Enumerative induction

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Enumerative induction or, as the basic form of inductive inference, simply induction, reasons from particular instances to all instances, thus an unrestricted generalization.[1] If one observes 100 swans, and all 100 were white, one might infer All swans are white. As this reasoning form's premises, even if true, do not entail the conclusion's truth, this is a form of inductive inference. The conclusion might be true, and might be thought probably true, yet it can be false. Questions regarding the justification and form of enumerative inductions have been central in philosophy of science, as enumerative induction has a pivotal role in the traditional model of the scientific method.



For a move from particular to universal, Aristotle in the 300s BCE used the Greek word epagogé, which Cicero translated into the Latin word inductio.[2] Namesake of empiricism, Sextus Empiricus—who in the 300s CE maintained that all knowledge derives from sensory experience—concluded in his Outlines of Pyrrhonism that acceptance of universal statements as true cannot be justified by induction.[2]


At 1620, Francis Bacon repudiated mere experience and enumerative induction, and sought to couple those with neutral and minute and many varied observations before to uncover the natural world's structure and causal relations beyond the present scope of experience via his method of inductivism], which nonetheless required enumerative induction as a component.


The supposedly radical empiricist David Hume's 1740 stance found enumerative to have no rational, let alone logical, basis but to be a custom of the mind and an everyday requirement to live, although observations could be coupled with the principle uniformity of nature—another logically invalid conclusion, thus the problem of induction—to seemingly justify enumerative induction and reason toward unobservables, including causality counterfactually, simply that modifying such an aspect prevents or produces such outcome.


Awakened from "dogmatic slumber" by a German translation of Hume's work, Kant sought to explain the possibility of metaphysics. In 1781, Kant's Critique of Pure Reason introduced the distinction rationalism, a path toward knowledge distinct from empiricism. Kant sorted statements into two types. The analytic are true by virtue of their terms' arrangement and meanings—thus are tautologies, merely logical truths, true by necessity—whereas the synthetic arrange meanings to refer to states of facts, contingencies. Finding it impossible to know objects as they truly are in themselves, however, Kant found the philosopher's task not peering behind the veil of appearance to view the noumena, but simply handling phenomena.

Reasoning that the mind must contain its own categories organizing sense data, making experience of space and time possible, Kant concluded uniformity of nature a priori.[3] A class of synthetic statements was not contingent but true by necessity, then, the synthetic a priori. Kant thus saved both metaphysics and Newton's law of universal gravitation, but incidentally discarded scientific realism and developed transcendental idealism. Kant's transcendental idealism prompted the trend German idealism. G F W Hegel's absolute idealism flourished across continental Europe and fueled nationalism.


Suggested in 1620 by Bacon, developed by Saint-Simon, and promulgated in the 1830s by his former student Comte was positivism, the first modern philosophy of science. In the French Revolution's aftermath, fearing society's ruin again, Comte opposed metaphysics. Human knowledge had evolved from religion to metaphysics to science, said Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology—in that order—describing increasingly intricate domains, all of society's knowledge having become scientific, as questions of theology and of metaphysics were unanswerable. Comte found enumerative induction reliable by its grounding on experience available, and asserted science's use as improving human society, not metaphysical truth.

According to Comte, scientific method frames predictions, confirms them, and states laws—positive statements—irrefutable by theology or by metaphysics. Regarding experience to justify enumerative induction by having shown uniformity of nature,[3] Mill welcomed Comte's positivism, but thought laws susceptible to recall or revision, and withheld from Comte's Religion of Humanity. Comte was confident to lay laws as irrefutable foundation of other knowledge, and the churches, honoring eminent scientists, sought to focus public mindset on altruism—a term Comte coined—to apply science for humankind's social welfare via Comte's spearheaded science, sociology.


