To get a baseline this is how Wikipedia describes circular reasoning:
Circular reasoning (Latin: circulus in probando, "circle in proving"; also known as circular logic) is a logical fallacy in which the reasoner begins with what they are trying to end with. The components of a circular argument are often logically valid because if the premises are true, the conclusion must be true. Circular reasoning is not a formal logical fallacy but a pragmatic defect in an argument whereby the premises are just as much in need of proof or evidence as the conclusion, and as a consequence the argument fails to persuade.
Here is an example of a valid circular reasoning proof using the reiteration (R) inference rule.
The proof checker claims the argument is valid, but it is circular. I am simply concluding what I accept as a premise. It is convenient to be able to do that.
The example from The Skeptical Scientist:
God exists because the bible says so, and the bible is true because God exists.
Following the OP, let us use the following symbolization key:
- P: "God exists."
- Q: "The Bible is true."
Then "if the Bible is true God exists" would be "Q → P" and "if God exists the bible is true" would be "P → Q". If we assume one of these implies the other, say, (P → Q) → (Q → P)". Then a truth table would show that this is not valid because the valuations for P and Q in the third line makes the whole statement false:
This would be an example of circular reasoning that is invalid. As the OP notes this is more specifically Affirming the Consequent fallacy. Here is Wikipedia's description:
Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dark,") and invalidly inferring its converse ("The room is dark, so the lamp is broken,") even though the converse may not be true.
The statements about God are caricatures. If someone believed P that "God exists" in addition to the implication P → Q, then the argument would be valid. Here would be a proof:
Again the proof is by reiteration (and conditional introduction at the end). This would not likely convince an unbeliever, but for a believer in P, the argument is valid.
Michael Rieppel. Truth Table Generator. https://mrieppel.net/prog/truthtable.html
The Skeptical Scientist. Retrieved on August 19, 2019, at http://www.timvanderzee.com/circular-arguments/
Wikipedia contributors. (2019, June 14). Circular reasoning. In Wikipedia, The Free Encyclopedia. Retrieved 20:02, August 19, 2019, from https://en.wikipedia.org/w/index.php?title=Circular_reasoning&oldid=901826531
Wikipedia contributors. (2019, August 14). Affirming the consequent. In Wikipedia, The Free Encyclopedia. Retrieved 20:22, August 19, 2019, from https://en.wikipedia.org/w/index.php?title=Affirming_the_consequent&oldid=910759549