# Is circular reasoning always a fallacy?

Suppose the following dialogue:

...

"I accept only one notion of land property. Namely, 'I am doing my stuff here, therefore I am here".

"But this means," he responded, "you can break into any place and stay there, using this rule."

"There is another rule, though. Right to act in its most natural sense, 'I am here, therefore I am doing my stuff here."

"But do not these rules put together become circular reasoning?"

"Of course, they do," I put my head closer to him, "and this is exactly that rare case when circular reasoning is not only sufficient, but is necessary."

"But why is that? Circular reasoning never was a correct argument."

"There are many circular things happening in nature. Life forms, for example, exist only in order to produce new life forms or to prolongate their own lifetime. Is there any reason to say this can't be the case for reason itself? Why must reasoning not be circular?"

...

Can circular reasoning ever be justified? For example, we know all the words in our languages can't be defined using other words without the use of circularity. We know logic can't prove itself without circularity. Are these arguments rational? Are there any other arguments where circular reasoning is still rational?

• This isn't precisely, or at least merely, circular reasoning, though. It implicitly equates "I am here" and "I have the right to be here." Further, ignoring that, an equally valid assertion is that "I am doing my stuff here" and "I am here" are in fact equivalent (it's physically impossible to be somewhere without doing something there, and vice versa). – Obie 2.0 Sep 17 '18 at 18:24
• You later argue...is this an excerpt from some book you're writing? – Obie 2.0 Sep 17 '18 at 18:42
• Yes, it can if the purpose is not a foundational justification but explanation and mutual corroboration in a system, see virtuous circle and hermeneutic circle. E.g. one may try to explain why certain laws of logic are reasonable, even though doing that will inevitably use those same laws of logic. This is the only mode of justification available for the so-called "properly basic beliefs". – Conifold Sep 17 '18 at 21:21
• Just to reiterate @Obie2.0 here - this isn't circular reasoning by the formal definition of it. It's not even an argument. In formal logic terms, the first speaker is stating two premises. You can state all the premises you want, but it doesn't make them true. The truth of premises is often outside the scope of pure logic. – Alex H. Sep 18 '18 at 17:59
• Is circular reasoning always a fallacy? If it were, this would therefore be a fallacy. – WBT Sep 18 '18 at 20:45

## 12 Answers

Geoffrey refers to feedback loops as a valid example of circular reasoning. This is not correct: they can be valid but they are not circular. Instead they are an example of reasoning by induction.

It is not a implies b and b implies a. Instead it is an implies bn and bn implies an+1. That is valid so long as you can start it off with a0. For example, it is fine to say that people will distrust the economy if inflation takes off, and inflation will take off absent trust in the economy. However, you need some other reason for inflation or distrust to get going in the first place.

It is not justified to accept any proposition if the only reason for accepting it cannot be justified without assuming the original proposition.

• Well, then should I say in the case of my dialogue, it's not circular reasoning as well? We can't say it is recursion without context. – rus9384 Sep 17 '18 at 18:37
• If it is properly circular, it is not valid. Even the example of animals existing because animals exist is a recursive one rather than a circle. (although care should also be taken to avoid conflating causes and logical implications.) – Josiah Sep 17 '18 at 19:24
• No, it shortly is "Life exists because life existed". But it is not about reasoning. However, this case assuming rights and laws is also very interesting when asking about circularity. – rus9384 Sep 17 '18 at 19:34
• @Conifold, I think the last sentence is exactly as broad as it should be. Proponents of logic would do well to remember that there is no reason to accept logic apart from logic. Thus when you seek to convince people to be purely logical, using only logic, you are doomed to failure. And thus it is that no one has ever marketed being logical, because so many logicians disdain marketing for its effective use of emotional reasoning to gain acceptance. – Wildcard Sep 18 '18 at 1:24
• @The_Sympathizer, you'll have to answer that yourself. I don't accept things on the basis of logic; I accept things on the basis of proven effectiveness in application. There is a large overlap between the two, but they are not identical. (To put it another way: the scientific method is senior to pure logic.) – Wildcard Sep 18 '18 at 2:12

Circular reasoning is an informal logical fallacy. Wikipedia describes it as follows:

Circular reasoning...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.

To see this is not a formal logical fallacy note that a circular argument is valid by considering the following proof with the reiteration (R) inference rule:

If whatever English sentence P is symbolized to represent can be determined to be true, then the argument is also sound.

Nonetheless, the argument is an informal logical fallacy and it should not persuade those hearing it.

As a side note this may be an example for my question whether there exists a sound and valid argument that is informally fallacious: Can an argument be formally valid with sound premises and still be informally fallacious?

