@lextenebris Wow, thanks a lot for this contribution to the topic! I have some comments:
scientifically, we know that any study can be false, and that any study has a pretty good chance of being faults, and that truth is discovered by multiple studies, reproduced in multiple places, by multiple groups of scientists – and anything less is not truth, it's just lower on the spectrum than pure lie.
I agree completely that the ability to being reproducible is what makes a study or finding solid and reliable, however I wouldn't go as far as to say that anything that hasn't been replicated many times is slightly better than a lie. It might well be true but lacks robustness and scrutiny in order to be taken as a generally applicable principle (or maybe this is what you meant?).
Try to communicate that to the True Believers and you're going to get blank, glassy stares, and the persistent belief that you are probably some sort of hideous agent of evil sent to corrupt their minds with doubt.
This is what I find most difficult dealing with: when people are shutting you down a priori because they decided that they don't trust anything that comes from science, then trying to reason and explain the evidence at hand becomes an impossible task.
Short of that, the only thing I can think of is to keep plodding onward, keep trying to get a deeper understanding of how science works in front of people, show them that it works, show them why it works, show them that the continuous chain of it working has led to lives which are neither nasty, brutish nor short compared to our forbearers.
Personally I don't know what the best way to deal with this recalcitrant attitude could be. Maybe your approach (which requires a lot of patience) could help to plant some seeds of real knowledge, although in my experience people like this would to anything to avoid admitting they might be wrong even if they start having doubts.
If it walks like a duck, and it talks like a duck – you are probably not best served by making decisions as if it were a chicken.
Something hasn't been replicated many times, it is structurally indistinguishable from a lie. It might be true, just as any live might be true, but without verifiability and someone taking the steps to verify it, a clever and wise person should never use that information for judgment.
This is a much higher bar than "generally applicable principle." It is indistinguishable from a lie. This is far more insidious if it happens to be something that you want to believe.
That's the trap that a lot of these people fall into.
I have simply taken to accepting that people who don't trust anything that comes from science have tacitly agreed that I can take from them anything that comes from science. If it doesn't exist, or it's an agency of evil, or it's an invalid way to make decisions, then clearly I'm protecting them and they should support me in protecting them – by taking their computers, their cell phones, their health, their primary, secondary, and tertiary means of socialization, the ability to move further than a swift jog…
You know, things. Derived from science. I'm helping!
Well, there is a real and true alternative.
We could kill all of them.
It's not like they would have particularly good weapons. Comparatively. Once you toss scientific thought, your weapons development strategy goes right out the window. I'm pretty sure we could take them.
That is a lot of work, however, so I am generally disinclined to bother. In the long run, their belief system is effectively self-defeating because it demands that they take inefficient options at every turn. So ultimately, science wins. In the short term, we just have to learn how to mock them openly and without hesitation, twisting the knife whenever possible. If nothing we say can really make a difference, enjoy the performance art.