During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science, William Whewell found enumerative induction not nearly so simple, but, amid the dominance of inductivism, described "superinduction".[4] Whewell proposed recognition of "the peculiar import of the term Induction", as "there is some Conception superinduced upon the facts", that is, "the Invention of a new Conception in every inductive inference". Rarely spotted by Whewell's predecessors, such mental inventions rapidly evade notice.[4] Whewell explained,

"Although we bind together facts by superinducing upon them a new Conception, this Conception, once introduced and applied, is looked upon as inseparably connected with the facts, and necessarily implied in them. Having once had the phenomena bound together in their minds in virtue of the Conception, men can no longer easily restore them back to detached and incoherent condition in which they were before they were thus combined".[4]

These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed consilience—that is, simultaneously predicting the inductive generalizations in multiple areas—a feat that, according to Whewell, can establish their truth. Perhaps to accommodate prevailing view of science as inductivist method, Whewell devoted several chapters to "methods of induction" and sometimes said "logic of induction"—and yet stressed it lacks rules and cannot be trained.[4]


Originator of pragmatism, C S Peirce who, as did Gottlob Frege independently, in the 1870s performed vast investigations that clarified the basis of deductive inference as mathematical proof, recognized induction but continuously insisted on a third type of inference that Peirce variously termed abduction or retroduction or hypothesis or presumption.[5] Later philosophers gave Peirce's abduction, etc, the synonym inference to the best explanation (IBE).[6]


Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer—but then figured not quite.[7] Bertrand Russell found Keynes's Treatise on Probability the best examination of induction, and if read with Jean Nicod's Le Probleme logique de l'induction as well as R B Braithwaite's review of it in the October 1925 issue of Mind, to provide "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics".[8] Two decades later, Russell proposed enumerative induction as an "independent logical principle".[9][10] Russell found,

"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must therefore be, or be deduced from, an independent principle not based on experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are forbidden. These, however, are not questions directly raised by Hume's arguments. What these arguments prove—and I do not think the proof can be controverted—is that the induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible".[10]


In a 1965 paper, now classic, Harman explained that enumerative induction is not an autonomous phenomenon, but is simply a masked consequence of inference to the best explanation (IBE).[6] IBE is otherwise synonym to C S Peirce's abduction.[6] Many philosophers of science espousing scientific realism have maintained that IBE way that scientists develop approximately true scientific theories about nature.[11]


Karl Popper in 1963 had declared, "Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure".[12] Popper's 1972 book Objective Knowledge—whose first chapter is devoted to the problem of induction—opens, "I think I have solved a major philosophical problem: the problem of induction".[12]

Within Popper's schemaProblem1 → Tentative Solution → Critical Test → Error Elimination → Problem2—enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during the problem shift.[12] An imaginative leap, the tentative solution is improvised, lacking inductive rules to guide it.[12] The resulting, unrestricted generalization is deductive, an entailed consequence of all, included explanatory considerations.[12] Controversy continued, however, with Popper's putative solution not generally accepted.[13]

By now, inductive inference has been shown to exist, but is found rarely, as in programs of machine learning in Artificial Intelligence (AI).[14] Popper's stance on induction is strictly falsified—enumerative induction exists—but is overwhelmingly absent from science.[14] Although much talked of nowadays by philosophers, abduction or IBE lacks rules of inference and the discussants provide nothing resembling such, as the process proceeds by humans' imaginations and perhaps creativity.[14]