Reference

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

• just a note; the wikipedia page on this issue has been particularly heavily edited over time. information has been greatly reduced and heavily altered by athiest/religious debates while a few years ago the article was more relevant to philosophy and logic. – Garet Claborn Jul 18 '19 at 9:58
• @GaretClaborn I agree. Wikipedia has the potential to change. I don't know what the version was that I quoted back then. What I do now is use the "Cite this page" link on the left panel of the Wikipedia article to generate a link to that specific version to (hopefully) make sure I can find the same version again. Thank you for pointing it out. – Frank Hubeny Jul 18 '19 at 10:40

It is a fallacy insofar as is doesn't generate support for a proposition, and thus is not a proof of its validity (while assuming the shape of a proof to the eye of the layman - that's why it is considered to be an error most of the time).

On the other hand, it is an instrument to explore which other propositions are consistent with an initial assumption A.

• If A->B and B->not A (or any longer circle like that), you have proven A is outright impossible.

• If A->B and B->A (or again any longer circle like that), you have proven that A and B are compatible. This means they may eventually coexist and both be true in an imaginary world (not necessarily in our world).

Remember, A might still be impossible if you take some yet untried C into consideration, so it's not a conclusive proof for A. But if you can't think of any such C, you might decide to take A as an axiom... one is still unable to prove A, of course, but now you can possibly generate valuable propositions like "under the assumption of A, the following interesting (formerly non-derivable) relationship holds: ...".

So circular reasoning has some application in deciding (most of the time: refuting) if any proposition A may be used as an axiom.

• Just to clarify, by `A->B->not A`, you mean `A->B and B->not A`, not `A->(B->not A)`, right? – Ray Sep 18 '18 at 14:28
• @Ray absolutely, yes, I will edit to make that clear - thank you :) – jvb Sep 18 '18 at 21:15
• " you have proven that A and B are compatible." Not just compatible, but logically equivalent. – Acccumulation Sep 18 '18 at 22:38
• Hmm, what if B -> C and C -> not A though? Then A, B still wouldn't be compatible (but still equivalent)? – gmatht Sep 19 '18 at 1:49
• @Blueriver Again, the fallacy is to believe it would generate support for a specific proposition. In modal logic, when discussing possible worlds, starting with A and trying to derive ~A is a reasonable check if expanding some set of propositions by A will introduce inconsistencies, and the technique to do so is rational. If you can derive A->~A, it's not a fallacy, but proof that a world with the chosen set of axioms isn't possible. If you cannot, it's neither a proof of A, but just an indication a world where A holds would be possible. Assuming it's proof of A is a fallacy ;) – jvb Sep 20 '18 at 6:15

## tl;dr

Circular reasoning means that a premise of your argument was the same as your conclusion. That means you haven’t established anything new. This can be useful if you’re stating a definition. Those need to give you back exactly what you started with, that is, be purely circular and tautological. You’re not supposed to introduce any new facts when you rephrase or simplify a statement!

You will sometimes see people argue that the good kind of circular reasoning isn’t really circular reasoning but something else, because their definition of circular reasoning says it must be unsound. That is a good example of circular reasoning.

If a more complicated argument turns out to be circular, all you did was waste your readers’ time. If people can’t agree on definitions, they’re actively harmful.

## The Long version

You normally use definitions to rewrite a statement in an equivalent way. Rewriting your premise as your conclusion is circular reasoning. But that’s sometimes a useful thing to do! So: “Let us define man to mean adult male human. Socrates is a man. Therefore, Socrates is adult, male and human,” is logically sound circular reasoning. Not a very interesting argument, but a historian might legitimately need to explain that the primary sources say that Socrates was an anthropos, not a woman, a child, a supernatural being or a pony.

A tautology like that is normally thought of as trivial, even suspiciously so (which is probably why “tautology” has picked up the secondary definition, “playing word games to hide nonsense” or “overcomplicating things that much is a big red flag.”)

### Some Examples from Mathematics

Back when I was young and brash, I used to ask on the first day of each math class, “Is this where we learn why one plus one equals two?” Finally, a classmate of mine asked me, “You do know that one plus one is defined as two?” (When I told this story over on Math.SX, people objected that he should’ve said, “Two is defined as one plus one,” and it started a little argument about whether both are equally valid from a certain point of view.) So, for a real-world example, we say “2+2 = 4” because it’s more convenient than “(1+1) + (1+1) = 1+1+1+1,” but if I were asked to prove that 2+2 = 4, I would replace all the terms with their definitions, then invoke the axioms that addition is associative and equality is reflexive. That is, we would show that “2+2 = 4” is a tautology under the definitions of 2, 4, + and = that the people we’re writing the proof for accept.

