Aristotle's View of Induction: A Summary

[On Induction]: “The soul is so constituted to be capable of this process.” [Aristotle, Posterior Analytics 2.19, 100a14]

In the history of induction, Aristotle features prominently as the first person to explain what it was. While Socrates practiced induction and sought universal definitions, Aristotle was the first to discuss the process of inductive thinking itself. And even though Aristotle thought that “[what] sort of thing induction is, is obvious,” he nevertheless took some effort in explaining its origin, its logical process, and the benefits that could be gained from using it (Topics 8.1, 157a8).

Now that I've spent a few months studying him and other inductive thinkers, I'd like to present a summary on his view of induction. In this summary, I'd like to cover four things, specifically: (1) Aristotle's mention of Socrates as the originator of induction, (2) his overall portrait of the process of induction given in Posterior Analytics 2.19, (3) the more specific elements of induction which he adopts, and (4) what induction essentially is, as opposed to what it is not.

Induction: with a Tip of the Hat to Socrates

Aristotle credits Socrates with the invention of induction:

Socrates was occupying himself with the excellences of character, and in connection with them became the first to raise the problem of universal definition. . . . It was with good reason that he should be seeking the essence, for he was seeking to argue deductively and the beginning of deductive arguments is the essence. . . . For two things may be fairly ascribed to Socrates—inductive reasoning and universal definition, both of which are concerned with the starting point of science. (Metaphysics 13.4, 1078b24-30)

Socrates was seeking deductive conclusions, such as whether virtue could or could not be taught, and in order to argue his point, he needed to know what the term “virtue” meant, its definition (in the Meno, 71a-b, d). He realized that deduction would be useless for discovering this, since deduction relies upon pre-existing knowledge, and he had no knowledge of what “virtue” even was as yet, only tentative guesses. (For instance, he reasoned that one of virtue's properties was that it could be granted to men by the gods. But this was only tentative knowledge; to be certain of this, he had to know what “virtue” is (Meno, 100b).)

In order to reach the essence of “virtue” and similar moral topics (e.g. courage, friends), to define it, he set out on inductive quests, questioning others about what particulars hold in common, which would allow him to discover this identical attribute, or essence. Aristotle agreed with and adopted Socrates's inductive method, and proceeded to explain the role it plays in our lives, in passages spread out across his works.

Posterior Analytics 2.19: From Perception to Memory to Experience to the Universal

One of the most important explanations of what induction is comes from Aristotle's Posterior Analytics. The work follows the Prior Analytics, which is fundamentally about deduction, specifically the elements of the syllogism such as “premise” and “term,” the syllogistic figures (moods), and how we can gain knowledge through syllogistic/deductive reasoning, something which Aristotle calls “demonstration.” The Posterior Analytics is fundamentally about the starting points of knowledge, the primary and universal principles, and whether or not a process exists by which we can reach such principles.

Book 1 of the Posterior Analytics starts with the claim that all learning and teaching by means of argument proceed from pre-existent knowledge, and that all subjects that can be taught or learned follow this principle. Even the syllogistic and inductive forms of reasoning follow this, he adds; deduction assumes the audience already knows and accepts its premises, and induction assumes the particular from which an implicit universal is exhibited (Posterior Analytics 1.1). While deductions give us justified conviction when properly carried out, they all rest on primary premises, and exactly how we gain these is an open issue. In the third chapter, Aristotle rejects two theories, (1) that all knowledge is false because of an infinite regress, and (2) that all knowledge is true because we can use circular demonstration to prove anything. He notes that not all knowledge is demonstrative (produced by deductive thinking), and that there is a process or “originative source” from which we gain definitions (chapter 3) and primary premises or principles. In chapters 4 through 12, he describes this process: by some means, we form universals (which serve as the “middle term” and constituent terms of premises and of syllogisms) and come to know primary and universal principles, which are true by the essential nature of the subject matter. In turn, these primary principles can become the premises of deductions (though not always).

Book 2 is an extended treatment of issues that were brought up in Book 1, on the relations between causes, definitions, essential natures, and demonstrative reasoning. In chapter 7, Aristotle denies that the discovery of essential natures can be carried out by definition or by demonstration, and in chapter eight he states that to know the essential nature of something is to know the cause of the thing's existence. Chapters 11 through 18 discuss the relations between causes, effects, forming universals, and definitions. The final chapter, 19, brings us back to issues which confronted us at the beginning of Book 1, concerning the starting points of knowledge, and how we come to know primary premises. He concludes that these premises cannot be innate in us (from birth) or developed from higher kinds of knowledge (such as from demonstrative knowledge, which he denied earlier), but instead is developed from sense-perception.

Thus from perception there comes memory . . . and from memory (when it occurs often in connection with the same thing) experience; for memories which are many in number form a single experience. And from experience, or from all the universal which has come to rest in the soul . . . there comes a principle of skill or of understanding. (Posterior Analytics, 2.19, 100a3-9, trans. Jonathan Barnes. Quoted in McCaskey, Regula Socratis: The Rediscovery of Ancient Induction in Early Modern England, 44)

Almost immediately after, he restates this process, but this time drops his earlier mention of memories and experiences, and focuses instead on the progression from particulars to universals:

When one of a number of logically indiscriminable particulars has made a stand [been perceived], the earliest universal is present in the soul: for though the act of sense-perception is of the particular, its content is universal-is man, for example, not the man Callias. A fresh stand is made among these rudimentary universals, and the process does not cease until the indivisible concepts, the true universals, are established: e.g. such and such a species of animal is a step towards the genus animal, which by the same process is a step towards a further generalization. (Posterior Analytics, 2.19, 100a12-100b1, trans. G.R.G. Mure. Brackets mine.)

It is in connection to this progression from particulars to universals that Aristotle mentions “induction,” as if the whole of the Posterior Analytics has been a build-up to this statement: “Thus it is plain that we must get to know the primitives [premises] by induction; for this is the way in which perception instils universals” (Posterior Analytics, 100b2-3, trans. Barnes).

