Thanks for the encouragement, Jim.
~~~~~~~~~~~~~~~~—Varieties of Induction
In valid deductions, premises provide absolutely conclusive reasons for conclusions drawn from them. Conclusions deduced validly are contained implicitly in the premises. To suppose premises true and a conclusion validly deduced from them false would itself entangle one in a contradiction. The necessity with which the valid deduction follows is just the force of the impossibility of contradictions in reality.
In the Topics,
Aristotle speaks of induction as “a passage from individuals to universals” (TP 105a12
). Such an induction proceeds “from the known to the unknown” (156a5
). Such an induction “extends our knowledge beyond information contained in the premises” (Kelley 1988, 167
). Induction in this sense is traditionally called incomplete
induction. It is better called ampliative
induction, to signify simply that the conclusions go beyond the premises (Wright 1979, 8–9
In the Posterior Analytics,
Aristotle speaks of induction as “exhibiting the universal as implicit in the clearly known particular” (APo 71a8
). In this sense, induction is termed abstractive
. The abstractive is surely close to the sense of induction to which Rand refers in saying that “the process of observing the facts of reality and of integrating them into concepts is, in essence, a process of induction” (1990, 28
). I have only a little to say distinctively about this type of induction, so I shall say it right now and shall not again take up abstractive induction in later sections.
Our concepts are abstracted from the experiences we have had and the concepts we have previously formed. One desideratum for our concepts is that they effect economies in our ways of thinking about past experience and the nature of the world. That is why concepts have intensions and why they are organized hierarchically. Another desideratum is that the concepts we form be ones productive of good ampliative inductions in the future.
But in what sense is the process of abstraction itself an induction? Looking at abstraction rather formally, there is induction in moving from “this item is an x
and that item is an x
” to “all such items are x’s
.” The reliance here on identity is obvious: “such items” are identical items within the bounds of what it takes to be an x
. From this formal perspective, abstractive induction seems akin to recursive, or mathematical, induction (which will be discussed later in this essay).
In actual abstractions, of course, the x
gets filled in with something definite: “This is hue, that is hue, . . . all such qualities are hue. This is an inertial force, . . . all such forces are inertial. This is a billiard ball, . . . all such things are billiard balls.” Where “all such” is purely hypothetical, we have still only the recursive induction we had when x
was left unspecified. Where the “all such” is meant to say “all such (and there have been, are, or will be specific such that we have not examined),” then it seems we are back to ampliative induction. In judging that two items are of a single kind, we may also rely on induction where we have not examined all characteristics relevant to the classification. This, too, is ampliative induction. I am inclined to think that induction commonly denominated abstractive
is an amalgam of straight ampliative inductions and a recursive induction akin to mathematical induction.
Ampliative inductions are not absolutely conclusive.[4
] By this common saying, we mean not that they are always rationally uncertain, but that taking the starting points of these inductions to be true and the inductions false would not itself entangle one in a contradiction.
We typically think, with Aristotle, that ampliative induction carries one from a certain number of examined cases to an indefinitely large number of unexamined cases. That is, we typically think of ampliative inductions as establishing generalizations, whether from everyday experience for practical conduct or from systematic observation and controlled experimentation for scientific theory. Ampliative induction, however, may instead carry us from a certain number of examined cases to the next case which turns up. This is commonly termed eduction
. It is inductive inference from particulars to particulars. Like generalizing induction, eduction proceeds from the known to the unknown.
We routinely make inferences from particulars to particulars, without explicit generalization. Such inferences are also made by animals incapable of generalization. “Animals have, on the one hand, certain congenital propensities, and on the other hand, an aptitude for the acquisition of habits. Both of these, in a species which succeeds in surviving, must have a certain conformity with the facts of the environment” (Russell 1948, 429
). Even if an animal cannot generalize explicitly, we might think of its behavior as guided by implicit generalizations, such as fire burns
. The child, too, “knows from memory that he has been burnt, and on this evidence believes, when he sees a candle, that if he puts his finger into the flame of it, he will be burnt again. He believes this in every case which happens to arise; but without looking, in each instance, beyond the preset case. He is not generalizing; he is inferring a particular from particulars” (Mill 1973, 2.3.3
Eduction goes far back with us; not only to the toddler stage, but to infancy. Jean Piaget (1954
) concluded from his investigations that infants first acquire a notion of enduring objects at nine months of age. Until then, if a toy attractive to the infant were covered with a cloth, the infant would make no attempt to lift the cloth and grasp the toy, even though she were capable (after the fifth month) of performing each of these actions. Piaget concluded that, until the ninth month, the infant does not regard the toy as an enduring entity that continues to exist while not in view; only after the ninth month does the infant begin to infer the continued existence of objects.