However, in some courses I’ve taken, that answer would not get me full points on an exam because the real purpose is to test my knowledge of a formal system we’ve been studying. I might instead be expected to substitute the definitions from, say, John Von Neumann’s model of arithmetic based on set theory, or Alonso Church’s model based on the Lambda Calculus. My argument from before would still be logically valid on its own terms, but would become unsound in this context because I’m no longer allowed to take the same definitions for granted. That is, “1+1 = 2 because 2 is defined as 1+1? That’s just circular reasoning. You were supposed to say that 1 is defined as {{}}, 2 is defined as { {}, {{}} }, and x+1 is defined as x∪{x} for all x. But not in the other course I’m teaching on Constructive Mathematics, which takes the concept of counting as a given and doesn’t accept set theory. If I give you the same question there, do the opposite.”

So that’s one way the same “circular reasoning” might or might not be accepted even in mathematics. There are still some forms of reasoning-from-definitions that are always considered fallacies, though: mathematicians would (almost always?) not accept a definition that introduced new premises. So, for example, you would not be allowed to define “the upper and lower floop of n” as “the lowest integers a and b, respectively, such that a/b × a/b = n” and then claim that the square root of 2 is rational by referring to “the floops of 2” as if such things existed. You’re no longer just making the exact same statement in an equivalent way; you’ve smuggled in a new premise, that the number 2 has floops, and it’s the exact same thing you’re trying to prove (that some fraction a/b is the square root of 2).

Likewise, everyone agrees that it’s a logical error to use Theorem 1 to prove Theorem 1, or—more likely to be an honest mistake—prove Theorem 1 from Theorem 2 and then use Theorem 1 in the proof of Theorem 2. Most math professors will therefore insist that you cannot use a theorem introduced later in the course to prove a theorem introduced earlier, unless you can prove the later theorem from earlier theorems and only earlier theorems. This is to save you from this form of circular reasoning. Once you’re no longer a student, this doesn’t work (or rather, it might be possible to enumerate the theorems of mathematics and enforce a partial ordering on them, but mathematicians haven’t), so it becomes especially important to declare the premises of your results explicitly, not smuggle any more in, so that other mathematicians can tell if using your result will make their own proof circular.

### In the Wild

When you follow a debate that gets beyond formalism into value judgments (or, bless your heart, get into one), there are a whole lot more ways for this to go wrong. A lot of the words we use about ethics are loaded. Everybody¹ agrees that words like fairness and justice mean that something is good, and words like murder and abnormal mean something is bad.

So you will often see an argument where (for example) both sides agree that FREEDOM™ is good, and then since all serious disputes are between parties that cannot all have everything they want, both of them implicitly define FREEDOM™ to mean “My freedom to get my way,” and ask the other “Why should I be forced to allow you to?” If you wrote this out as What-Alice-Calls-Freedom and What-Bob-Calls-Freedom, it would be really obvious that both of them are talking past each other, or less charitably, have as their first principle, “I win.”

¹ That is, everybody it’s possible to have a productive conversation with at all.

Circular reasoning is a fallacy because circular reasoning is defined as being fallacious. If it's not a fallacy it's not circular reasoning. How's that for circular?

If you have a conclusion that is true because the premise is true, and the premise is true because the conclusion is true, there is nothing outside the loop proving either true. This pure circular reasoning is a fallacy. If, however, the premise or conclusion is supported by some other logic, the circularity is irrelevant and this isn't an example of circular reasoning.

On the left you see circular reasoning with no foundation. On the right you see irrelevant circular reasoning based on some other foundation.

`````` c   c
( ) ( )
p   p
|
f
``````

Circular reasoning - reasoning circular in form - can represent feedback loops, among other things, with perfect correctness :

WHEN asked why the economy in a certain state is in a slump, an economist replies: "A lot of people are leaving the state. Things are very poor in the building industry, for example, because there is no need for new housing." Next question: "Why are people leaving the state?" The economist's answer: "Well, the state of the eco? nomy is poor. People just don't seem to be able to get jobs, with the economy being so slow at the moment." This sequence of questions and answers has taken us in a circle: the economy is depressed because people are leaving, and people are leaving because the economy is depressed. Isn't this just the sort of argument that might be cited in a logic text as an instance of petitio principii, the fallacy of arguing in a circle? If so, it seems that the economist's argument must be fallacious.

