Inductive reasoning

== History ==

===Ancient philosophy===
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''.<ref name="Gattei">Stefano Gattei, ''Karl Popper's Philosophy of Science: Rationality without Foundations'' (New York: [[Routledge]], 2009), ch. 2 "Science and philosophy", [https://books.google.com/books?id=oPPu1JvMBFoC&pg=PA28#v=twopage pp. 28–30].</ref>

====Aristotle and the Peripatetic School====
Aristotle's ''[[Posterior Analytics]]'' covers the methods of inductive proof in natural philosophy and in the social sciences.  The first book of [[s:Organon (Owen)/The Posterior Analytics|Posterior Analytics]] describes the nature and science of demonstration and its elements: including definition, division, intuitive reason of first principles, particular and universal demonstration, affirmative and negative demonstration, the difference between science and opinion, etc.

====Pyrrhonism====
The ancient [[Pyrrhonism|Pyrrhonists]] were the first Western philosophers to point out the [[Problem of induction]]: that induction cannot, according to them, justify the acceptance of universal statements as true.<ref name="Gattei" />

====Ancient medicine====
The [[Empiric school]] of ancient Greek medicine employed ''[[epilogism]]'' as a method of inference. 'Epilogism' is a theory-free method that looks at history through the accumulation of facts without major generalization and with consideration of the consequences of making causal claims.<ref name=":1">{{Cite book|last=Taleb|first=Nassim Nicholas|title=The Black Swan: Second Edition: The Impact of the Highly Improbable Fragility|publisher=Random House Publishing Group|year=2010|isbn=978-0812973815|location=New York|pages=199, 302, 383}}</ref> Epilogism is an inference which moves entirely within the domain of visible and evident things, it tries not to invoke [[unobservable]]s.

The [[Dogmatic school]] of ancient Greek medicine employed ''analogismos'' as a method of inference.<ref>[[Galen]] ''On Medical Experience'', 24.</ref> This method used analogy to reason from what was observed to unobservable forces.

===Early modern philosophy===
In 1620, [[Early modern philosophy|early modern philosopher]] [[Francis Bacon]] repudiated the value of mere experience and enumerative induction alone. [[Baconian method|His method]] of [[inductivism]] required that minute and many-varied observations that uncovered the natural world's structure and causal relations needed to be coupled with enumerative induction in order to have knowledge beyond the present scope of experience. Inductivism therefore required enumerative induction as a component.

====David Hume====
The empiricist [[David Hume]]'s 1740 stance found enumerative induction to have no rational, let alone logical, basis; instead, induction was the product of instinct rather than reason, a custom of the mind and an everyday requirement to live. While observations, such as the motion of the sun, could be coupled with the principle of the [[Uniformitarianism|uniformity of nature]] to produce conclusions that seemed to be certain, the [[problem of induction]] arose from the fact that the uniformity of nature was not a logically valid principle, therefore it could not be defended as deductively rational, but also could not be defended as inductively rational by appealing to the fact that the uniformity of nature has accurately described the past and therefore, will likely accurately describe the future because that is an inductive argument and therefore circular since induction is what needs to be justified.

Since Hume first wrote about the dilemma between the invalidity of deductive arguments and the circularity of inductive arguments in support of the uniformity of nature, this supposed dichotomy between merely two modes of inference, deduction and induction, has been contested with the discovery of a third mode of inference known as abduction, or [[abductive reasoning]], which was first formulated and advanced by [[Charles Sanders Peirce]], in 1886, where he referred to it as "reasoning by hypothesis."<ref>{{Cite journal |last=Plutynski |first=Anya |date=2011 |title=Four Problems of Abduction: A Brief History |url=https://philpapers.org/rec/PLUFPO |journal=HOPOS: The Journal of the International Society for the History of Philosophy of Science |volume=1 |issue=2 |pages=227–248 |doi=10.1086/660746 |s2cid=15332806 |access-date=16 April 2022 |archive-date=11 April 2023 |archive-url=https://web.archive.org/web/20230411064730/https://philpapers.org/rec/PLUFPO |url-status=live }}</ref>  Inference to the best explanation is often, yet arguably, treated as synonymous to abduction as it was first identified by Gilbert Harman in 1965 where he referred to it as "abductive reasoning," yet his definition of abduction slightly differs from Pierce's definition.<ref>{{Cite journal |last=Mcauliffe |first=William H. B. |date=2015 |title=How did Abduction Get Confused with Inference to the Best Explanation? |url=https://www.jstor.org/stable/10.2979/trancharpeirsoc.51.3.300 |journal=Transactions of the Charles S. Peirce Society |volume=51 |issue=3 |pages=300–319 |doi=10.2979/trancharpeirsoc.51.3.300 |jstor=10.2979/trancharpeirsoc.51.3.300 |s2cid=43255826 |issn=0009-1774 |access-date=16 April 2022 |archive-date=16 April 2022 |archive-url=https://web.archive.org/web/20220416041215/https://www.jstor.org/stable/10.2979/trancharpeirsoc.51.3.300 |url-status=live }}</ref>  Regardless, if abduction is in fact a third mode of inference rationally independent from the other two, then either the uniformity of nature can be rationally justified through abduction, or Hume's dilemma is more of a trilemma.  Hume was also skeptical of the application of enumerative induction and reason to reach certainty about unobservables and especially the inference of causality from the fact that modifying an aspect of a relationship prevents or produces a particular outcome.