This connection between induction and the primary premises was suggested earlier, in Posterior Analytics 1.18, in which Aristotle describes a hierarchy regarding universals, perception, particulars, demonstrative reasoning, and induction. In this hierarchy, Aristotle claims that:

(1)Knowledge of particulars depends on sense-perception.
(2)Induction proceeds from particulars, and thus induction is impossible without sense-perception.
(3)Universals, which are predicated of particulars, cannot be grasped except through induction.
(4)Demonstration (the deductive form of argument and knowledge) is developed from universals.

Precisely like Francis Bacon, Aristotle holds that induction is the basis for our concepts, which in turn are the basis for our propositions, premises, and deductions. (I'm referring to Bacon's famous statement that “a syllogism consists of propositions, and propositions consist of words, and words are the tokens and signs of notions [ideas],” New Instrument, 1.14.) More importantly, book 2 chapter 19 explains that induction is our ability to reason from perceptions to conceptual generalizations, and is the process by which we gain our most basic conceptual knowledge, and form more complex conceptual products. It solves Aristotle's initial riddle as to what process, if any, is responsible for the starting points of knowledge.

“This is Clear from Induction”

So now that we know how induction comes to exist, what are the other features of induction, and guidelines for making them? What is induction really about, according to Aristotle?

Induction and Deduction
Induction is one of two ways to reason, persuade, learn, teach, come to have beliefs, and obtain premises, and the other way is syllogistic thinking/demonstrative reasoning (deduction) (e.g. see Posterior Analytics 1.1, 71a6 for his comment on induction and reasoning in general). And as we learned from Posterior Analytics 2.19, induction is the fundamental kind of reasoning when compared to deduction (syllogistic reasoning), as it provides the premises required by deductive thinking. (Part 3 of my series “Aristotle's 'Two' Views of Induction: McCaskey's Resolution” elaborates on how induction provides the premises for deductions, for those interested.)

More specifically, induction is a reasoning process of moving from particulars to universals, as Aristotle explained in Posterior Analytics 2.19. And as he implies in the earlier quote, the “particulars” are not simply the objects of sense-perception that induction starts with, but also less abstract universals (“man”) when considered in a progression to a more abstract universal (“animal”). Induction is thus a method of achieving higher and higher levels of abstraction and generalization.

Aristotle also notes that induction is easier to understand than deduction (Topics 1.12 105a15-19). Indeed, induction is more useful for persuading or convincing people who either haven't been exposed to the sort of issue you're presenting an argument for, or are unconvinced of a certain point in your argument. He makes this kind of point by suggesting that when arguing with others, induction should be used for addressing a common crowd, whereas deduction should be used against skilled debaters; induction with a youth, deduction for the more experienced, mature thinker; induction to add support to a point (Topics 8.2, 157a19-20; 8.14 164a13; 8.1 157a6).

Induction as Limitless
Another characteristic of induction is that it's an open-ended process: the application of an induction extends beyond the particulars that were originally used to form it. An example of induction Aristotle gives is that of what makes someone the “best”: “...Induction, however, is a proceeding from particulars to a universal. For instance, if the pilot who has knowledge is the best pilot, and so with a charioteer, then generally the person who has knowledge about anything is the best“ (Topics 1.12, 105a10-19). Here Aristotle is illustrating how one would go about generalizing from particular instances of someone being “the best” in some field, such as in piloting ships or being a charioteer, and suggests that it is possessing the most knowledge which is what makes someone the “best” in a given field. He makes this suggestion by generalizing “being the best” to all fields or professions, beyond those of piloting ships or of being a charioteer, to any profession and any person who has the most knowledge in his profession. He is able to do this because induction leads to universals, and universals are the sorts of things that can be applied to countless particulars.

As Dr. Greg Salmieri explains in his essay on Aristotle's account of universality,

Universality [...] is a characteristic of our thinking and, in some sense at least, of what our thinking is about. A universal is whatever holds of some multiplicity of things as a whole—it is something that 'by its nature is predicated of many things ([On Interpretation] 17a39-b2, cf. Metaphysics Z.13 1038b11)...' [Salmieri, Aristotle's Conception of Universality, p. 7]

Similarity and Induction
Induction is a progression to universals, but it isn't simply on the basis of the number of particulars that we actually think inductively, but their similarity. As Aristotle states, “[the] study of what is similar is useful for inductive reasoning . . .because it is by induction of particulars on the basis of similars that we claim to bring in the universal” (Topics 1.18, 108b7-12). Things are similar when any attribute that a group of things possess is the same (Topics 1.17, 108a18). The skilled inductive reasoner is one who is trained in drawing out parallel cases, of making comparisons and discerning what is similar (Topics 8.14, 164a16).

One example of this is Aristotle's discussion of goodness in the Eudemian Ethics, that goodness is the best disposition or faculty or state of something that has some functional use or work, for he says afterward:

This is clear from induction, for we posit this in all cases: for instance, there is a goodness that belongs to a coat, for a coat has a particular function and use, and the best state of a coat is its goodness; and similarly with a ship and a house and the rest. So that the same is true also of the spirit, for it has a work of its own. [1219a]

Here, Aristotle brings up a variety of objects, points out something similar about all of them—that they have some particular use or work—and reasons that their “goodness” is the best state of each object, as this allows the object to best perform its particular function. A good house is a house that is in the kind of condition that allows it to perform its function as a place of shelter, so it has features like a roof and walls with a strong integrity, a good foundation, and made of materials which allow it to withstand typical natural events like gusts of wind and rainwater. (Oddly enough, the nursery tale of the “Three Little Pigs” and the big, bad wolf, is very illustrative of my point.)

