Circular Reasoning is Not a Fallacy James Coley (jc@unc.edu) September 2004

On my view, contrary to the commonplace view taught in beginning logic courses, begging the question is not the same thing as circular reasoning and, while the former is a fallacy, the latter is not.

If circular reasoning is not a fallacy, we can make quick work of the infinite regress problem in epistemology. One way to portray the regress problem is as a contradiction between there being only finitely many beliefs we can consider in giving justifications, and all of the following, taken collectively.

• I am (rationally) justified in some belief (e., I can give adequate reasons for it).
• If I am justified in some belief, it rests (at least in part) on some justified belief.
• No justified belief rests on itself (e., the “rests on” relation is irreflexive).
• The “rests on” relation is transitive.

Why should this relation be transitive? I believe (4) is the least intuitively compelling and least defensible, and I therefore reject it. Without it, circles of two or more steps are allowed in the “rests on” relation, but nothing rests on itself. (John Post of Vanderbilt University has independently arrived at a nontransitive epistemology.)

I draw stipulative distinctions between circular reasoning, begging the question and petitio principii.

“Circular reasoning” applies to two or more arguments when it is possible, in going from a conclusion to one of its premises, and then from that premise (regarded now as a conclusion) to one of its premises, and so on, to wind up at the start, having gone full circle.

Here is a formal way to say essentially the same thing, except that it is transposed from the context of arguments to that of belief systems. (Note that changes in belief systems are under justificatory constraints.) Let an idealized formal model of a belief system be an ordered pair <B,R> consisting of a set of beliefs B taken along with a reasons relation

R Ì (Ã(B) ´ B).

The idea is that “Rxb” means that the beliefs in the set x are the reasons for belief b in the belief system. Now define R’ Ì B2 so that R’b1b2 if and only if \$x (Rxb2 & b1Îx). The belief system <B,R> contains a circular reasons-structure if and only if, for some natural number n ≥ 2 there is a one-to-one function f from {1,2,…, n} into B such that

R’f(n)f(1)

and such that, for any natural number i such that 1 £ i < n, it is the case that

R’f(i)f(i+1).

Begging the question” applies to a single argument with a premise that must be in question if the conclusion is in question. The term petitio principii applies to a single argument the conclusion of which is (perhaps reworded) among the premises. This is a special case of begging the question. Thus, petitio principii is also fallacious.

The term “in question” must be understood in the context of dialectic. Let an idealized formal model of a dialectical situation include interlocutors I1 and I2 with respective belief sets B1, B2. Their common ground is B1 Ç B2. Interlocutor I1 may make an attempt to rationally persuade I2 of some belief b, where bÏB2 and, presumably, bÎB1. In doing so, I1 must have premises for the argument(s) put forth for b, and may present these premises to I2 for assent. In reply, I2 might agree or not agree with any particular premise, and might demand that further reasons and arguments be given for it.

When this move of asking “Why?” is legitimate in the context of the dialectical situation, the challenged belief is said to be “in question.” By contrast, when a belief is one for which it is legitimate for the interlocutor to presume assent, it is said to be a “given” in the dialectical context. I will make the simplifying assumption that, in this idealized model, B2 contains all and only the “givens” I1 is entitled to expect. Under this assumption, we may say that b is in question for I1 in the context just in case bÏB1. Of course, all of this goes vice versa for I2 and I1.

Another simplifying assumption is that all the interlocutors are minimally rational: their belief systems are not manifestly and egregiously incoherent. Virtually everyone who takes part in philosophical discussion would meet this criterion. A person with a psychological disability would not meet it if, for example, they were to say it is time to feed the housecat although, in a perverse derangement, they would not assent to the claim that there even is a housecat.

Why is begging the question a fallacy? A basic test of an argument is whether it could convince someone for whom the conclusion is in question. An argument that begs the question in the sense defined can not pass this test, because the argument depends upon a premise assent to which could not be expected of anyone for whom the conclusion is in question. In this respect, the argument “begs for” acceptance of the conclusion, instead of earning it by honest toil.

Circular reasoning is a structural feature of a belief system or a collection of arguments, and it is compatible with none of them having premises that must be in question if the conclusion is in question. Begging the question is not entailed by circular reasoning, and vice versa. Circularity in belief systems or collections of arguments is not fallacious.

One might object that when a claim rests on a premise that rests on the claim, the claim rests on itself. Therefore, the objection continues, circular reasoning is a special case of petitio principii and thus fallacious. But this argument is good only if “rests on” is transitive. Because I deny (4) above, I do not accept this objection. The claim that rests on a premise that rests on the claim does not rest on itself. It rests on the premise, as has already been stated.

A more formal reprise of my distinctions makes use of a function Q from individual beliefs to sets of beliefs. Let Q[b] = the set of beliefs that must be in question if b is in question. This should be understood as implicitly relative to an interlocutor in a particular dialectical context. Given the simplifying assumptions of the formal model of dialectic, we may say the following: a Î Q[b] just in case any minimally rational interlocutor who believes a also believes b.

I say of an argument for claim b that rests upon a set of premises P that the argument begs the question just in case P Ç Q[b] ¹ Æ, and that the argument is a petitio principii just in case bÎP. Since bÎQ[b], petitio principii is a special case of begging the question.

Diagram I shows begging the question, while diagram II shows that the trick to avoiding this fallacy has nothing to do with circularity. The trick is to keep P clear of Q[b].

Circular reasoning, in the simplest case of two arguments (where b1 rests on P1 and b2 rests on P2) is when

b1 Î P2

b2 Î P1

This, as diagram III shows, is compatible with

P1 Ç Q[b1] = Æ

P2 Ç Q[b2] = Æ

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