Piaget’s conclusion was consistent with his view, and that of traditional empiricists (e.g. Helmholtz 1885
), that comprehension of a world extending beyond immediate sensory experience is the product of a long process of visual and manual exploration. From the fifth month to sometime in the second year, the infant attempts to manipulate what she sees and to see what she manipulates. These explorations were thought to lead to elementary comprehension of an objective enduring world.
Piaget’s observations have been reconfirmed many times, but Piaget’s interpretation is now widely questioned. Infants younger than nine months may fail to lift the cover and grasp the toy not because by lack a notion of enduring objects, but because they are not yet able to coordinate such a means-ends sequence (Baillargéon 1986, 38–39
). It is likely that this inability is due to immaturity of the frontal cortex (Diamond 1989
). Recent experiments, not demanding coordination of means-ends-sequences, indicate that infants as young as six months (Baillargéon 1986
), even four months (Baillargéon 1987
), understand not only that objects continue to exist when not seen, but that moving objects continue along trajectories when not seen and that those trajectories could not be through other solid objects. Perhaps this is the most primitive form in which we grasp the principle of non-contradiction (see also Peikoff 1985
Infants four months old can determine whether two objects (rings) are connected by a solid (but occluded) link through bimanual grasping and displacement of the two objects They can discriminate visually (without touching) which of two assemblies, linked rigidly or elastically, they previously explored haptically. Object perception, even in the earliest phases of our development, seems to be mediated by relatively central mechanisms; haptic and visual discriminations are coordinated even before the onset of active manual-visual explorations (Streri and Spelke 1988; Spelke 1989
). Notice, “counter to the views of Quine (1960
) and others, that the organization of the world into objects precedes the development of language and thus does not depend upon it. I suspect, moreover, that language plays no important role in the spontaneous elaboration of physical knowledge” (Spelke 1989, 181
We come to language and explicit generalization (Macnamara 1986, 114–17, 163–70
). John Stuart Mill argues persuasively that general propositions such as all men are mortal
are a record of our past inferences from particulars to particulars.
All inference is from particulars to particulars: General propositions are merely registers of such inferences already made, and short formulae for making more: The major premise of a syllogism, consequently, is a formula of this description: and the conclusion is not an inference drawn from the formula, but an inference drawn according to the formula: the real logical antecedent, or premise, being the particular facts from which the general proposition was collected by induction. (Mill 1973, 2.3.4; cf. Russell 1948, 431–32)
This does not mean that generalizations and syllogistic reasoning are useless. With registration of experience in the form of general propositions,
the particulars of our experiments may then be dismissed from the memory, in which it would be impossible to retain so great a multitude of details; while the knowledge which those details afforded for future use, and which would otherwise be lost as soon as the observations were forgotten, or as their record became too bulky for reference, is retained in a commodious and immediately available shape by means of general language. (Mill 1973, 2.3.5)
Another reason Mill gives for generalizing rather than inferring directly from particulars is subjection of our near-term expectations, hopes, and fears to objectivity. Rather than simply pronouncing on the next case, we should try to generalize; reluctance to do so indicates infirmity in the grounds of our eduction. This seems a valid psychological point, but Mill goes too far, saying, “whenever, from a set of particular cases, we can legitimately draw any inference, we may legitimately make our inference a general one” (ibid.
). Such a principle is not applicable to time-dependent processes. Most of us are justified in believing we shall be alive a month from now, but none of us would be justified in believing we shall be alive always. Again, we are justified in believing the earth will orbit the sun once more, but not justified in believing this will continue forever.
Ampliative induction of the generalizing sort has received the most attention from philosophers, and it will be receiving most of our attention in this essay. The ways in which this sort of induction is grounded on identity will be our chief concern. We shall be aiming at the fully human
modes of induction.
Man’s knowledge differs from that of the beasts: beasts are sheer empirics and are guided entirely by instances. . . . Beasts, so far as one can judge, never manage to form necessary propositions, since the faculty by which they make sequences is something lower than the reason which is in man. The sequences of beasts are just like those of simple empirics who maintain that what has happened once will happen again in a case which is similar in the respects that they are impressed by, although that does not enable them to judge whether the same reasons are at work. That is what makes it so easy for men to ensnare beasts, and so easy for empirics to make mistakes. (Leibniz 1981, 4.1.50)
I shall also be proposing a distinct ground, again on identity, for mathematical induction. Recall the principle of mathematical induction (Cuillari 1989, 41–45; Hamilton and Landin 1961, 78–99, 115–16
). A statement, P(n), to be proved for integers n >
t, where t
is a fixed positive integer, is true, provided:
(i) P(t) is true, and
(ii) If P(t), P(t+1), . . . P(n) are true, then P(n+1) is true.
First verify (i), that P(n) is true for n = t. Then try to show that assuming the antecedent of (ii) to be true implies that the consequent of (ii) is also true.