It is on the basis of the similarity of particulars--what characteristic they have in common--that we attempt to form a new universal, goodness for instance, and discover one of the senses in which anything at all can be “good.” And this kind of reasoning occurs for other universals in Aristotle's works, such as for “triangle,” “contrariety,” and “longevity” (found in the Posterior Analytics 1.5, Metaphysics 10.3 and 10.4, and Prior Analytics 2.23, respectively).

Discovering what is similar about particulars is also how we come to understand what universals--our very concepts and propositions--are about, and indeed what induction is even for.

The Essence of Induction, and What it is Not

As Aristotle said, Socrates was concerned with the essence, with finding what a group of things held in common, so that he could define a concept he had in mind and use it as a term in a deduction. Although he himself never said this explicitly, Socrates was seeking “inductive reasoning and universal definition”; it was Aristotle who would later point this out, and adopt this peculiar method.

Unfortunately, Aristotle doesn't focus on induction as nearly as much as other topics, such as the soul in general (On the Soul), on “being” as such (Metaphysics), and even induction's counterpart, deduction (Prior Analytics). Fortunately, what Aristotle does say about it (and what we've managed to recover of Aristotle's works) is enough to understand what Aristotle truly meant in using the term, what its “essence” is.

In my last few posts on Aristotle and induction, I've stressed (agreeing with Dr. John McCaskey) that we have to stop relying on Aristotle's discussion of it in Prior Analytics 2.23 as the passage on Aristotelian induction. No progress in understanding Aristotle's views on induction, and on coming to gain knowledge, can be achieved without understanding that passage in light of the Topics, Rhetoric, and Posterior Analytics 2.19.

To accept Prior Analytics 2.23 as the best source on induction for Aristotle, and thereby interpret him as advocating enumerative induction, is to ignore his comments from Posterior Analytics 1.5 and 2.7. In 1.5, he states (as I discussed previously) that knowing something to be true of all triangles individually is not the same as knowing something to be true of triangles essentially or truly universally, but rather is to know it in a sophistic manner; and in 2.7, he states that we can't prove an attribute to be essential to some kind/group by bringing in all instances of that group and finding the attribute to belong to each, as this only shows that the attribute is held in common, not that it is essential. (I really have to thank Dr. McCaskey for drawing my attention to this on page 46 of his dissertation.)

If read carefully, Prior Analytics 2.23 merely distinguishes between two different kinds of deductions, not of a “deduction” on the one hand, and an “induction” on the other. The chapter itself gives us a few indications of how this is so. For starters, the supposed “definition” of induction given in chapter 23 states that it's about a deduction: “Induction, then--that is, a deduction from induction--is deducing one extreme to belong to the middle through the other extreme” (68b15-16; my emphasis) While Aristotle repeatably states that induction is a progression from particulars to universals, his discussions of induction outside of chapter 23 never state that it's about deducing anything, or that an induction can be demonstrated through one of the syllogistic figures, as he states here of the “induction” in question.

Perhaps most importantly, those who view Aristotelian induction as an inference ignore the fact that Aristotle's examples of induction weren't inferences at all. His examples of induction discussed what is similar about the objects being discussed in regard to a universal, what property made them of a genus or kind, such as people being the “best” in their professions similarly having the most knowledge of their respective professions. Furthermore, the examples of induction suggested a definition of the universal/term being considered or at least proposed some feature that was unique to the particulars being considered; in other words, his examples of induction focused on clarifying what a given universal means, whether by suggesting an actual definition or at least discussing some distinguishing (though not definitive) characteristic of the relevant particulars. The examples were about what it means (in some sense) to be a “triangle” or a “contrary” to something else, or to be “long-lived.” In his accounts of induction, Aristotle consistently discussed or exhibited some general feature of the particulars subsumed under a universal in light of the examples given (the particulars he would discuss), but never argued that the inductive conclusion could be deduced or inferred by studying the particulars.

In an enumerative induction, the enumerated cases serve as premises, which, if true, allow an inductive conclusion to follow or be inferred which would be true in virtue of the true premises. But such an induction cannot belong to Aristotle's account, because if “inference is a kind of reasoning by which, if the premises are true, something else follows from them specifically because they are true, not only is this not Aristotle's view of induction, it is his very definition of induction's opposite, i.e., of deduction” (McCaskey, Freeing Aristotelian Epagôgê from Prior Analytics 2.23, pp. 365-366).

As opposed to coming to a definite inference, Aristotelian induction was essentially no different than Socratic induction. Socrates, we must remember, wanted to deduce certain things from his terms, like “virtue,” or “friend,” and before he could do that, he had to identify the essence of those universals, what made them the kind of things they were: in this way, he would have definite knowledge of “virtue” or “friend” and would be in the position to state other things about such terms. Like Socrates, Aristotle's practice of induction was an exercise in clarifying or exhibiting the meaning of a universal in light of analyzing particulars. The “essence” of induction is that it is an instrument of our reason, one which allows us to form clear/well understood conceptual generalizations (whether concepts or propositions) by identifying the essential nature of the things being conceptualized.

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.)

Conclusion

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.)

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.

Introduction to Induction: What is Induction and Why Study It?

The aim of this essay is to give a preliminary statement about what induction is, and to present reasons why we should be interested in figuring out the answer.

Three Meanings of “Induction”

The term “induction” has several senses, all of them heavily related to one another, and I think that all of them have to be explained in any viable theory of induction.

Induction as Process

The first sense refers to the process of induction, by which our minds reason from particulars to universal generalizations about particulars, as Aristotle was the first to point out (Topics, Book I, Ch. 12; Rhetoric, Book I, Chapter 2, 1356b1). For instance, we reason from the fact that some individuals have eyes and can see (especially our own first-person visual of this fact), to the generalization that beings with eyes possess the ability of sight. The process involves an element of abstraction, a selective separation and contemplation of something in the world by our minds, which really can only be separated mentally; you can’t, for instance, physically take "durability" from the car that has that feature in order to further study it, but you can think about the property of durability without also having to consider that particular car, or wall, or any given object. This abstraction is how we change our cognitive focus from the particulars we were considering to a general feature or fact that is common to those particulars (and possibly other kinds of particulars as well).

