Wednesday, December 16, 2009

Thomas Reid (and David Hume) on Induction, Causality


Philosopher Thomas Reid's significance in regard to induction does not derive from his own inductive theory, as in Aristotle's case or Francis Bacon's. In fact, he explicitly states that he has adopted Bacon's method of induction in his Inquiry into the Human Mind, and gives Lord Bacon nothing but the highest praise. What makes Reid so significant is that he understood Hume's criticism of causality (in An Enquiry Concerning Human Understanding), interpreted what it would imply about induction and inductive reasoning, and offered a sort of counterargument to Hume's skeptical doubts.

Monday, October 5, 2009

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).

Saturday, September 26, 2009

Induction by Enumeration and Sophistry

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

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.

Monday, September 14, 2009

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.

Sunday, September 13, 2009

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.

Monday, September 7, 2009

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.


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.

Friday, August 28, 2009

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.

Tuesday, August 25, 2009

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

Monday, August 24, 2009

Concepts from an Objectivist Perspective, Part 1

As I said in my first post, I’m an Objectivist interested in understanding induction, and in sharing what I find out with others. I’m aware that there’s a connection between induction and concepts, as I discussed in my post The Importance of Concepts for Bacon. So I’d like to briefly discuss concepts as they are presented in my philosophy, Objectivism. In doing so, I hope to show that it is a persuasive account that others should adopt (if they haven’t already), and that it has important implications for the subject of induction. (Implications that may have to wait for another time, unfortunately.)

Monday, August 10, 2009

The Importance of Concepts for Bacon

Though Bacon adopted the term notion, he took discussion of it in a new direction. [McCaskey, “Regula Socratis,” 239]
Since about April of this year, if someone had asked what was Francis Bacon’s most important concept in his philosophy, I would’ve replied “‘induction’ , of course.” It wasn’t until reading Dr. John McCaskey’s Regula Socratis that I decided to change my mind. I’ve decided that the word “notion” is the most important for Bacon. The following is my summary of Bacon’s approach to knowledge in light of reading McCaskey’s dissertation.

Monday, August 3, 2009

Aristotle's "Two" Views of Induction: McCaskey's Resolution (Part 1)

In my studies of Objectivism and Aristotle, I’ve always thought that I understood what I was reading about and contemplating afterward. One of the exceptions to that understanding was my extended reading of Aristotle’s concept of “induction.”

My earlier note, “Aristotle on Induction,” was really one interpretation of his position, supported by the vast majority of Aristotle’s use of the term “induction” (epagōgē) in his known works. What I didn’t include in that note was the alternative view, the contemporary and dominant view, defended chiefly by citing Prior Analytics, book 2, chapter 23 (PrA 2.23); an account which seemingly presents an entirely different, even contradictory, view of induction than what is presented in Aristotle’s other works (and even the understanding of induction presented within the first paragraph of that 23rd chapter ). Upon reading this chapter, I wrote in my MS Word document for Aristotle notes: “I have problems with the view that Aristotle thought induction required ‘complete enumeration of instances’; it only definitely shows up twice, and isn’t consistent with his other statements on induction.” I simply could not connect this position on induction with, for instance, the views of induction presented in his Topics or Rhetoric.

Philosopher John McCaskey, in the first chapter of his unpublished dissertation “Regula Socratis: The Rediscovery of Ancient Induction in Early Modern England,” attempts to reconcile these two conflicting views by arguing that one position is actually Aristotle‘s position, and that the other position is simply a misunderstanding and misreading of Aristotle.

Before discussing the resolution, let’s review the details of these differing views on induction.

The Two Interpretations

McCaskey notes that the term “epagōgē” (induction) appears 96 times in Aristotle’s known works. The first interpretation, the one McCaskey accepts (along with a very few others in the history of philosophy, including myself), can be understood by reading the majority of those 96 uses of “induction.” It is primarily presented in the Topics, Rhetoric, and Posterior Analytics 2.19. This interpretation is summarized as follows:

Induction is a form of reasoning that moves from particular instances and rises to general and universal knowledge. It is founded on sense-perception and memory: from our senses and memories we gain experience, and from these experiences we gain universal knowledge via induction. The application of an induction extends beyond the particulars that were used in its formation, meaning that induction is an open-ended process, rather than limited to the two or three particulars that were used to form the inductive generalization. It is different from deduction, and a counterpart to that other form of reasoning. More precisely, induction is the fundamental method of reasoning in comparison to deduction, because it supplies the premises for deductions. Induction gets its force or legitimacy from the similarity of particulars, not their number. Lastly, induction is a tool for making conceptual generalizations by identifying the essential nature of things (this last from McCaskey, page 35 of the PDF).

