Inductive reasoning

===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>