The end result of this process leads to several different kinds of conceptual products, with two of the more familiar ones being individual concepts and causal explanations. For instance, we see White Shepherds and Black Labradors, and notice what they have in common (along with other kinds of dogs), and we abstract from the particular animals we see in order to form a concept “dog,” a general term which applies to a plethora of individual dogs, including unobserved and future referents of the concept. Also, we can observe people fighting angrily, breaking things, or cutting people off on the freeway and generalize from that that “people can experience rage,” that a certain emotion of ours can cause us to do certain violent or drastic things that wouldn’t happen otherwise.

Induction as Generalization

The second sense refers to the generalization itself that results from reasoning from particulars to the general. Generalizations like “all men are mortal,” “the human body resists disease,” and “all chocolate cake is food” can be inductions from our observations and scientific reasoning, such as the realization that microscopic germs can cause diseases in human beings. (Generalizations can also be reached deductively, for instance, since all men are mortal, all farmers must perforce be mortal. Due to this, I’ll speak specifically of inductive generalizations.) Definitions are probably the most familiar inductions that people know, as they state something general and essential to a class of things that differentiate them from other objects.

Induction as a Method

The third sense refers to induction as a method, which differs in its specifics depending on the theorist or philosopher you consult. The method of induction, for Aristotle (and presumably for Socrates as well, given what Aristotle said in his Metaphysics), is a comparison of things which leads to a recognition of similarity, resulting in an identification of an essential nature of the class of objects being thought about. It begins with our particular perceptions and memories of things, and ends with a general, universal identification of what makes something the kind of thing it is, which applies to an unlimited number of things which possess this similarity (Topics; Posterior Analytics, book 2, chapter 19).

A method of induction refers to the descriptive elements of this conceptual activity, the mental process of abstraction, and the end-products of this process--the completed concepts and generalizations which come to exist. More importantly, the method (1) fleshes out what these descriptive elements are in more detail, and (2) provides a normative approach for forming correct inductions and practicing generalized thinking. Bacon’s theory of induction, while saying little on the process of abstraction itself, discusses the steps we should employ in forming inductions and concepts in much detail. (See Book 2 of Bacon’s Novum Organum for his distinctive method.)

Now that I’ve explained these meanings of “induction,” I’ll explain why we need the method of induction, as in a specific theory.

Induction: Why Bother?

The components needed to make inductions are with us at birth: sense organs, the faculties of memory and reason, and the world around us. We habitually make inductive conclusions and generalizations, and try to define things and form new concepts. Sometimes, we reach sound inductive conclusions (“all men are mortal,” “man is the rational animal,” “lightning is a kind of electricity”), and sometimes we don’t (“all swans are white,” “universals are independently-existing Forms”). This might simply be an aspect of the human condition: to generalize, right or wrong.

Whence, then, is the significance of a theory of induction?

A theory of induction is needed because the human way of life is about volition--it’s about our choices. We’re given the biological equipment to make choices, but have to determine for ourselves in what way our minds are supposed to operate; we have to learn how to properly utilize our reason. Forming valid inductions is not pre-programmed into us; if it were, there would be no need to discuss a method of induction, just as there’s no need to discuss a “method” of perception.

Without a method of induction, we risk coming no closer to understanding the world than Socrates did, a man who made it his business to tell people that he held no genuine knowledge. What Socrates needed (and implicitly searched for) was a methodical way to reach definite concepts and definitions, to note genuine similarities and properly abstract from things to reach sound generalizations, bringing about genuine knowledge from which he could then deduce.

The consequences of not explicitly practicing a method of induction (of reaching generalizations) have already become apparent. Ignoring the generalizations that could be formed relating totalitarian/socialistic regimes to economic collapse, Venezuela is quickly becoming the latest experiment in socialism. (Generalizations formed by economist Ludwig von Mises in his work Socialism, for instance.) Despite all evidence to the contrary, there are still people who believe that the configurations of the stars influence and even determine human behavior. And there are people who make generalizations without any sensible attempt at proof and scarce evidence, such as racists, sexists, and religionists.

The reaching of such genuine knowledge is perhaps the most difficult of human endeavors, evidenced by the fact that, over two thousand years after philosophy’s inception, we still aren’t clear on how we actually reach general, universal knowledge that is true.

Our only hope, then, is to determine a definite method or theory of induction. This is what I think is the significance of inductive theory.

Aristotle's "Two" Views on Induction: McCaskey's Resolution (Part 2)

In part 1, I explained that Aristotle is currently understood to have advocated two conflicting views on induction. I said that the interpretation of him adopting enumerative induction is the far more popular interpretation, despite the understandable confusion that results from anyone reading Prior Analytics 2.23 (PrA 2.23). I also said that John McCaskey has found an approach which does away with the popular “enumerative inductivist” interpretation. In addition, it gives even more support to the conclusion that Aristotle, when talking about induction, is almost always referring to the induction presented in his Topics and Posterior Analytics 2.19. Lastly, I said that this approach also presents to us how inductions become the general premises for deductions (syllogisms). Let us now turn to this approach.

Criticism of the Conventional Interpretation

Returning to something I said in part 1, the conventional interpretation of Aristotelian induction is that it is validated by complete enumeration of cases, that it is just a kind of deduction, and that its applicability doesn’t extend beyond the particulars which originally formed the induction. Are these claims really substantiated by all that Aristotle has to say on induction? McCaskey would say "no," and proceeds to show us why through the majority of his dissertation’s first chapter and in his essay "Freeing Aristotelian Epagōgē from Prior Analytics II 23."