On the other hand, perhaps the circularity in his argument could be due to the feedback loops inherent in human behavior. If people leave, things get worse. But if things get worse, people leave in even greater numbers. An analogy could be to the following case. The more overweight the diabetic gets, the more insulin there is in his blood. The more insulin there is in his blood, the more he tends to eat and thereby store up more fat. Here the process is circular, but there seems to be no fallacy. At least, from one point of view the circle is not vicious, since the diabetic gets fatter and fatter. Similarly, in the previous case, the state could become more and more economically depressed, as the cycle progresses.

In mathematics, it is common practice to start at proposition A and then prove B, then start again at B and prove that A follows. An equivalence proof in mathematics, of the if and only if type, often takes this form. Although the form of proof is circular, in many instances such a proof is rightly thought non-fallacious. And some notions that are circular, like Russell's "set of all sets that are not members of themselves," were found troublesome not altogether because of the circularity involved, but because they contain a contradiction.

These examples may suggest that circular reasoning is not always fallacious or vicious. Some philosophers have even carried this further to argue that scientific reasoning itself may be inherently circular. Hull (1967) examines the questioning of their own methodology by evolutionary taxonomists. The taxa, or categories of organisms used by biologists are applied to the study of particular organisms to represent evolutionary descent. But as more is learned about the principles of descent with modification by such studies, the taxa are refined and improved. This process has seemed circular to some scientists, and it has been called "groping" and "reciprocal illumination" to indicate the suspicion of circularity. According to Hull (1967) the process is circular only to the extent that scientific verification of hypotheses is always circular.

Perhaps what Hull is suggesting is something like the following sort of process. First, a hypothesis is formulated on the basis of some initial evidence. As new evidence comes in, the hypo? thesis is clarified and refined. However, once stated more clearly and precisely, the hypothesis points to new evidence that has thereby become "relevant" or "significant." This new evidence improves the hypothesis once again. (Douglas N. Walton, 'Are Circular Arguments Necessarily Vicious?', American Philosophical Quarterly, Vol. 22, No. 4 (Oct., 1985), pp. 263-274 : 263.)

___________________________________________________________________________

References

Douglas N. Walton, 'Are Circular Arguments Necessarily Vicious?', American Philosophical Quarterly, Vol. 22, No. 4 (Oct., 1985), pp. 263-274.

David L. Hull, "Certainty and Circularity in Evolutionary Biology," Evolution, logy," Evolution, vol. 21 (1967), 174-89.

• But scientific process is the opposite of that. You propose a falsifiable and testable hypothesis and then test it looking for contradictory evidence. Only when you fail to find any in large amount can you infer that the hypothesis is probable and, with sufficient bodies of evidence, we might assume it to be true enough, but never with any certainty. Hypotheses are never proven with the rigour of logical argument, they merely survive many attempts to be disproven. – Danikov Sep 18 '18 at 10:16
• @Danikov. Walton only says what 'perhaps' Hull had in mind was ... My answer did not assume the truth of that claim. But we have your comment on the record, so it will serve as a useful corrective. Thank you. Best - GT – Geoffrey Thomas Sep 18 '18 at 10:51
• Well I also take umbrage with trying to apply a logical argument, which is a static structure, with continuous processes. Feedback loops don't imply that a -> b -> a, but rather than over time, the growth of a leads to the growth of b that reciprocally causes further growth of a. If the effects were instantaneous, all diabetics would explode, but instead the process is iterative rather than truly circular. Discovering and refining taxa also is an iterative process. – Danikov Sep 18 '18 at 11:04
• And also the math thing isn't circular, but rather because A -> B does not infer B -> A and therefore both must be proven. I can prove a white swan is a swan, but unless I prove all swan are white, I can't say the two are equivalent. If proven, then due to equality, it does become circular, because of identity. – Danikov Sep 18 '18 at 11:06
• No, I still think there's value in wrong answers, as long as they're demonstrated as such. Often they can illuminate why a popular course of thinking isn't fruitful as long as they're sufficiently detailed. – Danikov Sep 18 '18 at 11:51

Both circular and inductive reasoning suffer from the same flaw: where do you start? After all, the purpose of logic is to create new truth that follows from prior truths, also known as 'proof'.

With pure induction, it's as is said 'turtles all the way down'; that is to say any premise can be required to, itself, be induced from something else, leaving no ground to start from.

With circular logic, the start is arbitrary as you're constructing a ring. All points can be considered starts and ends equally, but ultimately it can be reduced to a very simple form: A -> B only if B -> A, but B -> A only if A -> B. The combination of two dependent conditionals regresses infinitely much like induction, just down the circle rather than to infinity.

Both systems lack an antecedent that does not require further justification: an axiom. Axioms are not without examination within the grand scheme of things, but for the sake of any argument are required. Axioms are often used without being identified as such, but they are critical to any reasoning that isn't fallacious, which is what many failed arguments fall into: not a failure of reasoning, but a failure to identify any common ground from which to start from.

