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I would say their logic is at fault, not that the arguments for lack logic.
It seems that the primary reason people refuse is mistrust ... and logic rarely figures once that's set in.
Waiting to see what the side-effects are isn't illogical. That's how science is done. There's just a lot of money in manufacturing and selling the vaccine right now, so the potential for negative side-effects is being dismissed by pharmaceutical-paid scientists.
That's not really mistrust. This is actually very common in science. It's one of the reasons why it's important to be scientifically literate, know who is funding what studies, and to read the studies yourself rather than rely on "science communicators."
Waiting to see what the side-effects actually are is common practice. So is large corporations downplaying the risks of new inventions to the public in order to make a quick buck. Mistrust here is completely rational and should be expected. Not having any mistrust seems to border on gullibility, like expecting a conservative news blog to accurately describe the beliefs of Left Libertarians.
Again, in science, the only way to know if something works and what its side-effects are is to test it and see; that's called empiricism. Science also has a long history of being distorted by large corporations selling particular products. Look at all of the research Exxon-Mobile funded to deny climate change, for instance. The first thing they taught me about science in 6th grade was to read the studies and not listen to these sort of pundits, and this has only been reinforced more as I'm pursuing my PhD.
It would be truly illogical to believe that the new vaccine is risk-free, or that it poses less of a risk than the virus, because we can't support either of these beliefs until we've seen the long-term effects of the vaccine.
I say all of this with a full belief that people should still get vaccinated and that businesses like restaurants have a responsibility to not allow unvaccinated anti-maskers into their establishments. It's forced vaccination that I'm opposed to, partially because the risk of serious long-term side-effects has not been ruled out; for all we know, getting the vaccine could be worse than getting COVID, especially in people under 50. That's the truly logical and scientific perspective.