In the first draft of this post, I spent an inordinate amount of time explaining the concepts of "axioms", "propositions", "logical equivalence", and so on, only to come to the realization that the people who are actually going to care enough to read this post probably already know what all those terms mean anyway. So let's skip to the real meat of the point I'm trying to make.

What's up with the "circular reasoning" fallacy?

See, if you and I both accept a given axiom A, and the proposition I'm trying to prove happens to coincidentally be A, then yes I'm assuming the conclusion I'm trying to prove. But so what? The logical derivation is still valid. The proof just happens to be a "1 step proof". People who cry "circular reasoning" presumably reject the axiom A, but if so, why not just clearly say "I reject axiom A", rather than claiming that a logically valid proof is somehow fallacious?

A sub issue I wanted to get into comes from the fact that it's possible for two axioms to be equivalent, e.g. the parallel postulate of Euclidean geometry, and the axiom that the angles of a triangle add up to 180 degrees. If two propositions are equivalent, then they contain the exact same logical information (i.e. if you already know the truth value of one, then you learn nothing new if I tell you the truth value of the other), and so if we both accept the parallel postulate, and I am trying to prove that the angles of a triangle add up to 180 degrees, then I am employing circular reasoning again (and again, non-fallaciously): I'm assuming the conclusion I'm trying to prove, because by assuming that there exists exactly one parallel line through a point, I am implicitly assuming that the angles of a triangle add up to 180 degrees. The two propositions are equivalent. I might need to show, as a corollary, that the two are equivalent, but once you accept that, I've got my 1 step proof again.

Now here's my second point: If I am "telling" you something, i.e. I am giving you new information that you did not have before, then I am essentially giving you an axiom. If you ask me where Bob is, and I tell you "Oh, he's at the bowling alley.", then I am offering a new axiom "Bob is at the bowling alley" to our mutual belief/discussion-system. Maybe you'll reject the axiom (perhaps because given the other axioms already in the system, accepting this axiom would lead to a contradiction), but it's still an axiom, in the sense that I am not proving that it is true, merely asserting that it is true (or I believe that it is useful for it to be true for the purposes of our discussion).

So every time anyone ever answers a fact-based question, they are employing circular reasoning, in that they are trying to demonstrate that a given proposition ("Bob is at the bowling alley") is true, by introducing that same proposition as an axiom. It's a 1 step proof.

What then is the usefulness of the term "circular reasoning"? I believe the term still has value in that it can be used to explain the need to be more explicit about the set of axioms being assumed in a given discussion. "The angles of this particular triangle add up to 180, because the angles of all triangles always add up to 180." "Well, hold on a minute here, maybe we shouldn't take 'the angles of all triangles always add up to 180' as an axiom, less we run into circular reasoning."

What I'm really ranting against is the assertion that if an argument uses circular reasoning, then it is invalid. That assertion is false. The argument may be unsound (you disagree with the axioms used), but it's almost certainly valid, because it's hard to screw up a 1 step proof.

Unfortunately, there's probably too much stigma associated with circular reasoning that the moment you willingly admit that you are using it, your arguments will immediately be dismissed.