Wednesday, September 23, 2009

Aristotle's "Two" Views of Induction: McCaskey's Resolution (Part 3)

McCaskey’s Revision of Prior Analytics 2.23

Throughout this series, I’ve maintained that there are two conflicting interpretations of Aristotelian induction, and that Dr. John McCaskey has discovered a way to resolve the issue, to the detriment of one of those views. His resolution is essentially a revisionist interpretation of Aristotle’s Prior Analytics, book 2, chapter 23 (PrA 2.23); an interpretation that, if correct, will make the eight uses of the term “induction” (that is, those uses that originally posed the controversy) consistent with the other eighty-eight uses of the term that support McCaskey’s interpretation.

PrA 2.23 is composed of three paragraphs and is found near the end of the book, after Aristotle finishes a lengthy exposition on the syllogism and conversion of terms, and the chapter starts as he compares the role of conversion with types of argument such as “example” or “objections.”

The first paragraph says nothing that damages McCaskey’s interpretation of induction, and the last sentence of it is consistent with that interpretation: “[f]or we have conviction about anything either through deduction or from induction.” (PrA 68b13-14. Compare with Aristotle’s other claims that there are two ways of reasoning or arguing, one is induction and the other is deduction.)

It is the second paragraph that poses the difficulty. To understand the basis for the conventional interpretation, let’s follow McCaskey’s approach, and summarize how Aristotle is interpreted in light of this paragraph.

The second paragraph begins, “Induction, then--that is, a deduction from induction--is deducing one extreme to belong to the middle through the other extreme.” Afterwards, Aristotle gives this example (I‘m abbreviating Aristotle‘s argument for sake of length):

(1) Man, horse, and mule are long-lived.
(2) Man, horse, and mule are bileless.
By conversion of (2): (3) Bileless animals are man, horse, and mule.
By (1) and (3): (4) Bileless animals are long-lived.

Here, Aristotle is drawing a universal conclusion (“B is A”: bileless animals are long-lived) by deducing one extreme (“A”: long-lived) to belong to the middle (“B”: bileless) by means of the other extreme (“C”: particular types of animals, specifically man, horse, and mule). The deduction is valid if the conversion from (2) to (3) is valid (that is, if “man, horse, and mule are bileless,” can be restated validly as “bileless animals are man, horse, and mule.”); and the conversion is valid only if the only bileless animals in the world are men, horses, and mules. According to the conventional interpretation, Aristotle is asking us to presume that this is true for the sake of illustrating his point. The paragraphs ends with, “One must understand C as composed of every one of the particulars: for induction is through them all.”

Therefore, he is saying that the only valid induction is a complete enumeration (“for induction is through them all [the particulars]”); that induction is ultimately a kind of deduction (a “deduction from induction” that “[deduces] one extreme to belong to the middle through the other extreme”); and that induction is reducible to a deduction, since the “inductive” argument here is really a syllogistic argument that enumerates all the particulars. This is what the conventional interpretation concludes about Aristotle’s view of induction.

The “Deduction from Induction”

Now, how does McCaskey challenge this interpretation? Answer: by using the surrounding text to elucidate what Aristotle means by a “deduction from induction.” McCaskey says that an “alternative interpretation can be found by reading the chapter from the outside in rather than from the inside out.” [p. 50] I said earlier that PrA 2.23 consisted of three paragraphs; McCaskey is suggesting that we imagine that the second, substantive paragraph is missing and has to be reconstructed from the surrounding passages (the first and third paragraphs).

The first paragraph states that all knowledge becomes such through the syllogistic figures presented earlier (the chapters before 23), and ends with the statement that we have belief about anything through deduction or from induction. None of this suggests a new understanding of induction is to follow. Now we ignore the second paragraph, assuming it exists but pretending that its contents are unknown, and focus on the third paragraph. It begins “This is the sort of deduction that is possible of a primary and unmiddled premise,” which indicates that the second paragraph must have been about a “deduction of an unmiddled premise.” The next sentence plainly states that there are two kinds of deductions: (1) deductions of middled premises in which the premise is the conclusion of a syllogistic argument with a middle term, and (2) deductions of unmiddled premises, in which the role played by a middle term is carried out by an induction. McCaskey decides to call the first a “deduction-from-a-middle” and the second a “deduction-from-induction,” and notes that the second paragraph must have been an example of a “deduction-from-induction,” instead of the “deduction-from-a-middle,” which had been treated substantially in earlier chapters. McCaskey argues that paragraph three is consistently about the differences between the “deduction-from-a-middle,” and the “deduction-from-induction.” Afterward, he shifts our focus back to the second paragraph, taking what we’ve learned with us.

Based on the third paragraph, we expect to see an example of a “deduction-from-induction” in the second paragraph, and we are not disappointed. Again, the second paragraph begins “Induction, then--that is, a deduction from induction--is deducing one extreme to belong to the middle through the other extreme.” [My emphasis] The example given is the argument that all bileless animals are long-lived, which will be addressed in the next section.

