There's a (possibly apocryphal) story about Bertrand Russell giving a lecture on logic.
In the lecture, he mentions that in the sense of material implication, a false proposition implies any proposition.
A student raised their hand and said "In that case, given that '1 = 0', prove that you are the Pope."
Russell immediately replied, "Add 1 to both sides of the equation: then we have '2 = 1'. The set containing just me and the Pope has 2 members. But 2 = 1, so it has only 1 member, therefore I am the Pope."
Whether the story is true or not, aside from being funny, it suggests how precarious holding false ideas can be, especially when we don't realize the ideas are false. We probably won't assert '1 = 0' as a base axiom, but there are certainly other, less obvious, falsehoods that we unknowingly hold. In those cases, our lives play out much the same way as the story: predicated on misguided assertions, we accept absurd conclusions with full confidence. In the end this shows that proper premises are essential for arriving at truth, and that proper logic alone is not a sufficient condition for truth.
The truth table for an implication:
| A | B | A → B |
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | False |
Russell's anecdote is about the lower two rows: a false premise can have any conclusive value (either true or false) and the implication is still valid. This general observation leads to the issue we mentioned earlier. What's interesting, and often counter-intuitive, is that false premises and true conclusions can coexist at all within the same implication! One explanation is that a conclusion that is genuinely true can be true for reasons outside the contents of the implication. For example: If I eat an apple, then I will not be hungry. But if I'm not hungry it won't necessarily be because I ate an apple, it might be that I'm not hungry because I ate an orange.
These scenarios highlight, what I consider to be, some of the most common lapses in reasoning that people use or encounter. Either, through unknowingly accepting a false premise, they accept all sorts of false conclusions, or because they have a true conclusion, they believe it is because of an incorrectly held premise.
I can only speak in generalities about how to respond. We may not know precisely what false beliefs we currently hold, but unless we can identify them, we are in constant danger. The best antidote I can think of, which comes closest to a panacea is humility. If we are quick and willing to consider how we might be wrong, I do think we will find that a greater breadth of truth and wisdom becomes open to us.