====Immanuel Kant====
Awakened from "dogmatic slumber" by a German translation of Hume's work, [[Immanuel Kant|Kant]] sought to explain the possibility of [[metaphysics]].  In 1781, Kant's ''[[Critique of Pure Reason]]'' introduced ''[[rationalism]]'' as a path toward knowledge distinct from ''[[empiricism]]''.  Kant sorted statements into two types.  [[analytic-synthetic distinction|Analytic]] statements are true by virtue of the [[syntax|arrangement]] of their terms and [[semantics|meanings]], thus analytic statements are [[tautology (logic)|tautologies]], merely logical truths, true by [[logical truth|necessity]]. Whereas [[analytic-synthetic distinction|synthetic]] statements hold meanings to refer to states of facts, [[contingency (philosophy)|contingencies]]. Against both rationalist philosophers like [[René Descartes|Descartes]] and [[Gottfried Wilhelm Leibniz|Leibniz]] as well as against empiricist philosophers like [[John Locke|Locke]] and [[David Hume|Hume]], Kant's ''[[Critique of Pure Reason]]'' is a sustained argument that in order to have knowledge we need both a contribution of our mind (concepts) as well as a contribution of our senses (intuitions). Knowledge proper is for Kant thus restricted to what we can possibly perceive (''[[phenomena]]''), whereas objects of mere thought ("[[Thing-in-itself|things in themselves]]") are in principle unknowable due to the impossibility of ever perceiving them.

Reasoning that the mind must contain its own categories for organizing [[sense data]], making experience of objects in ''space'' and ''time ([[phenomena]])'' possible, Kant concluded that the [[Uniformitarianism|uniformity of nature]] was an ''a priori'' truth.<ref name="Salmon" />  A class of synthetic statements that was not [[contingency (philosophy)|contingent]] but true by necessity, was then [[synthetic a priori|synthetic ''a priori'']].  Kant thus saved both [[metaphysics]] and [[Newton's law of universal gravitation]]. On the basis of the argument that what goes beyond our knowledge is "nothing to us,"<ref>Cf. {{Cite book|last=Kant|first=Immanuel|title=Critique of Pure Reason|year=1787|pages=B132}}</ref> he discarded [[scientific realism]].  Kant's position that knowledge comes about by a cooperation of perception and our capacity to think ([[transcendental idealism]]) gave birth to the movement of [[German idealism]].  [[Georg Friedrich Wilhelm Hegel|Hegel]]'s [[absolute idealism]] subsequently flourished across continental Europe and England.

===Late modern philosophy===
[[Positivism]], developed by [[Claude Henri de Rouvroy, comte de Saint-Simon|Henri de Saint-Simon]] and promulgated in the 1830s by his former student [[Auguste Comte]], was the first [[Late modern philosophy|late modern]] [[philosophy of science]]. In the aftermath of the [[French Revolution]], fearing society's ruin, 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 had become scientific, with questions of [[theology]] and of [[metaphysics]] being unanswerable. Comte found enumerative induction reliable as a consequence of its grounding in available experience. He asserted the use of science, rather than metaphysical truth, as the correct method for the improvement of human society.

According to Comte, [[scientific method]] frames predictions, confirms them, and states laws—positive statements—irrefutable by [[theology]] or by [[metaphysics]].  Regarding experience as justifying enumerative induction by demonstrating the [[Uniformitarianism|uniformity of nature]],<ref name="Salmon">Wesley C Salmon, [https://www.jstor.org/stable/210401 "The uniformity of Nature"] {{Webarchive|url=https://web.archive.org/web/20180818182105/https://www.jstor.org/stable/210401 |date=18 August 2018 }}, ''Philosophy and Phenomenological Research'', 1953 Sep;'''14'''(1):39–48, [39].</ref> the British philosopher [[John Stuart Mill]] welcomed Comte's positivism, but thought [[scientific laws]] susceptible to recall or revision and Mill also withheld from Comte's [[Religion of Humanity]].  Comte was confident in treating [[scientific law]] as an [[foundationalism|irrefutable foundation for all knowledge]], and believed that churches, honouring eminent scientists, ought to focus public mindset on ''[[altruism]]''—a term Comte coined—to apply science for humankind's social welfare via [[sociology]], Comte's leading science.