Students and commentators have had difficulty with those first four words (“Induction, then--that is“), as the phrase seems to indicate that “induction” is really a shorthand for the more specific “deduction from induction,” which implies that our understanding of induction in his other works should be corrected by considering them as “deductions from induction.” McCaskey rejects this conclusion, and claims that the “induction” shorthand only applies to the few sentences that follow in the second and third paragraphs; this would mean that Aristotle is only using “induction” in those paragraphs as a lecturer’s shortening of the long-winded phrase “deduction from induction,” and therefore is not to be confused with the “induction" discussed in the Topics, for example. As McCaskey aptly states, either his own interpretation of “induction” here being a shorthand is correct, or we must accept the absurd conclusion that, “without warning, Aristotle has proposed a new understanding of induction, inconsistent with the rest of the corpus [that is, his other works] and inconsistent even with the immediately preceding sentence.” [p. 54]

Converting a Deduction from Induction

Even if the phrase “deduction from induction” doesn’t mean that induction is a form of deduction, doesn’t the example given justify the conventional interpretation, that induction is a complete enumeration that can be turned into a deduction? Here is where McCaskey suggests an alternative interpretation for PrA 2.23, specifically the second paragraph, one in which we learn how inductions become the premises for deductions.

He begins with a broad overview of what the first two paragraphs are about:

From the opening of the [second] paragraph and from what Aristotle said in the preceding, introductory paragraph we know he wants to exhibit how a deduction-from- induction ‘comes about through the figures previously mentioned,’ that is, through the syllogistic figures. His tool for doing so will be conversion, the subject of discussion in the preceding chapter and the subject Aristotle mentioned right at the beginning of this one. His subject for the chapter’s middle paragraph, then, is how conversion is used to effect a deduction-from-induction. Aristotle will first present the relevant syllogistic figure using a simple example, an example in which the conversion is justified by a method other than induction, in this case by surveying one or a few particulars or kinds of particulars. He will then expand the example by replacing a conversion justified by survey with a conversion justified by induction. He will spend the bulk of the paragraph setting up the simple example and discussing the role that conversion plays. He will execute the expansion in the paragraph’s final words.

He then notes that the example which follows is an application of a conversion rule Aristotle brought to our attention--and proved--in the preceding chapter, PrA 2.22. “When A and B belong to the whole of C and C converts with B, then it is necessary for A to belong to every B.” This is exactly what Aristotle is arguing to be the case with a deduction from induction--that it has this syllogistic figure:

(1) All C is A.
(2) All C is B.
By conversion of (2): (3) All B is C.
By (1) and (3): (4) All B is A.

Aristotle lets “A” stand for “long-lived,” “B” stand for “bileless,” and “C” stand for particular long-lived animals, such as a man, horse, or mule. We are left guessing if he means one particular man, horse, or mule, or several of them, or whether he means specific men, horses, or mules, or particular kinds of long-lived animals. But what we do know is that Aristotle is not saying that men, horses, and mules are the only long-lived animals in the world: he is only using those three animals as a surveyable and illustrative list of long-lived things--a “sample” of such things, as McCaskey puts it.

The argument then becomes:

(1) All particular things on the list are long-lived.
(2) All particular things on the list are bileless.
By conversion of (2): (3) All bileless things are particular things on the list.
By (1) and (3): (4) All bileless animals are long-lived.

Here, we have a deduction from a surveyable list: all of the samples of particular things Aristotle introduced are both bileless and long-lived, and the conclusion is that everything bileless is long-lived, since the conclusion only extends as far as the surveyable list. This is not yet a deduction-from-induction, but Aristotle sees no difficulty in expanding it to be so. To do so, he redefines C, “But one must understand C as composed of every one of the particulars: for [a deduction-from-]induction is through them all.” [page 58 of McCaskey’s PDF] Earlier, Aristotle defined C as particular long-lived things (with man, horse, and mule as examples), but now means C to be all particular long-lived things, because a deduction-from-induction is not a deduction from a surveyed list, but through all the particulars. With this, Aristotle proceeds to the next paragraph, and his expansion of C finishes his demonstration of how a deduction-from-induction is presented in a syllogistic figure and how it is properly converted.

Unfortunately, Aristotle doesn’t completely explain his line of reasoning: from the looks of it, those of us reading him have seemed to miss a step. With his redefinition, we now have:

(1) All particular long-lived things (men, horses, mules, and others) are long-lived.
(2) All particular long-lived things (men, horses, mules, and others) are bileless.
By conversion of (2): (3) All things bileless are particular long-lived things (men, horses, mules, and others).
By (1) and (3): (4) All things bileless are long-lived.

Aristotle justified the earlier conversion by surveying the particulars in his sample; he now justifies this expanded conversion by means of induction.

