This comment received a lot of backlash. Admittedly, my comment wasn't as diplomatic as it might've been nor did I elaborate on my own reasoning. In my defence, I didn't think the original article was making much of an effort to get at the truth (see other criticisms above). Rather, it is a (very) one-sided account advocating that we should not consider the other side of the story (ie, it is an attack on both-sidesism).

The attack on both-sidesism is consistent with findings referenced in the video below. Both sides are prone to such anti-democratic behaviour, but the findings also suggest that one side is "slightly more willing to sacrifice democracy (by supporting actions that benefit their own party at the expense of democracy)". This might be a case in point.

Feel free to watch the whole thing, but the tldr part starts at 4:05.
https://www.youtube.com/watch?v=PVqjH6MaqRY

The events of Nov 6/7, 2024 might support the argument that the original argument was indeed self-defeating; ie, an argument against the argument against both-sidesism---effectively, an argument for both-sidesism.

Good arguments support some claim. Bad arguments provide little/no support. Some arguments are so bad that they support the opposite claim. This is one of those.

I think the issue might be that I'm interpreting circular reasoning as something stronger than you; ie, in the pernicious sense which explains why "The idea that circular reasoning is bad is widespread".

I suspect that according to your interpretation all valid deductive reasoning is circular in some way, circularity thus being necessary for valid deductive reasoning. In this regard, circularity would be a desirable attribute.

In contrast, my interpretation is one in which in the process of affirming a belief, one presupposes something that would require to have already affirmed the same belief; what is sometimes called "begging the question".

In this context, I don't regard A |- A (circular in your sense, but not in mine) as problematic, as it just involves inferring something that has already been affirmed.

"Although I didn't think the distinction was so important to focus on in this essay." The distinction is important.

Do you agree that in logic, A |- A says something akin to "from a true A (a non-logical axiom), you may infer the truth of A"? If so, I don't see any circularity there; anything on the left hand side of the |- is an 'additional' axiom of your theory, hence accepted as true. Circular reasoning, by my interpretation, is where you infer/assume as true what you're in the process of trying to prove to be true. This is not the case with A |- A, in which the occurrence of A on the left of the |- means you are taking it as true to begin with. That's why I assumed you meant A -> A when you said "A implies A".

The law (A -> A) -> A would allow you to infer A from A -> A, which would, indeed, allow you to infer A without it having being established as true, doing so, rather, solely from the premise that if it were true, it would be true, thereby concluding that it is true. This is closer to the interpretation of circular reasoning I gave above.

"It also doesn't fit the geometric intuition behind 'circular' well. $A⊢A$ leads back around to where it started". I think this appeal to geometric intuition is misleading, which might explain the confusion. All A |- A it is saying is that if you already know/believe A, then you may infer it. In a formal system it may take several steps to derive ('lead to') A from A, but I don't think that circular reasoning should be tied to proof theory.

"The first important thing to note is that circular reasoning is logically valid. A implies A. If circular arguments are to be critiqued, it must be by some other standard than logical validity."

I challenge that premise: "A implies A" is not circular; this confuses a logical law with a valid deductive inference---reasoning, circular or otherwise, is about argument/inference. The logical law "A implies A" doesn't say that ones believes A; no position on the belief/truth of A is being made. This is different from having the belief A, and inferring A, which is just saying that you can infer what you already believe. What would be circular would be the law "(A implies A) implies A", as that would allow one to infer A from being able to infer A from A. That, however, isn't a logical law.