A great example in support of this interpretation of induction is Aristotle’s comment on goodness (aretē), in the Eudemian Ethics. There, he states that it is “the best disposition or state or faculty of each class of things that have some use or work.” He then gives us reason to believe that he has identified the nature of goodness by using the example of a coat’s goodness, which belongs to a coat in virtue of it carrying out its particular function or use. The inductive generalization about goodness states the essence of the subject: what makes “goodness” the kind of thing it is; is open-ended in that it extends to all kinds of particulars that have uses (not simply the one coat or house that could be used to form this induction); points out the element of similarity which justifies the induction (things having particular functions or uses); and is related to sensory data and memories about things having various functions.

The second interpretation on Aristotelian induction has been supported by the majority of philosophers who have commented on Aristotle, including contemporaries such as philosopher of science John D. Norton. (Norton has originated a “material theory of induction,” and thinks that Aristotle supports the enumerative induction viewpoint (page 3 of the PDF).) It is supported by eight of the 12 uses of the term “induction” in the two-book Prior Analytics, specifically (PrA 2.23).

In this interpretation, induction is a kind of deduction that is validated by complete enumeration of cases. It cannot extend beyond the particulars used in forming the generalization, and if it does not include all of them then it is invalid. It is primarily a matter of deducing a property (major extreme) or feature to belong to all the particulars of a class (middle term) by arguing that the same property belongs to one or some of the class’s particulars (the minor extreme). This is then perfected by adding “etc.” or “and these are all the particulars.”

PrA 2.23 has baffled students of Aristotle and commentators alike, while at the same time it has been regarded as the chief chapter on Aristotelian induction. In the first paragraph, Aristotle states his often-repeated claim that there are two kinds of ways of having convictions: induction and deduction. In the very next sentence, the start of the second paragraph, he seems to contradict this by claiming that induction really is just a type of deduction.

McCaskey’s resolution of this dilemma not only clears Aristotle’s name from the list of enumerative inductivists, but also teaches us the process by which induction supplies the premises for deductions.

I’ll cover the approach of McCaskey’s resolution in Part 2.

Monday, July 27, 2009

Rand on Concepts, Relation to Induction (Part 1)

In her Introduction to Objectivist Epistemology (ItOE), Rand presents a theory of concepts, which describes what concepts are (as opposed to what they are not) and how they are properly formed (and how they are improperly formed). A concept, Rand maintains, is a “mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.” (ItOE, 2nd Edition, p. 13)

Sunday, July 12, 2009

Bacon's "More Certain and Guarded Method" of Induction

"[...]our determination is that of trying, whether we can lay a firmer foundation, and extend to a greater distance the boundaries of human power and dignity."[Bacon, New Instrument, Book 1, Aphorism 116]
"Our only hope, then, is in genuine induction."[ibid., Aphorism 14. See here: The Ideas that Have Influenced Civilization, in the Original Documents]
Francis Bacon (1561-1626) was the first modern philosopher of science, and was instrumental in the development of what we now call the “scientific method.” Here are the essentials of his method of induction, which unfortunately was never completed.

Induction's Bad Reputation

[Originally posted on Facebook]

In my note "Aristotle on Induction," commenter Brian Tinker raised a good question:
Given what you and [Edwin] Locke [in this paper] have said about induction, why does it have such a bad reputation?
I think some of its bad reputation is earned: a very small part, that is. Let me explain.

Most people understand induction to be "by enumeration." Something like: "All the enumerated cases of swans I've observed are white, therefore all of them are white," or "the sun has always risen, therefore it will rise tomorrow (and the next day, etc.)."