In general, McCaskey proceeds through every use of the word “induction” (epagōgē) in the known Aristotelian works, beginning with passages whose overall meaning is clear, and proceeding to the more confusing ones; this method allows us to learn about the meaning of induction by understanding the passage. For instance, McCaskey begins his journey through the 96 uses of “induction” in the Topics, book 1, chapter 12, lines 105a10-19, in which four claims are clearly made about induction:

[Induction] (1) is different from and a counterpart to deduction, (2) is a proceeding from particulars to a universal, (3) results in a universal generalization that extends beyond the particulars that went into its formation, and (4) is generally easier for people to grasp than deduction. [McCaskey, “Regula Socratis,” page 23]

These four claims about induction are repeated multiple times throughout Aristotle’s works, and make it highly doubtful that he would suddenly adopt a view of induction that contradicts one or more of these claims, as would be the case if the conventional understanding of PrA 2.23 is correct. Indeed, McCaskey uses this survey of “induction” to not only elucidate the meaning of the concept “induction,” but also to point out how erroneous the conventional interpretation must be.

Two of his counter-arguments to the conventional view should suffice before I move on to his revisionary interpretation of PrA 2.23.

(1) McCaskey notes that in the vast majority of Aristotle’s inductive arguments, the particulars subsumed in the generalizations are countless and cannot be enumerated successfully before making the generalization, such as his argument that what makes someone the “best” in a profession is their knowledge (in the Topics) or his argument about the nature of goodness I mentioned in part 1. Aristotle never presents (and defends) a completely enumerated list of cases and forms an inductive generalization from them, nor states that an induction can only apply to the cases enumerated and not, for instance, presently unobserved cases.

(2) In Posterior Analytics 1.5, Aristotle gives possibly his only example of a complete enumeration. As McCaskey summarizes it:

He says that knowing something to be true of scalene, isosceles, and equilateral triangles is not sufficient for knowing it to be true of triangles qua triangles [as in their essential nature]. It may be known of each triangle taken singly, but not of triangles ‘primitively and universally,’ not ‘of triangles as [triangles].’” [McCaskey, page 46]

Here, there should have been a perfect case for a defense of enumerative induction, which Aristotle supposedly supports according to the conventional interpretation, and yet Aristotle flat out denies that something can truly be known about triangles by considering each individual triangle--the very method that an enumerative inductivist would be forced to use by his own doctrine. Here, he suggests that truly knowing something about triangles has something to do with identifying the essence of triangles, rather than completely enumerating cases. This gives support to the other interpretation of induction, because it is closely related to the inductive arguments in Aristotle’s Metaphysics, Physics, and Eudemian Ethics that strongly suggest that induction is a tool for identifying the essence or nature of something.

McCaskey has several more counter-arguments in his arsenal, but none more powerful than his reinterpretation of PrA 2.23, the key passage cited in support of the enumerative induction interpretation.

The Justification for Induction--Or Lack of It

‘If we demand a proof for everything, he [Aristotle] had said, ‘we shall never be able to prove anything, since we shall not have a starting point for any proof. Certain things are obviously true and do not require proof.’

‘Prove it,’ his nephew Callisthenes had said. Aristotle was glad Callisthenes had gone off with Alexander. He was not sorry to learn he’d been killed.

Obviously, Aristotle saw, it is impossible to prove that anything is obviously true.

Even that.

He enjoyed the paradox. [Joseph Heller, Picture This, 1988, p. 288]

In my first post, I said that I had two misgivings about whether a theory or method of induction could be successfully presented; the point of this post is to discuss one of these misgivings.

The issue is: does induction need a justification?

A “justification” is a conclusive reason (or a number of such reasons) for believing that something is proper or warranted. So, if there are conclusive reasons for believing that induction (as a cognitive process employed by us) is proper/warranted, does the process of induction need such reasons? Presently, I don’t think that induction needs a justification, and will now explain why.

Perhaps the best way to make my point is through analogy. With that thought in mind, let’s look at a few examples, starting with Aristotle’s “Principle of Non-contradiction.” (Hereafter, “Principle of Non-contradiction” is shortened to PNC.)

Aristotle’s “Principle of Non-Contradiction”

The PNC has several features, among them is that it states that two opposite assertions cannot be true at the same time: that this is impossible. (See Metaphysics Book 4, Chapter 6, 1011b13-20) For instance, one can state that “this bird is looking at me,” and alternatively state that “this bird is not looking at me,” but cannot state both assertions as happening at the same time in some way as to make it true. The combination of both claims would be ascribing a predicate (“looking at a particular person, that is, ‘me’”) and ascribing the very opposite of that predicate (“not looking at me”) to the same subject (the bird) at the same time, which is no different from ascribing nothing at all to a subject; cognitively, it is no different from refraining from making any assertion at all.

Someone, perhaps a skeptic of knowledge or someone unacquainted with Aristotle’s metaphysics or theory of logic, could ask the question: what justifies the PNC? Why is it the case that two contradictory assertions cannot be true at the same time? The attempt to then justify why this is would lead to predicating certain features as belonging to the PNC, like some noteworthy point about “contradictions,” or some fact about the human mind or reason. At the same time, however, in the very attempt of asserting these predicates of the PNC, the person is utilizing the PNC (perhaps unknowingly): in using these predicates to justify the PNC, he does not intend to assert that these predicates are true of the PNC and do not belong to the PNC at the same and in the same respect.

Aristotle regarded the PNC as axiomatic (as a starting point) for our very thoughts, as inescapable for anyone who chooses to think or use reasoning; as he says it, it is a principle which “is necessary for anyone to have who knows any of the things that are.” (Metaphysics Book 4, Chapter 3, 1005b15) Accordingly, he held that we can’t even engage in an argument without first accepting and relying on the PNC (if only tacitly if not explicitly). This reasoning, I’d like to point out, would apply to our purported justification for the PNC, since it, too, would be an argument.

The Justification of Perception

Another helpful example may be perception, whether or not our senses convey anything about reality, specifically about the external world. (Whether seeing, for instance, gives us any awareness of objects being seen, such as dogs, trees, or houses.)