Ingeniously, McCaskey remarks that the justification lies somewhere else, outside of this paragraph; Aristotle isn’t saying that he’s justified in extending his argument to all particulars because of the list of long-lived things surveyed in the earlier argument. (Contrary to the conventional interpretation.) “He is saying that because of some induction performed elsewhere, he is justified in claiming that not only are all particular long-lived things bileless (2), but that every particular thing (or kind of thing) that is bileless is also long-lived (3).”

The question then is: what would justify premise 3? Premise 3 would be justified if Aristotle believed that lacking bile was the essential cause of longevity in all animals; if this were the case, then the conversion would be valid, and the universal statement (premise 4) would be true. This relates back to what Aristotle said about triangles, to the fact that knowing something to be true of all triangles does not make it valid to conclude that it applies to triangles as triangles: justifying the conclusion would require identifying the essential nature of the universal’s subject. And this, identifying the essential cause of something being the kind of thing it is, is what Aristotle believes induction is, as McCaskey’s survey of his works has persuasively argued. As McCaskey points out, “It was an ancient view that lack of bile was the essential cause of longevity in animals, and Aristotle agreed. That belief is the step that Aristotle presumed we knew, and that he presumed we knew was a discovery reached by induction.” [page 59-60]

(Aristotle affirms that lack of bile was the essential cause of longevity in animals in Parts of Animals, book 4, chapter 2, lines 677a30-35. He calls bile a “purifying excretion,” in one translation, suggesting that its leaving the body extended the health of animals.)

A “deduction from induction,” then, utilizes the same syllogistic figure and is validated by the same law of conversion as the earlier deduction from a surveyed list, but the justification for the conversion itself is understanding that the universal statement is valid for all the particulars due to their essential nature. What’s important to know here is that “(4) All things bileless are long-lived.” is not the inductive generalization, but rather is the deductive conclusion. “Induction operates in the premises, not in the conclusion,” as McCaskey aptly remarks.

Aristotle is not here arguing for an inductive generalization, but rather is demonstrating that once one knows the premises by induction, it is possible to then form a syllogism, a deduction from induction; the induction here does the work that a middle term does in a deduction-from-a-middle. The third paragraph said that the second was about a “deduction from induction,” in which a deductive conclusion results from an induction operating in the premises, and if read correctly (McCaskey’s interpretation), that is what the second paragraph is about.

If McCaskey’s revision is correct, then PrA 2.23 is not about an induction being proved by complete enumeration, or that an induction can be changed into a deduction by assuming that men, horses, and mules are the only bileless and long-lived animals that exist. It has nothing to do with coming to an inductive conclusion whatsoever.

As to what PrA 2.23 actually is about, I‘ll let McCaskey have the last words, as I don‘t think I can put the point any better myself:

[The passage in PrA 2.23] is about the reason and method by which inductive conclusions, once reached, can provide the premises for syllogisms. The reason they can be is that conclusions reached inductively are universal. They apply to all particulars of a kind, not just those surveyed in performing the induction. The method by which they can be is the swapping of subject and predicate by conversion.

Such conversion is the goal of identifying essence. If one can determine by Socratic induction that the essence of being the best is having the most knowledge, then one can convert ‘All men who are the best in a profession are the ones who have the most knowledge of that profession’ with ‘All men who have the most knowledge of that profession are the best in that profession.’ If, as in the Metaphysics, it can be claimed that contrariety is the maximum difference of two ends of a continuum, then ‘contrariety’ and ‘maximum difference of two ends of a continuum’ can be interchanged in a syllogistic premise. Induction, for Aristotle, is a process by which such equivalences can be reached, and thus premises for deductions generated. [p. 61]

Comments would be appreciated.

1 comment:

  1. I will need to go and read this properly in Aristotle, but it occurs to me that there could be another interpretation open here. Aristotle could just be saying what must be the case *when the induction is valid.* If the induction is valid, then the argument Aristotle gives will be a syllogism. That is to say that the validity of the induction is reducible to the validity of the conversion. This need not mean that induction itself is just a kind of deduction, or even tell us anything at all about whether the conversion is in fact valid. This brings a degree of uncertainty to the conclusion, of course, but things stand not significantly different with demonstrations, since they are only good when the middle term we use is a cause, and this depends on our having the right definition, and we cannot be certain of having the right definition. Hence, somewhere in the Posterior Analytics we find Aristotle saying that we cannot demonstrate definitions -- cannot show by logic that they are correct -- but demonstrations *exhibit* definitions. I think you need to distinguish epistemic uncertainty from logical uncertainty. You can know a necessary truth without knowing that its truth is logically necessary. I think that you can also know the conclusion of a demonstration even without being certain that the demonstration is in fact a demonstration (because uncertain whether the original definition was correct.) As long as the definition is correct or, in the case of induction, the complete enumeration is true, then this suffices for knowledge, irrespective of whether you are justified in believing it to be true.