Inductive reasoning

===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" />