Philosopher Bertrand Russell gives another example, through parable, of "induction by enumeration" in his book The History of Western Philosophy, page 543. The parable begins: A census officer is questioning homeowners in a village, all of whom seemed to have the same name, William Williams. Finally, the officer decides that every homeowner in the village is named William Williams (an enumerative induction), records the names, and takes a holiday. But he was mistaken: there was one man named John Jones who owned a home there, but the officer had missed him.

This means that the induction is faulty, and illustrates the problem--that inductions by enumeration are never certain; one counter-example can ruin even the strongest of inductive generalizations.

Francis Bacon, the first modern philosopher of science, points out the basic problems with enumerative induction (criticism which I agree wholeheartedly with):
The induction which proceeds by simple enumeration is puerile, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too small a number of facts, and those only the most obvious. [Bacon, Novum Organum (New Instrument), Book 1, Aphorism 105.]
Bacon summarizes why most philosophers (in my opinion) denigrate induction, and why it has a bad reputation. Since what he points out about enumerative induction is true, it is my reason for thinking some of induction's bad reputation is earned.

Also, I agree with Dr. Leonard Peikoff's comments on enumerative induction:
Induction in my judgment is not in any sense a matter of quantity. There's a type of induction called induction by simple enumeration, which means induction simply by enumerating or counting instances. [...]Now, I regard quantity as such as insignificant. It entirely depends on what happens to that quantity, which can be a good suggestive beginning if you see something happening. What happens when you integrate it with everything else you know that's relevant? (Art of Thinking, Lecture 2)
While enumerating cases can be instructive in forming ideas, noting similarities and differences, and in figuring out how rare or widespread a certain phenomena may be (say, meteoroid impacts), I think it's a poor candidate for valid inductive thinking.

Aristotle on Induction

[I'll reference any relevant Aristotle material at the bottom of this note, as it took a bit of digging through his works to find what I managed to find here.]

According to Aristotle, Socrates is the first person known to discuss induction and general definitions:
...for two things may be fairly ascribed to Socrates—inductive arguments and universal definition, both of which are concerned with the starting-point of science [knowledge])...[Aristotle, Metaphysics, Book XIII, Chapter 4, 1078b25-30]
Induction is the foundational reasoning activity, and is built upon sense-perception. More specifically, induction is (following Socrates's practice) reasoning from particular cases or individuals to general or universal knowledge.

An example would be forming the concept "animal": we can observe with our senses the similarities among individual species (humans, dogs, mules, etc.) and how different they are from both inanimate objects and other life-forms which don't seem to be conscious (plants would largely be our data for this conclusion)--all of this could eventually lead to forming the concept "animal" through induction. (In addition, it might lead to concepts such as "consciousness," "awareness" "life," "mobility" and concepts of particular animal species.)

Relatedly, he thought that induction is part of the means of forming general concepts ("genus") and, from there, building even more generalized concepts utilizing the knowledge gained from the earlier-formed ones. An example Aristotle gives is the inductive forming of the genus "animal" from the various animal species, and this kind of reasoning being the first leads to the formation of an even wider generalization; in our current case, we can integrate plants and microscopic lifeforms with our knowledge of the "animal" genus into a wider genus "organism." Regarding induction and concept-formation, Edwin Locke summarizes Aristotle's position this way:
His view was that one groups entities according to their perceived similarities and identifies their essential characteristics, the essence of a kind ... [t]his included the formulation of definitions based on genus and differentia [a genus--integrating the concept into a wider category—and a differentia—differentiating the concept from other existents in that genus, namely, man is the rational animal—meaning he is the animal who has the capacity to reason].
[Edwin Locke, The Case for Inductive Theory Building, Journal of Management, Vol. 33, No. 6, page 870 and 881 in brackets (2007)]

Lastly, as I noted earlier, Aristotle believed that induction was the basic or founding rational activity; the other main reasoning process, deductive thinking, was held to be a product of inductive thinking. Induction was thus logically prior to deduction, as it supplied the premises from which one could deduce.


On perception as validly giving knowledge/experience of reality:
On the Soul
(Latin: De Anima), Book II, Chapters 6-12, and Book III, chapter 3, 427b27-428a18.
In the latter chapter, Aristotle even notes: "for perception of the special objects of sense [like "color" for the sense of sight] is always free from error, and is found in all animals..." (427b11-13) Also, his biological treatises, such as
History of Animals (Historia Animalium) and Parts of Animals (De Partibus Animalium), are filled with evidence that he affirmed sense-perception as a means of knowing reality.