During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science, [[William Whewell]] found enumerative induction not nearly as convincing, and, despite the dominance of inductivism, formulated "superinduction".<ref name="Toretti-p220">Roberto Torretti, ''The Philosophy of Physics'' (Cambridge: [[Cambridge University Press]], 1999), [https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA219 219–21] {{Webarchive|url=https://web.archive.org/web/20220509192130/https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA219 |date=9 May 2022 }}[https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA216 &#91;216&#93;] {{Webarchive|url=https://web.archive.org/web/20220509192129/https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA216 |date=9 May 2022 }}.</ref>  Whewell argued that "the peculiar import of the term ''Induction''" should be recognised: "there is some Conception ''superinduced'' upon the facts", that is, "the Invention of a new Conception in every inductive inference". The creation of Conceptions is easily overlooked and prior to Whewell was rarely recognised.<ref name="Toretti-p220" /> Whewell explained:

{{Blockquote|text="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."<ref name=Toretti-p220/>}}

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 the prevailing view of science as inductivist method, Whewell devoted several chapters to "methods of induction" and sometimes used the phrase "logic of induction", despite the fact that induction lacks rules and cannot be trained.<ref name="Toretti-p220" />

In the 1870s, the originator of [[pragmatism]], [[Charles Sanders Peirce|C S Peirce]] performed vast investigations that clarified the basis of [[deductive inference]] as a mathematical proof (as, independently, did [[Gottlob Frege]]). Peirce recognized induction but always insisted on a third type of inference that Peirce variously termed ''[[abductive reasoning|abduction]]'' or ''retroduction'' or ''hypothesis'' or ''presumption''.<ref>Roberto Torretti, ''The Philosophy of Physics'' (Cambridge: [[Cambridge University Press]], 1999), [https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA226 pp. 226] {{Webarchive|url=https://web.archive.org/web/20220509192128/https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA226 |date=9 May 2022 }}, [https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA228 228–29] {{Webarchive|url=https://web.archive.org/web/20220509192127/https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA228 |date=9 May 2022 }}.</ref>  Later philosophers termed Peirce's abduction, etc., ''[[Inference to the Best Explanation]]'' (IBE).<ref name="Poston" />

===Contemporary philosophy===
====Bertrand Russell====
Having highlighted Hume's [[problem of induction]], [[John Maynard Keynes]] posed ''logical probability'' as its answer, or as near a solution as he could arrive at.<ref>David Andrews, ''Keynes and the British Humanist Tradition: The Moral Purpose of the Market'' (New York: [[Routledge]], 2010), pp. [https://books.google.com/books?id=1VsYOwRsOVUC&pg=PA63 63–65].</ref> [[Bertrand Russell]] found Keynes's ''[[A Treatise on Probability|Treatise on Probability]]'' the best examination of induction, and believed that if read with [[Jean Nicod]]'s ''Le Probleme logique de l'induction'' as well as [[R. B. Braithwaite|R B Braithwaite]]'s review of Keynes's work in the October 1925 issue of ''Mind'', that would cover "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics".<ref>{{cite book |last1=Russell |first1=Bertrand |title=An Outline of Philosophy |date=1927 |publisher=Allen and Unwin |location=London and New York}} reprinted in Bertrand Russell, ''The Basic Writings of Bertrand Russell'' (New York: [[Routledge]], 2009), "The validity of inference"], pp. 157–64, quote on [https://books.google.com/books?id=jqun5YJGt-wC&pg=PA159 p. 159] {{Webarchive|url=https://web.archive.org/web/20220509192130/https://books.google.com/books?id=jqun5YJGt-wC&pg=PA159 |date=9 May 2022 }}.</ref>  Two decades later, [[Bertrand Russell|Russell]] followed Keynes in regarding enumerative induction as an "independent logical principle".<ref>{{harvnb|Russell|1948|pp=396–450}}.</ref><ref>Gregory Landini, ''Russell'' (New York: Routledge, 2011), p. [https://books.google.com/books?id=Simvmwn2m2gC&pg=PA230 230] {{Webarchive|url=https://web.archive.org/web/20220509192127/https://books.google.com/books?id=Simvmwn2m2gC&pg=PA230 |date=9 May 2022 }}.</ref><ref name="Russell1945" />  Russell found:

{{Blockquote|text="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 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."<ref name="Russell1945">Bertrand Russell, ''A History of Western Philosophy'' (London: George Allen and Unwin, 1945 / New York: Simon and Schuster, 1945), pp. 673–74.</ref>}}

====Gilbert Harman====
In a 1965 paper, [[Gilbert Harman]] explained that enumerative induction is not an autonomous phenomenon, but is simply a disguised consequence of Inference to the Best Explanation (IBE).<ref name="Poston">Ted Poston [http://www.iep.utm.edu/found-ep "Foundationalism"] {{Webarchive|url=https://web.archive.org/web/20190926003413/http://www.iep.utm.edu/found-ep |date=26 September 2019 }}, § 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 belief]]s.  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 '[[abductive reasoning|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 [[molecules|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 form of [[inference to the best explanation]]".</ref>  IBE is otherwise synonymous with [[C S Peirce]]'s ''[[abductive reasoning|abduction]]''.<ref name="Poston" />  Many philosophers of science espousing [[scientific realism]] have maintained that IBE is the way that scientists develop approximately true scientific theories about nature.<ref>Stathis Psillos, [https://www.jstor.org/stable/2956303 "On Van Fraassen's critique of abductive reasoning"] {{Webarchive|url=https://web.archive.org/web/20180818183602/https://www.jstor.org/stable/2956303 |date=18 August 2018 }}, ''Philosophical Quarterly'', 1996 Jan;'''46'''(182):31–47, [31].</ref>