A simple way to see one aspect of the fragility of human reasoning is through joint probabilities.
P(1 .. n) = P(1) * P(2) * … P(n)
So, if I have an argument whose conclusion relies on 3 premises, and the argument is valid, then the chance that the argument is sound is:
probability (first premise being true) * probability (second premise being true) * probability (third premise being true)
(Assume independence of probabilities.)
Let’s say I’m ‘fairly sure’ about each premise, assigning each one a percentage of 80. That is, at each point along the argument, I’m fairly sure that it is right. At the end, I may then have the feeling that I should be fairly sure about the argument. Yet, this is not the case.
0.8 * 0.8 * 0.8 = 0.51, i.e., almost as likely to be unsound as sound.
Also see here.