If people don't reason in a Bayesian way, but they do reason, it implies there is a non-Bayesian way to reason which works (at least a fair amount, e.g. we managed to build computers and space ships).
There is. That does not mean that it is without error, or that errors are not errors. A&B is, everywhere and always, no more likely than A. Any method of concluding otherwise is wrong. If the form of reasoning that Popper advocates endorses this error, it is wrong.
Someone told me that humans do and must think in a bayesian way at some level b/c it's the only way that works.
Whoever that was is wrong.
Here's someone saying it again by quoting Yudkowsky saying it:
http://lesswrong.com/lw/56e/do_people_think_in_a_bayesian_or_popperian_way/3w7o
No doubt Yudkowsky is wrong, as you say.