  1. Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 355. ISBN 0-312-02353-7. OCLC 21216829. In a typical enumerative induction, the premises list the individuals observed to have a common property, and the conclusion claims that all individuals of the same population have that property.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  2. 2.0 2.1 Stefano Gattei, Karl Popper's Philosophy of Science: Rationality without Foundations (New York: Routledge, 2009), ch 2 "Science and philosophy", pp 28–30.
  3. 3.0 3.1 Wesley C Salmon, "The uniformity of Nature", Philosophy and Phenomenological Research, 1953 Sep;14(1):39–48, p 39.
  4. 4.0 4.1 4.2 4.3 Roberto Torretti, The Philosophy of Physics (Cambridge: Cambridge University Press, 1999), pp 216, 219–21.
  5. Roberto Torretti, The Philosophy of Physics (Cambridge: Cambridge University Press, 1999), pp 226, 228–29.
  6. 6.0 6.1 6.2 Ted Poston "Foundationalism", § b "Theories of proper inference", §§ iii "Liberal inductivism", Internet Encyclopedia of Philosophy, 10 Jun 2010 (last updated): "Strict inductivism is motivated by the thought that we have some kind of inferential knowledge of the world that cannot be accommodated by deductive inference from epistemically basic beliefs. A fairly recent debate has arisen over the merits of strict inductivism. Some philosophers have argued that there are other forms of nondeductive inference that do not fit the model of enumerative induction. C S Peirce describes a form of inference called 'abduction' or 'inference to the best explanation'. This form of inference appeals to explanatory considerations to justify belief. One infers, for example, that two students copied answers from a third because this is the best explanation of the available data—they each make the same mistakes and the two sat in view of the third. Alternatively, in a more theoretical context, one infers that there are very small unobservable particles because this is the best explanation of Brownian motion. Let us call 'liberal inductivism' any view that accepts the legitimacy of a form of inference to the best explanation that is distinct from enumerative induction. For a defense of liberal inductivism, see Gilbert Harman's classic (1965) paper. Harman defends a strong version of liberal inductivism according to which enumerative induction is just a disguised from of inference to the best explanation".
  7. David Andrews, Keynes and the British Humanist Tradition: The Moral Purpose of the Market (New York: Routledge, 2010), pp 63–65.
  8. Bertrand Russell, The Basic Writings of Bertrand Russell (New York: Routledge, 2009), "The validity of inference"], pp 157–64, quote on p 159.
  9. Gregory Landini, Russell (New York: Routledge, 2011), p 230.
  10. 10.0 10.1 Bertrand Russell, A History of Western Philosophy (London: George Allen and Unwin, 1945 / New York: Simon and Schuster, 1945), pp 673-74.
  11. Stathis Psillos, "On Van Fraassen's critique of abductive reasoning", Philosophical Quarterly, 1996 Jan;46(182):31–47, p 31.
  12. 12.0 12.1 12.2 12.3 12.4 Donald Gillies, "Problem-solving and the problem of induction", in Rethinking Popper (Dordrecht: Springer, 2009), Zuzana Parusniková & Robert S Cohen, eds, pp 103–05.
  13. Ch 5 "The controversy around inductive logic" in Richard Mattessich, ed, Instrumental Reasoning and Systems Methodology: An Epistemology of the Applied and Social Sciences (Dordrecht: D. Reidel Publishing, 1978), pp 141–43.
  14. 14.0 14.1 14.2 Donald Gillies, "Problem-solving and the problem of induction", in Rethinking Popper (Dordrecht: Springer, 2009), Zuzana Parusniková & Robert S Cohen, eds, p 111: "I argued earlier that there are some exceptions to Popper's claim that rules of inductive inference do not exist. However, these exceptions are relatively rare. They occur, for example, in the machine learning programs of AI. For the vast bulk of human science both past and present, rules of inductive inference do not exist. For such science, Popper's model of conjectures which are freely invented and then tested out seems to me more accurate than any model based on inductive inferences. Admittedly, there is talk nowadays in the context of science carried out by humans of 'inference to the best explanation' or 'abductive inference', but such so-called inferences are not at all inferences based on precisely formulated rules like the deductive rules of inference. Those who talk of 'inference to the best explanation' or 'abductive inference', for example, never formulate any precise rules according to which these so-called inferences take place. In reality, the 'inferences' which they describe in their examples involve conjectures thought up by human ingenuity and creativity, and by no means inferred in any mechanical fashion, or according to precisely specified rules".

See also