What exactly is circular reasoning?

Question

The way I used to be getting it was that circular reasoning occurs when a proof contains its thesis within its assumptions. Then, everything such a proof "proves" is that this particular statement entails itself; which is trivial since any statement entails itself.

But I witnessed a conversation that made me think I'm not getting this at all.

In short, Bob accused Alice of circular reasoning. But Alice responded in a way that perplexed me:

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. Thus each set of axioms implicitly contains all theses that can be proven from this set of axioms. As we know, each theorem in mathematics and logic is little more than a tautology: so is mine.

Not sure what should I think? On the one hand, Alice's reasoning seems correct. I, at least, can't find any error there. On the other hand, this entails that... Every valid proof must be circular! Which is absurd.

What is a circular proof? And what is wrong with the reasoning above?

Answer 1

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:

1. The steps in the argument need to be logical (valid .. the conclusion follows from the premises)

2. 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.

Answer 2

(Not really an answer, but too long for a comment. Refer to amWhy's comment and Bram28's answer for a direct answer to your question.)

From what I understand, tautologies have a reputation in philosophy as being a waste of time. Things like "if I am cold and wet, then I am cold" don't seem to contain any new information, and have little value. But, the philosophers wonder, why is mathematics, which traditionally is 100% tautology, so non-trivial and produces unexpected results?

Well, I'm sure there are many answers to this, but the way I see it, it's finding a path from premises to conclusions that can be new information. Human beings aren't perfect; they cannot instantly account for every statement they hold to be true, and combine them in every possible way, to effortlessly see every consequence one can derive from them.

This is the point of proofs/arguments. I can read over and properly comprehend the axioms of ZFC, the definition of $\mathbb{R}^3$ and its Euclidean unit sphere, but that doesn't mean I instantly know how to prove the Banach-Tarski paradox!

Alice's reply seems to take the opposite stance. She seems to be assuming, implicitly, that there is no such gap between premises, consequences, and hence conclusions, rendering arguments useless. It's a view that I think most people would consider extreme (and wrong).

Answer 3

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.

Answer 4

Supposing that Alice tells the truth about using circular reasoning, it can get demonstrated that Alice has made an error. Alice says:

"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. Thus each set of axioms implicite contains all thesis that can be proven from this set of axioms."

Alright, let's suppose the first two sentences correct. The third is false though. Why? Because without rules of inference we can't conclude anything from any axioms. So, we can't conclude that each set of axioms implies all of the theses, because axioms don't imply theses in the first place. Ever since Frege it has been clear that rules of inference and axioms are not the same thing. Only axiom and rules of inference taken together imply a thesis. But then, the set of axioms doesn't implicitly contain all theses, because it's axioms and rules of inference that imply all theses provable in the theory.

Edit: Also, it really doesn't work out that "logic is little more than a tautology" or "logic is little more than a collection of tautologies". Logic often involves showing something like A $\vdash$ B, where A is not a tautology, nor is B, and A $\vdash$ B is not a tautology.

Answer 5

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.

Answer 6

If her thesis is true, you could argue she is somehow right. But we dont know that her thesis is true. Lets say A is the set of axioms, and S(A) all true statements derivable from this set, and t her thesis. Then she wants to show that t in S(A). Only if this were true she could use that t is in S(A). But she doesnt know it.

To make it clear to her: you could ask why not use also not t to show that the axiom system is contradictory. How does she know not t is not in S(A)

Answer 7

We can take the extreme case, where the axioms not only imply the conclusion, but contain the conclusion. Suppose I'm trying to prove it's raining outside. The following piece of reasoning is perfectly valid:

It's raining, therefore it's raining.

Nothing wrong with that, and no one would call it circular. It's just pointless. But suppose someone skeptical, without access to a window to look outside, were to challenge me to prove it's raining. If I now start my reasoning with "Well, we know that it's raining, so...", now I'm committing circular reasoning.

Notice that in both cases my proof and my axioms were the same, only the context changes. Circularity is not a property of a proof, it's a property of the context in which that proof is given. When we prove something from axioms, we should choose the axioms to be statements that are known to be true. If they're not known to be true, this is a Bad Thing. A special case of this kind of Bad Thing is where one of the axioms is the conclusion we're trying to prove (which is not known to be true): this special case is called circular reasoning.

Answer 8

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.

Answer 9

The apparent dilemma arises from two words which do not mean the same thing.

You are talking about proofs. Alice is talking about truth. These are not the same thing.

A statement which is always true is a tautology, so in a sense, every such statement, including a true theorem, is a tautology.

But truth is not a proof.

Proofs are simply the re-expression of statements as other statements without relying on other statements (i.e., no circular reasoning).

But the result of a correct proof is no more true than the original statements, so a proof is likewise not always the truth.

Answer 10

Circular reasoning is when two statements depend on each other in order to be true (consistent with the model).

• The inside angles of a triangles are supplementary
• The co-interior angles of a line intersecting parallel lines are supplementary

Both statements can be used to prove the other but neither of the above statements can be proven from the existing set of axioms in Geometry unless one of them is assumed as true without proof (an axiom). This "straightens" the argument so as to begin at this assumption. This is not fallacious at all unless it produces a contradiction with existing assumptions. It just limits or rather refines the set of statements that can follow as a logical consequences of the additional assumption.

Alice's Argument that Mathematical Proofs are Circular can be understood as:

• Each and every proof must be based on axioms.

• Axioms implicitly contains all theses that can be proven from the set of axioms.

To prove any one of these two statements will require the other one. This is circular. To "straighten" it we must assume one of the two to be true and the other can follow by the rules of inference/logic.

Is Mathematics then a tautology? If you define a tautology as a system of statements that is true by necessity or by virtue of its logical form, then I think it unashamedly is, and has to be based on the rules of inference/logic and the assumption just made: Each and every proof must be based on axioms or restated: All theses must follow by the rules of inference (proof) from the set of axioms.

Is Alice's argument valid that, in form, it is equivalent to that of axiomatic system of Mathematics? Is it consistent with the model produced by the assumption: all theses must follow by the rules of inference (proof) from the set of axioms.

Proof by contradiction: Assumed it is true that a proof contains its thesis within its assumptions.

Two situations, 1) The thesis is an existing axiom (assumption without proof).

If the thesis is an existing axiom, then of course her model is equivalent. A leads to A.

2) The thesis is not an existing axiom (assumption without proof).

If the thesis is not an existing axiom but it is an axiom because it is assumed without a proof, then is leads to a logical contradiction.

Which means her proof is only valid if her thesis is indeed an existing axiom.

Alice's solution: So if her argument is indeed circular, then her solution is to admit that one of the statements that she is making must be is an necessary assumption to develop the model/argument/worldview etc. Then it could be equivalent to the axiomatic system of Mathematics if it does not contradict any existing assumptions/axioms.