowed us to gather knowledge faster and faster. We learned to outsource memory via the written word, and pass down knowledge to future generations. We eventually learned how to create tools, like the scientific method, which allowed us to quality-check our knowledge and refine it to ever higher degrees of accuracy. Unfortunately these leaps happened rather quickly in terms of evolution, and so our default mode of forming beliefs is to still rely heavily on things like personal experience and gut feelings.
So, if you go around expecting everyone to examine and prove their beliefs you're not going to make much of a difference. It's not something people can do just because they were told, and they may not be capable of being honest with themselves using the default mental processes they've used all their lives. If you truly want them to think more about things, you'll have to get them to see the benefit in doing so, and give them the tools they need to do it correctly. You'll need to minimize defensiveness and cognitive dissonance, and encourage the basic principles of skepticism and critical thinking.