What exactly is circular reasoning?
Solution 1:
Of course my proof contains its thesis within its assumptions. Each and every proof must be based on axioms, which are assumptions that are not to be proved.
Hold it right there, Alice. These specific axioms are to be accepted without proof but nothing else is. For anything that is true that is not one of these axioms, the role of proof must be to demonstrate that such a truth can be derived from these axioms and how it would be so derived.
Thus each set of axioms implicite contains all thesis that can be proven from this set of axioms.
Implicit. But the role of a proof is to make the implicit explicit. I can claim that Fermat's last theorem is true. That is a true statement. But merely claiming it is not the same as a proof. I can claim the axioms of mathematics imply Fermat's last theorem and that would be true. But that's still not a proof. To prove it, I must demonstrate how the axioms imply it. And in doing so I can not base any of my demonstration implications upon the knowledge that I know it to be true.
As we know, each theorem in mathematics and logics is little more than a tautology:
That's not actually what a tautology is. But I'll assume you mean a true statement.
so is mine.
No one cares if your statement is true. We care if you can demonstrate how it is true. You did not do that.
Solution 2:
I believe this has a simple resolution:
When we say informally that Alice is required to prove a result, it is sloppy language; she is actually required to prove the implication axioms $\implies$ result. So of course, she can have the axioms in her premises. However, she cannon have axioms $\implies$ result as one of her premises; that would be circular reasoning.
Solution 3:
All reasoning (whether formal or informal, mathematical, scientific, every-day-life, etc.) needs to satisfy two basic criteria in order to be considered good (sound) reasoning:
The steps in the argument need to be logical (valid .. the conclusion follows from the premises)
The assumptions (premises) need to be acceptable (true or at least agreed upon by the parties involved in the debate within which the argument is offered)
Now, what Alice is pointing out is that in the domain of deductive reasoning (which includes mathematical reasoning), the information contained in the conclusion is already contained in the premises ... in a way, the conclusion thus 'merely' pulls this out. .. Alice thus seems to be saying: "all mathematical reasoning is circular .. so why attack my argument on being circular?"
However, this is not a good defense against the charge of circular reasoning. First of all, there is a big difference between 'pulling out', say, some complicated theorem of arithmetic out of the Peano Axioms on the one hand, and simply assuming that very theorem as an assumption proven on the other:
In the former scenario, contrary to Alice's claim, we really do not say that circular reasoning is taking place: as long as the assumptions of the argument are nothing more than the agreed upon Peano Axioms, and as long as each inference leading up the the theorem is logically valid, then such an argument satisfied the two forementioned criteria, and is therefore perfectly acceptable.
In the latter case, however, circular reasoning is taking place: if all we agreed upon were the Peano axioms, but if the argument uses the conclusion (which is not part of those axioms) as an assumption, then that argument violates the second criterion. It can be said to 'beg the question' ... as it 'begs' the answer to the very question (is the theorem true?) we had in the first place.
Solution 4:
The fallacious version of circular reasoning is an appeal to the proposition
$$(A \to A) \to A \tag{C}$$
That is, to establish a proposition $A$ from no premises, you first establish the proposition $A$ under an assumption of $A$. Example:
- Accuser: "You stole that."
- Defendant: "No I didn't, it was mine."
- Accuser: "It couldn't have been yours because you can't steal something you own."
In that case the accuser is correct as far as pointing out that "stealing" implies "not yours" which implies "stealing", the $A \to A$ part, but the fallacy comes from dropping the assumption and concluding "stealing" under no assumptions, which is the final $\to A$.
Guessing what Bob's side of the argument was : "Because the semantic meaning of a theorem is contained in the semantic meaning of the assumptions, you used circular reasoning". Here Bob is making 2 mistakes. First, he is equating circular reasoning with the proposition $A \to A$. But that isn't what circular reasoning is, because $A \to A$ always holds, how can you object to that?
Second, he is not addressing the argument that Alice made. Even if $(A \to A) \to A$ (for her specific claim) applies to the assumptions and conclusions of her argument: unless she appealed to that theorem as an inference, she hasn't made any mistake. $(A \to A) \to A$ does hold in the case that $A$ itself is provable, such as when $A$ is a tautology. Observing this isn't deductively equivalent to assuming $(A \to A) \to A$ holds in all cases and using that as an inference.