Unlike ampliative induction, mathematical induction is absolutely conclusive. The conclusion of an argument by way of mathematical induction is implicitly contained in the premises. Mathematical induction is, overall, a deductive argument. The inductive character of (ii) is a peculiarity of the serial order property of the infinite set of natural numbers.
The challenge to ground mathematical induction is the challenge to ground, in the world or perhaps in the operation of the mind, denumerable infinite sets (on Gottlob Frege and Richard Dedekind, see Kneale and Kneale 1984, 467–71, 492–93; see also Poincaré 1983, 398–402; Hilbert 1983, 191–94; and Gödel 1983, 463,–64
). One of Frege’s objections to Mill’s proposal to ground arithmetic in empirical reality was that this would jeopardize the existence of the infinite set of natural numbers, to which there might be no corresponding empirical collection. (Saying that the set of naturals is just counting “1,2,3, and so forth forever” will not do. Where in empirical reality is the basis for such infinite
It might be thought that the ground of such sets and mathematical induction would be at hand once we have firm ground for abstractive induction, particularly in its recursive aspect. In other words, it might be thought we have the objective basis of denumerable infinite sets once we have a theory of universals that bases them in objective reality. Let us sort this out in some detail.
There are overlaps, but also disjunctions between the problem of grounding (denumerable) infinities and the traditional problem of universals. Recall the latter problem: the world is made up of only particular, individual things. (This idea won overwhelming acceptance among the philosophically minded, which it enjoys to this day, between the times of Abelard [1079–1142] and Ockham [1290–1349].) Yet thoughts about the world are largely in terms of universal ideas, or concepts. The challenge, then, is to discover how we form our universal ideas from experience of only particulars and, so, to spell out the relationship of concepts to the particulars belonging under them. In taking on this challenge, philosophers really are taking on not much less than the issue of the relationship between thought and reality.
It will be useful to make the common distinction between the extension and the intension of a concept. The extension consists of just those individuals the concept refers to, and the intension is a criterion, possibly complex, by which just those individuals might be grouped together.
The extensions of concepts are always denumerable: “Here is one spoon,
there is another; here is one lake,
there is another; one sister
lives here, another lives there; one performance
will be this week,
another, next.” The extensions of concepts are (aside from concepts of pure numbers detached from particular physical application) indefinitely large but may be finite or infinite. Is the number of spoons that ever will exist finite or infinite? We do not know, and we do not need to know in order to have a perfectly good concept of spoons. It might have seemed at first blush that we use infinite sets whenever we use universals, but that is incorrect (even when individuals in the extension are infinitely decomposable). The problem of universals can be solved without the problem of denumerable infinities being solved therewith. [6
Let us probe a little further, just to be sure. When using a concept, one treats particular instances as
instances of the items falling under the concept. Rand emphasized that in doing this one regards the particular item as a unit among other units (the other items) falling under the concept. This is the same as regarding a member of a set as
a member (see further Armstrong 1989, 133–35
) except that the criteria by which one selects the members of a set may be purely arbitrary whereas the criteria by which one selects which particulars should be united in a single concept must be natural and useful.
Intensions of concepts and rules specifying the members of sets are both rules of selection. In both cases, listing of included individuals can be supplanted by the selection rule. In both cases, the selection rules unify the individual items and, for large collections, effect economies in thinking. Both in our referring to extensions of concepts by their selection rules and in our referring to memberships of sets by their selection rules, we step back from the itemizing mode of reference and adopt a more global and abstract mode of reference. The selection rules for sets, though, may well be recursively generative of the sets (Hamilton and Landin 1961, 120–28
); the sets may be nothing but what (items grouped in thought) would result from repeated application of their definitions (see also Boolos 1983
). Such a set could be denumerably infinite. However, in referring to such a set by its definition (or its name) we do not generate the infinite set. The idea of infinite collections cannot receive its grounding simply from the act of abstraction itself, not even from the act of abstraction on a purely generative set.
Before ending this short tour through the garden of induction, we should settle whether abstractive inductions in their recursive mode are absolutely conclusive. In their ampliative mode, of course, they are not.[4
] In their recursive mode, they are. If I say “this is a robin, that is a robin, . . . but not all things (just) like these (in the requisite ways) are robins,” then I contradict myself within that very saying. (On perceptual judgments generally, see Kelley 1986, 208–28
—Ockham – Contingency
—Nicolaus – Experience of Substance
—Nicolaus – Reasoning to Substance
—Hume – Experience of Cause and Effect
—Hume – Reasoning to Cause or Effect
—Hume – Necessity
—Hume – Uniformity
—Existence is IdentityReferences
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Armstrong, D. M. 1989. A Combinatorial Theory of Possibility
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Baillargéon, R. 1986. Representing the Existence and the Location of Hidden Objects: Object Permanence in 6- and 8-Month-Old Infants. Cognition
——. 1987. Object Permanence in 3.5- and 4.5-Month-Old Infants. Developmental Psychology
Benacerraf, P., and H. Putnam, editors, 1983 . Philosophy of Mathematics
. 2nd ed. Cambridge: Cambridge University Press.