A justification of perception would amount to a defense of our particular sense organs and sense-activities like seeing, hearing, and feeling. While someone could presumably try this, the resulting defense would contain an underlying fraud: it would assume the validity of the senses, even as it tries to justify them. The defense would argue that we should form our ideas of “touch,” “smell,” etc., from particular cases of touching and smelling, but the point at issue is whether we “touch” or “smell” anything at all.

The reason why a justification would be needed is that, for whatever reason, someone is unsure of whether they even have these senses. What the skeptical person needs is not a conceptual argument, which would be circular reasoning as I explained earlier, but perception. Nothing shows us that our senses are valid, besides the fact that they allow us to perceive; in losing senses, we also lose our ability to perceive, as people with normally functioning eyes discover when their eyes are damaged (through disease or some accident) and they become blind.

Now, let’s return to induction.

Can We Justify Induction?

To answer the question proposed by this section’s title, we need to turn once again to Aristotle. It was Aristotle who once aptly remarked that there are principally two ways of coming to have convictions (beliefs), two ways of reasoning or of argumentation: induction and deduction. If we were to suppose that induction required a justification, then the only two ways to provide such would be through inductive or deductive arguments.

The problem with the first approach--an inductive argument--is that it would consist of building from particular observations to piecemeal generalizations, presumably resulting in a universal, general account of induction. But the very issue at hand is whether such generalizing from particulars is valid in the first place. To utilize inductions to justify induction generally is to commit the “petitio principii,” the fallacy of “begging the question.”

On the other hand, defending induction by means of a deductive argument is impermissible because deductions can only justify non-ampliative inferences. Ampliation is our mental power of extending knowledge we already have to new cases, beyond the ones we originally used to gain that knowledge, and often leads to our possessing universal knowledge about some subject. A non-ampliative inference is one that doesn’t extend our knowledge, so to speak, but merely applies it to a new case or makes explicit something that was implicit in our argument’s premises (or thoughts).

An induction is principally an ampliative inference, and the supposed need to justify induction stems from this ampliative character; what justifies our purported knowledge of the future by means of our past knowledge, or our knowledge of the whole by means of our knowledge of some parts? What is the process of ampliation going on here, and how does it allow us to properly reason from observed cases to unobserved cases, from the past to the future, from particular areas of the world to the entirety of the universe? There is no general account of generalizing or of a universal kind of thinking from which one could produce a deductive argument about ampliation and induction--most likely because an account of induction would have to explain the role of “generalization,” “universal thinking,” and “ampliation” before deductions could be produced, and so would be just as questionable as induction, in this context.

So then what are we supposed to conclude?

I’ve maintained that induction can’t be justified through argument or reasoning, whether inductive or deductive. Rather than becoming an inductive skeptic, I ask that we return to Aristotle, specifically his point that our two ways of gaining conviction and reasoning are done by either induction or deduction. Inductions and deductions, we should come to realize, are our methods of justifying things, of coming to reach conclusions about things. By their peculiar nature, they can’t prove or justify themselves; they can’t be used to prove each other, and they can’t prove themselves (except in a trivial manner, such as “This is a man, therefore this is a man”). Rather, they are the starting points of the whole notion of justification: justification assumes the validity of induction and deduction, as these are the principal ways by which things can be justified at all, and this notion cannot be applied to them without sophistic results, such as circular reasoning. (Indeed, any attempt to prove a starting point or axiom must end in a trivial statement or circular reasoning, as Aristotle was the first to notice, see Posterior Analytics, Book 1, Chapter 3.)

I don’t think this reveals any problems with induction, as if the lack of justification reveals some hidden, underlying arbitrariness at the heart of inductive thinking. I’ve reached the conclusion that this isn’t the kind of thing that can be justified. To make the attempt results in either failing miserably, or in assuming that which one is attempting to prove, which amounts to the same thing as failing.

Conclusion

That said, I obviously don’t think that any and all inductions are therefore valid as a result, just as Aristotle’s thinking that there were “first principles” (starting points) of deductions didn’t lead him to conclude that all deductions were valid. There were proper and improper forms of deductions, and of using the syllogistic forms of demonstrations, he held, and carried out the Herculean task of explicating proper deductive thinking. There are proper and improper forms of inductions, of reaching generalizations, and of conducting the process of ampliation and abstraction, I hold. What we need is not a justification for induction, but a full-scale explication of what exactly induction is.

Concepts from an Objectivist Perspective, Part 3

Part 1

Part 2

Unit-Economy, Words, and Definitions

In part 1, I said that concepts are typically represented or symbolized by words in a language. In addition, we typically have definitions for the words we use, or seek definitions when we don’t understand or need clarity on some idea or word.

But why is this? Do words and definitions serve some important purpose in our quest for knowledge? Or could we do without them? To answer these questions, we must understand an important fact about concepts, and about the human mind.

The concept-formation process is truly multifaceted and complex. It involves observation of particulars, noting of features, differences, and similarities; it specifically is about thinking of certain particulars as members or units of a class (group) one is trying to form, abstracting certain distinctive features by omitting particular measurements, and finally integrating all of this knowledge into a single mental unit or thing, a basic unit or block of thought (or perhaps several units). The product is a kind of awareness that extends beyond what a given person could observe in his lifetime, and thus grants him a wealth of knowledge he otherwise could never achieve.

On this point, Rand states that “[c]onceptualization is a method of expanding man’s consciousness by reducing the number of its content’s units--a systematic means to an unlimited integration of cognitive data.” [Ayn Rand Lexicon: Unit Economy]

These facts about concepts and their purpose can be summed up as a guiding principle of thought: “unit-economy.”

Our minds are limited in what they can be aware of at any given time--the human mind cannot perceive everything with its sense organs; it cannot focus on everything it has learned in considering its concepts. Not even the power of memory has the ability to hold all of our knowledge and keep particular units of knowledge (like the concept job) firmly separated in our awareness from others (such as drive or door).

