It surprises me how incredibly bad most people are at understanding even basic logic. I've always enjoyed logic because its an excellent example of creativity emerging from a deterministic system of rules and premises. Logical arguments are beautiful; even ones that I don't end up agreeing with are often fascinating to me. Take the ontological argument, devised by Anselm, for the existence of God (from Curtis Brown's website at Trinity University):
1. God is something than which nothing greater can be conceived. (definition of "God")
2. If someone understands the concept of God (i.e. the concept of something than which nothing greater can be conceived) then God "exists in the understanding" of that person. (definition of "exists in the understanding")
3. It is greater to exist in reality than in the understanding alone. (More precisely: if x exists in the understanding but not in reality, and y is exactly like x except that y also exists in reality, then y is greater than x.)
4. The fool understands the concept of God (= the concept of something than which nothing greater can be conceived).
5. Therefore (from 2 and 4) God exists in the understanding of the fool.
6. Suppose for the sake of argument that God exists only in the understanding of the fool (i.e. not in reality as well). (This assumption will form the basis of a reductio ad absurdum.)
7. Then we could conceive of something exactly like what exists in the fool's understanding except that it also exists in reality.
8. The entity that we conceived in 7 would be greater than the entity that exists only in the fool's understanding (by 3)
9. But in that case what the fool conceived was not after all something than which nothing greater can be conceived (after all, we've just conceived of something greater).
10. So we have a contradiction! (Between 5 and 9)
11. So the assumption we made in 6 must be mistaken (since it led to a contradiction).
12. So God exists in reality. (6 was the assumption that God does not exist in reality; since 6 is mistaken, God does exist in reality.)
As mistaken as I think this argument is (for a number of reasons I won't get into, because plenty of philosophers have already pointed them out), its still an impressive argument. To come up with this argument took incredible creativity on the part of Anselm. And while I do think it is mistaken, it is quite difficult, when first presented with this argument, to determine exactly where the problem lies.
Most of us are introduced to logic in math classes, when we first learn to construct truth tables, and then go on to prove things such as the congruency of two triangle. Unfortunately, most of us never learn what other practical benefits logic has in the real world. I suspect that if more people understood and could apply even a little bit of logic to the real world, clear, critical thinking would also be more prevalent.