Induction as the foundational form of reasoning:
(Ars Rhetorica), Book II, chapter 20, 1939a25-27.

Induction as based on sense-perception, and as reasoning from "particulars" to "general":
(Topica), Book I, chapter 12.

Induction and concept-formation:
Posterior Analytics
(Analytica Posteriora), Book II, chapter 19, 100b1-5.

Induction, as supplying premises used in deductive thinking and argument:
Posterior Analytics
, Book 2, chapter 19, 100b3-5, when compared with Book I, chapter 3, 72b23-29.

P.S. Chapter 1 of McCaskey's dissertation goes in-depth into Aristotle's conception of induction, so my summary here may be expanded in the future if I learn of anything significant in this different account.

First Suggestion for Induction

In a dissertation by Dr. John McCaskey that I'm currently reading, it is noted that Aristotle appears to know exactly what induction is whenever he discusses it in his works. Aristotle,
never suggests that his theory of induction is problematic, complicated, or controversial. He never presents a catalog of competing theories of induction (as he frequently does for other matters), never says there are multiple ways of understanding induction, never says he will consider a kind of induction different from that usually discussed, never even explains fully what induction is." (Page 17 of "Regula Socratis: The Rediscovery of Ancient Induction in Early Modern England." Available for free viewing here:
In short, Aristotle was neither confounded by induction nor ignorant as to what its practice consisted of. On the contrary, readers of Aristotle are told in his work Topics that "[w]hat sort of thing induction is, is obvious."

One day, I would like to have that level of confidence in my understanding of induction, even if it will probably differ from that of Aristotle's. In fact, we should all strive to achieve his level of understanding.

My first suggestion then for induction is to gain some basic knowledge of it from everyday life, and use this knowledge to become more familiar with its basic method, to make it more obvious. Taking an alternative approach, such as immersing oneself in the heated philosophical controversies over the issues of induction, might leave one bewildered by--and unprepared for--the many variants of the method that now exist.

To conclude, I suppose I'll give a rough definition of what induction basically is, the very one that Aristotle thought was obvious: a kind of reasoning that moves from particular cases or instances to general or universal knowledge about those kind of particulars.

Next, I offer what I think Aristotle says on induction.

First Post

I suppose introductions are in order.

I'm Roderick Fitts, and this is my blog, Inductive Quest.

As far as personal info:

I'm an Airman First Class (E-3) in the U.S. Air Force, and before that I attended (and dropped out of) the University of Michigan. I was formerly a philosophy major and am still very interested in the field; in fact, philosophy is my main motivation for creating this blog.

I consider myself an Objectivist, meaning that I understand and agree with the principles articulated in Ayn Rand's philosophy. My other philosophical influences are Aristotle, Francis Bacon, and John Locke.

I've studied the philosophy for about 3 years now, and I'm a sophomore student of the Objectivist Academic Center (currently on hiatus). The OAC has been a great place to enhance my understanding of the philosophy, and I highly recommend it for others who would like to learn about it (and other philosophies) from highly trained professors.

Why the title "Inductive Quest":

I named my blog "Inductive Quest" because I'm determined to understand induction, both its proper and improper forms, as much as possible--even to the point of developing my own theory. I have two misgivings about whether or not a theory of induction can successfully be presented, which I plan to discuss sometime in a future post.

For now, I want to know the features of induction. This has led me to read a number of philosophers in ways I haven't done previously, and I think most of my posts will be my reflections on what I've learned.

Also, expect posts pertaining to the philosophy of Objectivism, since I'm still learning about the philosophy (though I do consider myself somewhat advanced) and will want to discuss some elements of it every once in a while. I'll also write about the philosophy because understanding it regularly involves implementing inductive thinking; though I haven't listened to it yet, this seems to be the main point of Dr. Leonard Peikoff's "Objectivism Through Induction" series of lectures.

Lastly, I'll probably post anything I have in mind about rationalism and empiricism, and other topics in epistemology, as I've been more interested in that branch of philosophy than any other field.

I hope you, the reader, enjoy!