Elementary logic is arguably the most basic ingredient to fruitful argumentation. Nevertheless, you don't have to look far in the world wide web to figure most people have one or the other problem with it. I blame English grammar for it.
For example, in classical logic a double negation of a statement is equivalent to the statement itself. Ergo, "We can visit my parents," means the same as "We can not not visit my parents," which however means we have to visit them. Thus, we have to visit my parents whenever we can. But what's even better is that "I could care less" actually means "I couldn't care less." According to that logic "Yes, we can!" means the same as "Yes, we can't!" This clearly has extraordinary explanatory power when it comes to American politics. And let's not even get started on issues like boxing rings that are actually square, and that one fills in a form by filling it out, and so on.
Another logic relation that people often stumble over is that "From A follows B" is equivalent to "From not B follows not A" and not to "From not A follows not B," which would be equivalent to "From B follows A." Consequently, if you hypothesize that "All of reality is mathematics" it does not follow "All of mathematics is real." Neither does "All of mathematics is real" mean that "All what's real is mathematics."
Then, let's have a look at what it means for an argument to be circular. A circular argument is not necessarily wrong, it just doesn't have explanatory power. Consider you ask an alcoholic why he drinks: "I drink to forget," he tells you. "Well, what do you want to forget?" you ask. "That I drink." We laugh about that exactly because it doesn't explain anything. You could just as well not drink and not forget that you don't drink. However, there is nothing logically wrong with his explanation. Circular arguments are very common mistakes in proofs, when you accidentally make an assumption that already implies the outcome. For example, if you want to show free will exists, you better not use free will as an assumption in your argument.
Finally, let me expand on one of the few enlightened comments to our recent post, made by Neil B on what it means for something to be tautologically true: "if we thought everything that could be logically inferred directly was superfluous to state, then the entire body of what is analytically derivable from given evidence should just remain unsaid." In fact the value of logical conclusion is subjective. If it follows from the assumptions (or axioms) the question whether or not you find something "tautologially true" depends on how difficult it is for you to understand the conclusion. Given the standard model lagrangian, next-to-next-to-next-to leading order contributions to the top quark pair production are at least to me not obvious. Given the Maxwell equations, the continuity equation is "tautologically true." But that might not be obvious to everybody. Or maybe it might not not be not obvious. Now I'm confused. No, I'm not. Wait....