## Infinite Regress vs Circular Reasoning

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

### Infinite Regress vs Circular Reasoning

Are Infinite Regress and Circular Reasoning the same thing? How are they related?

This has come up in a recent discussion in which it was maintained that infinite regress was a form of circular reasoning. This is news to me so I'm wondering if others might have some insights to this and some background information on it.

mtbturtle
Banned User

Posts: 9742
Joined: 16 Dec 2005

### Re: Infinite Regress vs Circular Reasoning

My opinion is that they might be similar because both need an operator to cycle back on similar state of premises - form & content-identical for circular reasoning and form-identical for infinite regress. Infinite regress looks like circular when viewed at its breadth (structural), but may really be a spiral viewed at its longer dimension (details). If we would consider circularity of reasoning to include those of structures, then possibly, infinite regress is a form of circular reasoning.
Don Juan
Active Member

Posts: 1158
Joined: 17 Jun 2010

### Re: Infinite Regress vs Circular Reasoning

mtbturtle,

Circular Reasoning is a special case (subset) of Infinite Regress.

Infinite Regress is when a proposition $P_1$ depends on $P_2$ which depends on $P_3$ and so forth.

Circular Reasoning is the same thing except it also asserts $n$ periodic equalities $P_i=P_{i+j{\times}n}{\forall}\left{\left{1{\leq}i{\leq}n\right},\left{j>0\right}\right}$ where $n$ is the number of elements in the circular loop.

For example, if an instance of Circular Reasoning is $P_1$ depends on $P_2$ which depends on $P_3$ which depends on $P_1$, then this is exactly the same as an infinite regress where $P_4=P_1$, $P_5=P_2$, etc. So here we have $n=3$ elements ($P_1$, $P_2$, and $P_3$) with periodic equalities:
1. $P_1=P_4=P_7=P_{10}=...$;
2. $P_2=P_5=P_8=P_{11}=...$;
3. $P_3=P_6=P_9=P_{12}=...$.
Natural ChemE
Forum Moderator

Posts: 2744
Joined: 28 Dec 2009

### Re: Infinite Regress vs Circular Reasoning

Natural,

Sorry I don't follow most of that. Could you point towards a source? thanks.

mtbturtle
Banned User

Posts: 9742
Joined: 16 Dec 2005

### Re: Infinite Regress vs Circular Reasoning

mtbturtle,

I Wikipedia'd infinite regress, saw that Logic people apparently expressed the propositions of an infinite regress as $P_1$ depending on $P_2$ depending on $P_3$, then went from there.

Another way of seeing it would be where Infinite Regress is a straight line of propositions,
$P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_5{\rightarrow}...$,
while Circular Reasoning is a loop
$\begin{tabular}{ccc}P_1&{\rightarrow}&P_2\\{\uparrow}&&{\downarrow}\\P_4&{\leftarrow}&P_3\\\end{tabular}$.

The Circular Reasoning loop can be written in linear form as
$P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_1{\rightarrow}...$,
which is the same thing as
$P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_5{\rightarrow}P_6{\rightarrow}P_7{\rightarrow}P_8{\rightarrow}P_9{\rightarrow}...$
so long as you assert the periodic equalities
1. $P_1=P_5=P_9=....$;
2. $P_2=P_6=P_{10}=....$;
3. $P_3=P_7=P_{11}=....$;
4. $P_4=P_8=P_{12}=....$.
This demonstrates that any instance of Circular Reasoning is necessarily an instance of Infinite Regress.

Since Circular Reasoning can be expressed as an Infinite Regress plus additional constraints, i.e. periodic proposition equalities, then Circular Reasoning is a special case (subset) of Infinite Regress.
Natural ChemE
Forum Moderator

Posts: 2744
Joined: 28 Dec 2009

### Re: Infinite Regress vs Circular Reasoning

I think Natural ChemE explained it very clearly!
Venus

### Re: Infinite Regress vs Circular Reasoning

mtbturtle wrote:This has come up in a recent discussion in which it was maintained that infinite regress was a form of circular reasoning.

So they basically got it backwards; circular reasoning is a form of infinite regress.

Just to preempt it, I can see someone countering, "But an infinite regress can be modeled as circular reasoning with infinite unique propositions!" Their argument would be that you never get to the first loop back since there are infinite propositions to get through first.

This misunderstanding would be based on a common fallacy about how infinity works, i.e. the notion that things after an infinite series are moot. It's actually the same fallacy that telescoping arguments (those employing infinite regress) suffer from.
Natural ChemE
Forum Moderator

Posts: 2744
Joined: 28 Dec 2009

### Re: Infinite Regress vs Circular Reasoning

This is infinite regress:

You will never get back to the outer tea cup, your starting point.

This is circular reasoning:

You will return an infinite number of times to your starting point.
Venus

### Re: Infinite Regress vs Circular Reasoning

Venus,

I'm going to make millions selling that circular reasoning loop on T-shirts and bumper stickers on university campuses.