Where circular reasoning can trap people into thinking it works is with identity. Identity is circular as something will always equal itself. However, if you pose something else that /seems/ different as equal to the other, the reverse will be true and it might feel like you've proved something. Descartes broke the mould on that one with "I think, there for I am." By separating thinking and existing as two things and trying to reason a causality between them, you create a circle that doesn't prove anything.

It's also worth noting that circular reasoning being fallacious does not disprove the thing in question, but rather fails to prove it by those means. Circular definition of words do not start that way; we know that babies do not speak, so they must bootstrap their language in non-verbal ways. However, those foundations are lost once language builds a self-integral web. The justification for the veracity of the initial foundations may be forgotten, but that isn't to say it didn't exist at all.

• And deductive reasoning also suffers from the same flaw. – rus9384 Sep 18 '18 at 13:29

# No, with counter-example

I eat because I am unhappy

and

I am unhappy because I eat

can both be true at the same time. Both invidious effects are caused by each other in a never ending cycle of obesity and misery (and Mike Meyers movies).

• This is a helical argument. Although both can be true, neither the eating nor the unhappiness has been proved. – amI Sep 20 '18 at 10:32

Reasoning and logics are based on causality: if a [cause] occurs, then a [consequence] will occur.

Circular reasoning breaks causality: a consequence cannot be the cause of itself (in case of a singular causal proposition). You can chain several causes and consequences, but if you create a loop, you break causality: A->B->C->D->A is wrong, a cause cannot trigger a previous consequence.

Your example is a common error, a correct result obtained from bad reasoning. That happens.

• In this case Joshua's answer can be a counterexample, right? Because it is argued there that causal loops are possible. My example is not my reasoning. I tell how it works in nature. It is common reasoning. – rus9384 Sep 18 '18 at 5:21
• @rus9384: I suppose you're joking. In such case, not only circular reasoning, but all wrong reasoning is correct, because the multiple universes quantum interpretation allows it. Good thinking. – RodolfoAP Sep 18 '18 at 5:44
• do you mean laws of logic in other universes can differ from ours? – rus9384 Sep 18 '18 at 6:11
• 1/sqrt(2)|yes>+1/sqrt(2)|no> – RodolfoAP Sep 18 '18 at 7:53
• @rus9384 A causal loop is not a 2D circle. It's a 3D spiral -- with the extra axis being time. – user20 Sep 19 '18 at 19:04

Fallacies Only Refer to Arguments

For example, "His argument is wrong because he is stupid" is an example of the ad hominem fallacy. However, just because this is a fallacy doesn't mean he is smart. Similarly, "We shouldn't trust his claim to be an expert witness as to the accuracy of this complex proof" is not a fallacy.

In many of the cases you describe we aren't really talking about arguments, so fallacy wouldn't apply.

Sometimes reasoning isn't an argument, but a statement of a belief system. For example, someone may state that they only believe things that can be scientifically proven. If they respond to the question "Why do you believe science works?" with a scientific publication, this shows that this question does not show that their belief is inconsistent. If they do not also use it as an argument for science, it is not a fallacy.

Broadly a belief system may be described as foundationalist or coherentist. In foundationalism there are some axiomatic beliefs that need no justification, circular or otherwise. Coherentism, allows for any proposition to be questioned, but only has other beliefs against which to test the proposition. This isn't necessarily a fallacy. For example, showing that A -> B, B -> C and C -> A means that any evidence for B and C is now also evidence for A. In each type of belief system, some beliefs are not supported by non-circular justifications.

According to my 9th grade geometry teacher, axioms are unprovable, but everything is built on them. Hence, an axiom can be viewed as leading to circular reasoning (as can a dictionary, because eventually, all words are defined by other words, etc.).

This means, "no", it is not a fallacy.

However, "circular reasoning", by itself, doesn't prove a case either.

Also, many arguments have hidden circular reasoning within them. This must be watched-out for.

No. Here is non-fallacy:

If you were to successfully travel backwards in time, and then attempt to change something that would have changed your own past so that you would not travel back in time, something will happen so that you are not successful. In General Relativity, the result of attempting this is the new something that appears has a causal loop and physics becomes nondeterminsitic without invoking any quantum rules.

Arguments about what happens within such a causal loop are circular and there is no fallacy because that is the construction of the underlying physics. If the debate becomes "which form will it take" it literally cannot be resolved without trying it. (Pro tip: Don't try it. You are likely to not like the result.) One of the circular arguments may be correct or perhaps none of them.

But if the debate is within the path of a single such loop, the circular argument is resolvable despite being circular.