Boolos, G. 1983 . The Iterative Conception of Set. Reprinted in Benacerraf and Putnam 1983.
Cupillari, A. 1989. The Nuts and Bolts of Proofs
. Belmont, CA: Wadsworth Publishing.
Diamond, A. 1989. Differences Between Adult and Infant Cognition: Is the Crucial Variable Presence or Absence of Language? In Weiskrantz 1989.
Gödel, K. 1983 . Russell’s Mathematical Logic. Reprinted in Benacerraf and Putnam 1983.
Hamilton, H., and J. Landin 1961. Set Theory and the Structure of Arithmetic
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Helmholtz, H. 1962 . Treatise on Physiological Optics
. J. Southall, translator. New York: Dover Publications.
Hilbert, D. 1983 . On the Infinite. Reprinted in Benacerraf and Putnam 1983.
Kelley, D. 1986. The Evidence of the Senses
. Baton Rouge: Louisiana State University Press.
——. 1988. The Art of Reasoning
. New York: W. W. Norton.
Kneale, W., and M. Kneale 1984 . The Development of Logic
. Oxford: Clarendon Press.
Macnamara, J. 1986. A Border Dispute
. Cambridge, MA: Cambridge University Press.
Mill, J. S. 1973 . A System of Logic Ratiocinative and Inductive
. Toronto: University of Toronto Press, Routledge & Kegan Paul.
Peikoff, L. 1985. Aristotle’s “Intuitive Induction.” The New Scholasticism
Piaget, J. 1954. The Construction of Reality in the Child
. New York: Basic Books.
Poincaré, H. 1983 . On the Nature of Mathematical Reasoning. Reprinted in Benacerraf and Putnam 1983.
Quine, W. 1960. Word and Object
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Rand, A. 1990 [1966–67]. Introduction to Objectivist Epistemology
. 2nd ed. H. Binswanger and L. Peikoff, editors. New York: Meridian, Penguin Books.
Russell, B. 1948. Human Knowledge: Its Scope and Limits
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Spelke, E. S. 1989. The Origins of Physical Knowledge
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Streri, A., and E. S. Spelke 1988. Haptic Perception of Objects in Infancy. Cognitive Psychology
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~~~~~~~~~~~~~~~~This essay of 1991 had no notes. I will now add a few endnotes to indicate changes or emendations to the positions I took in this essay nineteen years ago. I will also add some hyperlinks within the text.
. I ended up retracting this generalization in the later section “Nicolaus – Reasoning to Substance.”5
. Mill’s conception of what would be an “empirical grounding” of denumerable infinities is not the only or best such conception. Turn to Kitcher 1984
. On empirical grounding of arithmetic, see also Jenkins 2008
The concretes falling as instances under a concept are a denumerable collection. That much is fine. However, for the concept of a spoon, it need not be prejudged whether some spoons existed in endless past ages or will occur in endless future ages. The indefinitely large number of actual spoons subsumed under the concept may have no first or last spoon in time, as far as the elementary concept is concerned. This would suggest that, as substitution units*
under the concept, possible particular spoons are isomorphic with the set of integers.*
Dimensions on which the concept spoon
consists are not basically temporal. They are ranges of shape and strength over minimal durations, all set by functional adequacy. In what era a spoon existed is irrelevant to its membership (its substitution-unit standing) in the concept class. The class is the class, however, when consideration of temporal order is dropped. So it remains that there may be profound logical and epistemological connection between (i) concretes as substitution units under their concepts and (ii) the set of integers.
When we turn to the measure-value of a spoon along a particular qualitative dimension, such as the spoon’s shape (see UM ¶’s
8 and 27), by which the concept can be analyzed, there is no reason to prejudge whether the applicable dimension’s linear order is a closed interval, whether it is dense, or additionally, whether it is continuous (LO
). So I should leave open as well the possibility that there is profound logical and epistemological connection between (i) concretes ranged with their concept-class measure values along a dimension and (ii) the numerical orders Q
Our widest analysis of abstractive induction, under Rand’s measurement-omission theory of concepts (as I have amplified it
)—in respect of both substitution units and measure values—is not more complete than our widest analysis of mathematical induction, linear orders, and ordered spaces.*
The “ground” of abstractive and mathematical induction is the world in its dimensions and magnitude structures when grasped by our conceptual cognitive systems, our measurement-omission systems.
Edited by Stephen Boydstun, 15 April 2011 - 06:02 AM.