If anyone complains, I can just tell them that $\begin{tabular}{ccc}&\text{copyright}&\\\rotatebox{90}{\text{ it's not}}&&\rotatebox{270}{\text{infringment}}\\&\rotatebox{180}{\text{because}}&\\\end{tabular}$.
Natural ChemE
Forum Moderator

Posts: 2744
Joined: 28 Dec 2009

### Re: Infinite Regress vs Circular Reasoning

Natural ChemE wrote:mtbturtle,
Another way of seeing it would be where Infinite Regress is a straight line of propositions,
$P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_5{\rightarrow}...$,
while Circular Reasoning is a loop
$\begin{tabular}{ccc}P_1&{\rightarrow}&P_2\\{\uparrow}&&{\downarrow}\\P_4&{\leftarrow}&P_3\\\end{tabular}$.

The Circular Reasoning loop can be written in linear form as
$P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_1{\rightarrow}...$,
which is the same thing as
$P_1{\rightarrow}P_2{\rightarrow}P_3{\rightarrow}P_4{\rightarrow}P_5{\rightarrow}P_6{\rightarrow}P_7{\rightarrow}P_8{\rightarrow}P_9{\rightarrow}...$
so long as you assert the periodic equalities
1. $P_1=P_5=P_9=....$;
2. $P_2=P_6=P_{10}=....$;
3. $P_3=P_7=P_{11}=....$;
4. $P_4=P_8=P_{12}=....$.
This demonstrates that any instance of Circular Reasoning is necessarily an instance of Infinite Regress.

Since Circular Reasoning can be expressed as an Infinite Regress plus additional constraints, i.e. periodic proposition equalities, then Circular Reasoning is a special case (subset) of Infinite Regress.

Good explanation Natural ChemE.
Don Juan
Active Member

Posts: 1158
Joined: 17 Jun 2010

### Re: Infinite Regress vs Circular Reasoning

Well, I couldn't find anything in any of my sources on fallacies, critical thinking that equated or even related circular reasoning and infinite regress. In fact none of them even mentioned infinite regress. On some webpages where I've come across circular reasoning and infinite regress mentioned together there is a difference between them in which an infinite regress of justifications are offered or circular reasoning. For example,

http://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma

Here, one has a mere choice between:

1. an infinite regression, which appears because of the necessity to go ever further back, but isn't practically feasible and doesn't, therefore, provide a certain foundation;
2. a logical circle in the deduction, which is caused by the fact that one, in the need to found, falls back on statements which had already appeared before as requiring a foundation, and which circle does not lead to any certain foundation either; and finally:
3. a break of searching at a certain point, which indeed appears principally feasible, but would mean a random suspension of the principle of sufficient reason.

mtbturtle
Banned User

Posts: 9742
Joined: 16 Dec 2005

### Re: Infinite Regress vs Circular Reasoning

mtbturtle,

I'm not sure that I understand; why do you need a source when you can prove it?

Also, tangential question, but what subjects do texts on critical thinking tend to discuss?
Natural ChemE
Forum Moderator

Posts: 2744
Joined: 28 Dec 2009

### Re: Infinite Regress vs Circular Reasoning

In the way you expressed it, of course, there's a way of making them similar, but in actual practice, the two are thought differently. In circular reasoning, the conclusion is justified by defining the terms in such a way that the conclusion follows. In justifications that are infinite in length, I'd conclude that the justification fails because it wasn't able to draw the conclusion (though perhaps if there were some sort of inductive proof mechanism, a conclusion could be made to follow).
owleye

### Re: Infinite Regress vs Circular Reasoning

Natural,

prove what? This is more a matter of semantics and proper definition, use for me. Perhaps I'm just being a bit of a pedant.

The couple of books I have deal with the kinds of argumentation found at Fallacy Files

mtbturtle
Banned User

Posts: 9742
Joined: 16 Dec 2005

### Re: Infinite Regress vs Circular Reasoning

Perhaps NaturalChemE needs more proof, for example, you need to show that P1->...->P1 is the same as P1->...->Pn. Of course they are both chains, but at what point the unique feature of the circular reasoning (the identical events) belongs to the form of infinite regress? Does Pn in infinite regress allow recursion if we will consider its definition? If both line of propositions occur in some context (structural-level thinking), then possibly there can be some points of similarities and differences by noting the structural contexts for both, ...->P1(n) and ...->Pn(n), but if they occur independent of context (and focusing on their internal structure), then the two line of propositions are unique from each other.

However I still see the good part in your explanation because if we consider reality, identity can rest absolutely in logical forms, but not in actual forms, so that P1 cannot be exactly the same as P1 on some actual contexts - no two events are exactly the same, making circular reasoning a form of infinite regress.

In some sense, infinite regress in context will loop back to the observer so that it becomes a form of circular reasoning. So they seem to kind of alternate, circular - infinite - circular, etc if one is in structural-level thinking.
Don Juan
Active Member

Posts: 1158
Joined: 17 Jun 2010