A Fallibility of Logic

Ex falso, quodlibet — you can logically prove anything given a false premise.
Insofar as humans are inherently incapable of omniscience, is it not then impossible for us to be truly without false premises in our worldviews? Would this not mean that taking any of our necessarily imperfect worldviews to their logical ends would inevitably eventually result in insanities? If so, then logic is imperfect as a tool for humans, and we, paradoxically, must temper it with imperfect heuristics and other such things in order to prevent escalation to logical-but-insane conclusions.
Accordingly, we should caution ourselves against compounding logic upon logic, as each subsequent conclusion is likely to drift us further and further from Truth.