“Words” are the solution to this dilemma, and are the reason why concepts can perform their role as “unit-economizers” or unit-reducers.

Words are perceptual symbols for our concepts. Except for proper nouns (such as my name Roderick Fitts), all of our words represent our mental concepts. We need them because they complete the concept-formation process. Without them (or some other symbol for our thoughts, like pictures), we would have no way of retaining and subsequently using the information we’ve gained, information that’s relevant for forming a given concept, and this problem would only worsen when trying to think about other subjects which one wants to form into a concept. If this were to continue, we would more likely than not lose the conceptual information we were trying to integrate, and be completely unable to deal with even more difficult, higher-level conceptual material.

They facilitate the concept-forming process by allowing us to retain the conceptual information we’ve put together by merely perceiving the concrete symbol. The symbol (and thereby the concept) is stored into our memory and subconscious, making the information it contains instantly available whenever we choose to focus on it, and allows us to pursue other, more complex knowledge which would be impossible without the more basic conceptual knowledge already gained. It perceptually separates the concept it symbolizes from other things that exist and from our other concepts, allowing us to focus on particular concepts in our minds at will. To put it another way, perhaps in words Aristotle would use, “words” make forming concepts “second nature.”

The same sorts of problems present themselves when we consider why we need definitions.

Definitions are statements that identify the nature of the units included under concepts. We need them because we need a way to retain and recognize the characteristics that are both distinctive only to the particular units of a concept (the individual dogs of the concept dog) and the characteristics shared by them and a larger group of things (such as mammals in general, or animals). They explicitly point to the distinctive characteristic(s) of a group of similars, and identify the characteristics relating them to a larger group. By doing this, they allow us to hold in mind the relationships of our other concepts to each other. For instance, the concepts human, dog, and camel are related because their constituent units (the particular men, dogs, and camels) have characteristics which allow us to group them together as animals in our animal concept (such as mobility in environments, metabolic processes, and sensory organs).

But a definition does not merely state any given characteristic of a concept’s units, and neither does it state every single one. Merely stating a characteristic, while presumably factual in relation to the units, is not helpful for differentiating the concept from other concepts. One could say humans are “rational” (the characteristic), but this alone isn’t helpful for understanding how humans are related to other objects, like lamps, planets, and other animals. On the other hand, stating all the characteristics of a concept’s units would defeat the purpose of a definition, since it’s supposed to facilitate an immediate understanding of the nature of a concept’s units in the form of a few concepts, a single statement, not in the form of a catalog of everything one knows about the units. Definitions then, like words and concepts, are designed by us to achieve “unit-economy,” this time in relating concepts to other concepts (in terms of higher-level, more abstract concepts and lower-level, less abstract ones), and in relating a distinctive characteristic (a differentia) to a characteristic distinctive of a larger, more generalized group (a genus).

They do so by stating the essential characteristic(s) of the concept’s units, those features that make the units the kind of things they are. This characteristic both causes the greatest number of other characteristics shared by the units to even be possible, and explains the greatest number of characteristics, why they exist as aspects of the units. Consider the concept human and particular men and women: we do many, many things, among which are building functional airplanes, theorize about weather and climate, consider our mortality and free will, employ strategies in war, and laugh at ironic situations. If we were to pick any one of the above as a “defining characteristic,” we would be unable to understand why we can do any of the others, which is because they stand in a logical relationship to another feature of humans, their rationality, the capacity of reason. We can build viable airplanes, for instance, because we can use our faculty of reason to understand things like gravity, thrust, and drag, the scientific principles of electricity and magnetism, and the properties of materials which make some materials better suited as aircraft parts than others.

This statement of the essential characteristic is the means by which a definition identifies the nature of a concept’s units. A definition of humans as the “rational animal,” as the essential characteristics of humans, points to the facts which make us what we are, as opposed to other animals, other living things, other existents. In turn, this is how definitions bring clarity to our ideas and concepts: it identifies the nature of units in a concept, thus establishing its connection to other elements in one’s knowledge (other conceptual material), bringing up the context by which we can recall what the concept meant in the first place.

Thus, words and definitions serve an important role in the concept-formation process identified in Objectivism: in their own ways, they complete the process and finally establish the “mental integration” or “retained unitary cognition” that is characteristic of concepts.

Conclusion, and a Comment about Induction

Many philosophers have discussed the relationship between concepts and the method of induction, if only indirectly. Aristotle once pointed out that if someone doesn’t yet possess a term or word for an inductive generalization he’s trying to make, then he should go ahead and coin a new term (a unit of thought, a concept) for it. Karl Popper famously denied the validity of induction entirely, and consequently denied the validity of concept-formation on similar grounds. Francis Bacon passionately stated that our very concepts would be worthless and corrupted (mere “idols”) without a viable method of induction.

Ayn Rand, though not possessing a theory on induction, noticed the significance of concepts in relation to them. She stated that:

The process of forming and applying concepts contains the essential pattern of two fundamental methods of cognition: induction and deduction.

The process of observing the facts of reality and of integrating them into concepts is, in essence, a process of induction. The process of subsuming new instances under a known concept is, in essence, a process of deduction.[Ayn Rand Lexicon entry: Induction and Deduction]

It was Rand who demonstrated to us the validity of concepts by drawing out the striking similarities between concept-formation and the field of mathematics.

Perhaps the way to reach sound inductions can be discovered by drawing out the method of induction's relationship to concepts.

Concepts from an Objectivist Perspective, Part 2

Part 1

Differences, Similarities, and the Unit-perspective

As far as we know, other animals lack concepts, and even the ability to form them. While they have their own ways to perceive the world (some snakes see through processing infrared light, for instance), as we do, they cannot do anything more with their perceptions than act on them. Using sight and hearing, a lion can hunt and kill its prey, but cannot do something that we do all the time: in general, other animals cannot organize their perceptual field, the objects they deal with every day.

