Saturday 10 December 2011

Circular Reasoning

Socrates. Louvre. [1]
Socrates is a man.
All men are mortal.
Socrates is mortal.

Let's begin from the beginning: what is a sentence? A sentence for our purposes is a sequence of subject, verb and predicate, in that order.

Above we have three examples; in the first one, the subject "Socrates" is in red, the verb "is" and the predicate "a man" are in blue.

Together these three sentences (aka statements) constitute a well-known example of valid deductive argument, where the first two sentences (the premises) provide evidence for the conclusion (the last sentence).

Considered as an argument, one could read the three sentences above like this:

Given that "Socrates is a man" and that
"All men are mortal",
it must be the case that
"Socrates is mortal".

Let's call this Version 1 of the argument.

And the argument is said to be valid because the conclusion ("Socrates is mortal") follows logically from the premises ("Socrates is a man" and "All men are mortal").

Note something in this example: the subject of the first premise ("Socrates") and the predicate of the second ("mortal"), both represented in red and joined by a variation of the verb ("is" replacing "are"), form the conclusion: "Socrates is mortal".

That is, in the example the conclusion is formed by recombining pre-existing elements of the premises. [2]

Let's sum things up: deductive logic allows to produce new knowledge (the conclusion), from a known set of premises which are assumed true. If the premises are indeed true, the validly deduced conclusion is inescapably true.

However, as the example shows, the conclusion is also limited by the premises (here, by the pre-existing elements recombined).

To explain this last point: from the two premises given one cannot conclude "Skippy is a bush kangaroo" (because, in the premises, neither "Skippy" is a subject, nor "a bush kangaroo" is a predicate); it's not possible to conclude either that "Socrates is not a mortal" (because it does not follow logically from the premises). And one cannot conclude "Socrates loves me" (even if you were a male), because "to love" is not a verb available.

In this sense, conclusions validly derived from a set of premises are always determined by those premises.

Let's now consider this: the premise "Socrates is a man" informs us that Socrates is an element of the set of men; this way in the premise "All men are mortal" the subject "All men" can be considered shorthand for a list of all individuals in the set of men, including Socrates! In other words, the subject "All men" is equal to the list "1st guy, 2nd guy, ..., Socrates, ..., and Last guy".

So, one can delete the first premise and, using our list, rewrite the second as "1st guy is mortal" and ... and "Socrates is mortal" and ... and "Last guy is mortal".

So, by replacing the old premises with the new ones, Version 1 of the argument, originally stated as:

Given that "Socrates is a man" and that
"All men are mortal",
it must be the case that
"Socrates is mortal".

Can be re-stated now as:

Given that
"1st guy is mortal" and ... and "Socrates is mortal" and ... and "Last guy is mortal",
it must be the case that
"Socrates is mortal".

Let's call this, Version 2 of the argument. Compare this second version with Version 1, above.

One finds that the sentence "Socrates is mortal" appears twice in Version 2: as premise and as conclusion. We concluded where we started from: the reasoning moves in circles.

When this happens in informal logic, as one finds in texts and conversation, it is a logical fallacy, not surprisingly called "circular reasoning": the argument is meant to prove that the conclusion is true, based on premises presumed to be true. And in this case one of the premises is the conclusion.

What is all the fuss, the reader could ask. It's only a matter of checking for repeated sentences, isn' it?

Not quite. As it happens, in informal logic, unlike the example, the same sentence can be re-written in many different forms with the same meaning and it can be difficult to notice that two apparently different sentences in reality are the same.

Further, as my example shows, even if the argument is fallacious (as it was in Version 2, if we consider it as an informal logic statement), this does not mean that the conclusion is false (because it's the same conclusion of Version 1). Further, my ingeniously crafted example shows an additional fact: a valid argument (Version 1, for instance) could be re-stated as circular reasoning (Version 2) and a circular reasoning evidently could be re-stated as a valid argument.

The main points are: (1) all valid conclusions are determined by the corresponding premises; (2) it can be tricky to correctly identify a fallacious circular argument, and (3) simply to claim that an argument is circular does not prove its conclusion false.

In consequence and generalizing a bit: the sure way to oppose an argument is to demonstrate that either (1) the conclusion does not follow from the premises (if it doesn't indeed) or (2) the premises are false (if that's really the case).

I don't know whether this reflects the readers' experience, but I have often encountered the "circular reasoning" wildcard applied rather indiscriminately, while surfing the web and even in formal readings.

It works like this: someone argues something you don't like, but you can't challenge the validity of the argument or its premises. Easy way out: Yell "circular reasoning" like crazy.

I may provide some concrete examples of this and related issues in the not-so-distant future.

Photo Credit:
[1] Socrates, Louvre Museum. Wikipedia.

[2] Strictly speaking, the examples here are part of Predicate Logic, but the main points apply regardless to other forms of logic, and particularly to informal logic, which one commonly finds in web discussions.


  1. @Magpie: I don't really agree with you here (but I won't shout "circular reasoning"). As it stands, your Version 2 is not in fact an example of circular reasoning. If I accept as true that "Socrates is a man" it may be trivial to conclude that "Socrates is a man" but it is not circular reasoning. Circular reasoning arises when you try to establish propositions A and B by arguing that A entails B and B entails A. Of course if you (and everyone else) accepts A and you establish that A entails B, then you can logically deduce that B is true (and, trivially, that A is as well...but that's not much fun).

    As a technical aside, in a mathematical context, you can not always switch from a version 1 to a version 2 form. For example, if your initial statement is "all real numbers have property P", you cannot switch to a list of statements for "x has property P" for every real number x (because real numbers are uncountably infinite).

  2. You are... I mean, this is... uh... err... "Circular reasoning! CIRCULAR REASONING!!!" ;-) Just kidding.

    Thanks for the input! I'll have to give your first paragraph a lot more thought. I think I get some of your idea.

    The second paragraph I think I understand your point and you are right. Although, if I understand your point, in the case of the set of men, of course, it is not an uncountably infinite set.

  3. Okay, let's see if this is what you meant with your first paragraph.

    Although the text uses Predicate Logic statements (cleverly disguised, of course), this logically valid proof of the Sentential Logic statement S => S could do the trick:

    1 ! ! S_____Assumption
    . ! !-----
    2 ! ! S_____1 Reiteration
    . !
    3 ! S=>S____1-2 =>Introd.

    This is a valid proof, therefore the statement cannot be fallacious. It is, however, trivial in most contexts: obviously, S always entails S (i.e. S => S).

    As this follows my own reasoning, Version 2 is not fallacious, but trivial.

    Is that what you meant?