Our evaluation of the logical strength of an argument tends to be biased by the believability of the conclusion, rather than how well the argument supports the conclusion. Be careful not to over- or undervalue an argument based on how strongly you may wish to believe the proffered conclusion.
"Fear, or the perception of risk, is subjective. It's a matter of how we feel about the facts we have, not just what the facts say." David Ropeik, How Risky Is It Really? Why Our Fears Don't Always Match the Facts".
This quote appeared in an op-ed piece this morning discussing the currently trending topic of vaccinations. I should disclose that I have strong personal opinions about the dangers posed by the voluntarily unvaccinated. But I also read another piece just yesterday that reported on a study conducted at Dartmouth College, in which "vaccine skeptics" were presented with four different messages in an attempt to determine whether any of them would change their minds. (These messages ranged from a fact-heavy presentation correcting misinformation regarding autism risks to images of the effects of preventable diseases.) None of them did; in fact, they resulted in parents becoming more entrenched in their existing positions.
When it comes to closely held beliefs - beliefs that impact someone's moral compass, their physical safety, their children's well-being, and the like - facts alone, no matter how convincing we may think they are, are not enough. In fact, using facts to disconfirm closely held beliefs tends to lead to the opposite effect, in a psychological phenomenon known as "cognitive dissonance". Humans strive for internal consistency; when confronted with inconsistency, we become psychologically uncomfortable and attempt to reduce this dissonance, by further affirming our beliefs and by actively avoiding both situations and information which are likely to increase it.
This is as true for vaccine skeptics as it is for religious and political beliefs, beliefs about parenting styles, food, and climate change, to name just a few.
So where does that leave us, if someone with whom we are in disagreement "just won't listen to reason"? As counter-intuitive as it may seem, throwing more facts at them in an attempt to disprove their position is likely to lead only to frustration for everyone involved. Instead, try to listen. Listen to the other person's fears and the underlying interests that give rise to those fears. Try to show genuine empathy, even if you vehemently disagree. Because often, empathy - and the relief of the other person at finally being heard - can lead to trust and respect... and maybe even the possibility of changing someone's mind.