Animals notice that things around them exist and act in certain ways, but they cannot reach the next step: the recognition of similarities and differences among the identities of things. We’re able to notice that some things are completely different from each other. Birds have the characteristic of flight, but trees do not; we see objects in colors and shapes, but our thinking about our own thoughts lack such features; some things in the universe are life-forms, but other things possess no life processes. In observing the world, we can’t help but notice the plethora of features and characteristics that objects have (or don’t have).

However, we’re not restricted to only noting differences amongst things. We can also notice the ways in which things are similar, or are less different, in comparison with other things. In realizing that some things are alive and some things aren’t, we can then relate these living things as having a certain attribute in common, namely “life.” Some animals have legs and can run, making them similar in comparison to, say, snakes or snails that cannot run.

These two facts, our noticing of differences and similarities, points to another significant fact about the human mind: we’re able to group or classify things according to shared characteristics (flying, color, weight, speed, etc.), considering them as units or members of a group of similars. This is the “unit-perspective,” which Rand insists is the key or beginning of the conceptual level of consciousness.

A unit is an “existent regarded as a separate member of a group of two or more similar members.” [ItOE, p. 6] While perception allows us to become aware of certain characteristics of objects around us, such as appearing or feeling like they possess a certain length or a rough surface, the unit-perspective allows us to be aware of things as existing in certain relationships with other things due to their characteristics, whether the things being compared are different from or similar to each other. (I’ll note that the concept unit doesn’t apply only to perceptual objects, such as balls and dogs. Political systems and scientific theories can be units too, in relation to the concept theory for instance, but it’s important to realize here that our first units are of perceptual objects.)

As I said in part 1, concepts are things that relate certain knowledge as applying to a plethora of things that we’ve grouped together. Such a phenomena as a concept would be impossible if we didn’t group things together in the first place, if we didn’t regard things as units. Of special significance is the fact that, without a unit-perspective, we would not be to “count, measure, identify quantitative relationships [such as some object weighing 10 pounds]; [we] could not enter the field of mathematics.” [OPAR, p. 76]

This seeming coincidence is, as Rand argues, actually the means by which we can understand the connection between concept-formation and mathematics, and thus understand the nature of concepts themselves. “The process of concept-formation is, in large part, a mathematical process.” [Introduction to Objectivist Epistemology, p. 7]

The point of the next section is to see why that is.

Measurement, and Measurement-omission

To truly understand concepts, we need to understand the mathematical idea of measurement, both what it is and the reason why we measure things.

In Rand‘s definition, measurement, “is the identification of a relationship--a quantitative relationship established by means of a standard that serves as a unit.” [Ayn Rand Lexicon entry: Measurement] Typically, measurement involves two things--the thing being measured, and the other thing which acts as the standard of measurement. By taking a foot as a standard of “length,” for instance, we can compare/measure other objects with it and determine if they are longer or shorter. A foot is itself a unit of length, so it can be used to measure other units of length and give us knowledge about certain attributes, specifically information about the magnitudes of various objects, whether of large or small magnitude. A similar process occurs when measuring weight, density, volume, time, and other units of measurements.

What’s important here is that the real purpose of measurement isn’t to simply relate objects that we deal with in everyday experience, but to expand the range of what we can consider and learn about beyond the perceptual level, beyond individual feet, or seconds which we can count. We can observe something that weighs one gram, for instance, but we can’t comprehend the weight of the Earth by merely looking at it; instead, we need to compare it to other objects that we can weigh and form new standards of measurement, such as a kilogram, which we can relate to a perceptual unit (the gram). Our perceptual field is the foundation and standard, and we relate our more sophisticated and abstract measurements to units that we can perceive with only our senses.

A similar thing happens when dealing with objects classed under a concept; the objects have the same characteristic (as we realize from observation), but differ in the exact quantities of these characteristics. Two birds may have the same characteristic “flight,” but may differ in certain quantities relating to flight, such as how high they can fly, how swift, how fast they can take off from the ground, and so on (for a striking comparison, look at eagles versus flamingos). Correspondingly, this will lead to differences in our measurements of these quantities. The world, we realize, is filled with objects which have the same characteristics, but differ in various ways in regard to the particular quantities of such characteristics or features, and our measurements will differ when relating these objects to our units of measurement.

To form concepts, we retain the characteristic, but omit our measurements of the various quantities of things’ characteristics. To form the concept flight, we specify the relevant characteristics (a self-propulsion through a certain medium, pushing against the force of gravity, etc.), but omit/not specify the particular measurements of these characteristics (for instance, the kind of atmosphere, the speed of propulsion, the instruments being used to fly, the amount of gravity being counter-acted). We must be careful to recognize that in “omitting” measurements, we’re not pretending that they don’t exist: without measurements, there is no one relating the quantities of things, and thus no comparisons which would lay the groundwork for forming a given concept. Instead, the “principle is: the relevant measurements must exist in some quantity, but may exist in any quantity.” [Ayn Rand Lexicon entry: Concept-Formation]

This “some-but-any” principle, known formally in Objectivism as “measurement-omission,” is the process of abstraction. Omitting the particular measurements from our consideration of a given characteristic is the same process as abstracting a feature from the particular circumstances we observed it in (or originally thought about it being in). In omitting measurements, we’re able to determine the characteristics that a group of things have in common (or do not), and thus apply knowledge gained about this characteristic to all the instances or particulars included in the (future) concept, regardless of any irrelevant circumstances or measurements carried out.

Thus, we come to Rand’s definition of the concept concept, and simultaneously a single-sentence summary of her theory of concepts. A concept, in her definition, is “a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.” [Ayn Rand Lexicon entry: Concepts]

Now that we’ve discussed the nature of abstraction, we can learn about how concepts are completed, which is the purpose of the third (and final) part of this series.

Part 3