Saturday, September 26, 2009

Induction by Enumeration and Sophistry

A person who upholds “induction by enumeration” is one who believes that, by counting instances, limiting one's reasoning to some finite list of particulars, or in some way including all the particulars that one is reasoning about (such as saying “etc.”) he can reach an inductive conclusion that is true.

One example is Peter of Spain's understanding of induction: “Induction is a progression from particulars to universal. For instance, Socrates runs, Plato runs, Cicero runs, et cetera; therefore every man runs.” (Dr. John McCaskey's translation of Peter of Spain, Tractatus (a.k.a. Summule Logicales), 56.12-5)

Another example is the modern one, about swans. Say that you've observed numerous white swans across several continents, and finally inductively conclude that “all swans are white.” In both examples, all that has been done is that one has made observations (or held that the unobserved will be like the observed, such as when saying “etc.”) and formed a generalization off of these observations.

For centuries, philosophers have cited problems with this method of reasoning. “It correlates, but doesn't show the cause,” or it makes a “hasty generalization,” we might hear from people today; in the 17th century, Francis Bacon said that such induction was “childish”; in the 15th century, Lorenzo Valla argued that all enumerative inductions are forms of circular reasoning, since the conclusion is merely a restatement of the premises.

In the fourth century B.C., Aristotle held that the thinking characteristic of inductions by enumeration was “sophistic.”

Knowing Triangles “Universally”

Despite being classified as an enumerative inductivist, Aristotle seemed to present only one inductive argument that used enumeration, one which hypothetically showed that a certain property or characteristic belonged to triangles as such by citing all of the kinds of triangles. And here he argues, against enumerative induction, that knowing something which applies to all types of triangles does not show that it applies to triangles universally, to the essence of triangles:
[E]ven if someone proved of each triangle, whether by a single demonstration or by different ones, that each has [angles which equal] two right angles ([proving this] separately of the equilateral, the isosceles, and the scalene), he would not yet know of the triangle that it had [angles equal to] two right angles, except in the sophistic manner; nor would he know it of triangles universally, not even if there are no other triangles besides these. For he would not know [it] qua triangle, nor of every triangle—or rather, [he’d know it of every triangle only] in number, but not of every [triangle] with respect to form, even if there were none of which he did not know [this]. (Posterior Analytics Book 1, Chapter 5, 74a25-32)

The term “sophistic” had two meanings for Aristotle, one pertaining to the actual Sophists of philosophy, and the other referring to the specious and defective ways by which the Sophists would reason about and argue their cases. Here, he means the latter, that the kind of reasoning he's discussing is of a fallacious character. He's telling us that an argument which reasons from particular statements about something (for instance, “this swan is white”) to the union of such statements into a universal one (“Swans are white”) is not establishing the conclusion it's trying to make. To observe that something holds for a variety of particulars under the same group, that they have it in common, is not the same as showing that it belongs essentially to those particulars, precisely what the inductive conclusion seeks to make plain.

Clearly, Aristotle thought there was more to induction than generalizing from counted instances, or even finding that all the particulars have the property that one is arguing to be essential to them. Induction, in Aristotle's works, is defined as a progression from particulars to a universal, but the enumeration of triangles in Posterior Analytics I 5 failed to reach a universal (it failed to give the person the knowledge of the characteristic applying to triangles “universally”). As Dr. McCaskey concludes in his essay, Freeing Aristotelian Epagôgê [induction] from Prior Analytics II 23, page 7 of the PDF, “[t]hus the complete enumeration in Posterior Analytics I 5 of three types of triangles is not an induction. If it were, it would have led to knowing 'universally,' and it did not.”

One of the Very Hardest Things

To complete some enumerative inductions, it was thought that a phrase including all the unobserved cases was needed, such as “etc.,” and “and so on in all cases like these” (the philosopher Jacopo Zabarella believed this, for instance).

When arguing for universals (concepts, universal statements), Aristotle notes that people often say “and such with all the cases,” as a way to firmly establish a universal that is under discussion. But Aristotle recognizes that it is “one of the very hardest things to distinguish which of the things adduced are ‘of this sort’, and which are not” (Topics, VIII 2, 157a25-27). He understood that adding such a phrase in no way affected the validity of the argument, because in arguing for a universal, the very question that is being disputed is whether or not particulars can be adduced which are of such a nature so as to justify the universal in the first place. Adding phrases like “etc.” seem to imply that both parties--the person raising the question and the other answering it--already have an idea as to what particulars are to be included under a universal and why, which simply is not the case. (This gives credence to Valla's point that enumerative inductions merely beg the question, since those arguments assume the universal being proved through the union of the particular statements, or the particular things.)


In truth, those who adopt enumerative induction take the task of generalizing too lightly. They attempt to justify their conclusions by merely noting similarities, when in reality there is something more that is needed in order to demonstrate their point. For Aristotle, this needed element was an identification of the essence or form of the subject that one was arguing about; this was the means by which one can reach universal knowledge about the subject, and thus reach true inductive conclusions. Until one has reached this essence, a generalization could only point out that a number of particulars happen to have a characteristic in common, in a coincidental manner, and thus that the generalization isn't warranted.

(I've neglected to discuss the enumerative induction that seems to occur in Prior Analytics II 23, because I've already discussed it in my series “Aristotle's 'Two' Views of Induction: McCaskey's Resolution” and due to the fact that McCaskey deals with this issue expertly in his essay that I mentioned above.)

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