View source for Module:Inductive reasoning

{{Short description|Method of logical reasoning}}
{{Redirect-distinguish|Inductive inference|Mathematical induction}}
{{Use dmy dates|date=March 2022}}

'''Inductive reasoning''' is a [[method of reasoning]] in which a general [[principle]] is derived from a body of observations.<ref>{{Cite book|last=Publishing|first=Walch|title=Assessment Strategies for Science: Grades 6–8|publisher=Walch Publishing|year=2004|isbn=0-8251-5175-9|location=Portland|pages=4}}</ref> It consists of making broad generalizations based on specific observations.<ref>{{Cite web |date=2022-01-10 |title=Deductive, Inductive Reasoning: Definition, Differences, Examples |url=https://mundanopedia.com/economics/microeconomics/deductive-and-inductive-reasoning-methods/ |access-date=2022-03-07 |website=Mundanopedia |language=en-US}}</ref> Inductive reasoning is distinct from [[Deductive reasoning|''deductive'' reasoning]]. If the premises are correct, the conclusion of a deductive argument is ''certain''; in contrast, the truth of the conclusion of an inductive argument is ''[[probable]]'', based upon the evidence given.<ref>{{cite book |last1=Copi |first1=I.M. |last2=Cohen |first2=C. |last3=Flage |first3=D.E. |year=2006 |title=Essentials of Logic |edition=Second |location=Upper Saddle River, NJ |publisher=Pearson Education |isbn=978-0-13-238034-8 }}</ref>

== Types ==
The types of inductive reasoning include generalization, prediction, statistical syllogism, argument from analogy, and causal inference.

=== Inductive generalization ===
A generalization (more accurately, an ''inductive generalization'') proceeds from a premise about a [[Sample (statistics)|sample]] to a conclusion about the [[statistical population|population]].<ref name=":0">{{Cite book|last=Govier|first=Trudy|title=A Practical Study of Argument, Enhanced Seventh Edition|publisher=Cengage Learning|year=2013|isbn=978-1-133-93464-6|location=Boston, MA|pages=283}}</ref> The observation obtained from this sample is projected onto the broader population.<ref name=":0" />

: The proportion Q of the sample has attribute A.
: Therefore, the proportion Q of the population has attribute A.

For example, say there are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. An inductive generalization would be that there are 15 black and five white balls in the urn.

How much the premises support the conclusion depends upon (1) the number in the sample group, (2) the number in the population, and (3) the degree to which the sample represents the population (which may be achieved by taking a random sample). The greater the sample size relative to the population and the more closely the sample represents the population, the stronger the generalization is. The [[hasty generalization]] and the [[biased sample]] are generalization fallacies.

==== Statistical generalization ====
A statistical generalization is a type of inductive argument in which a conclusion about a population is inferred using a [[Sample (statistics)|statistically-representative sample]]. For example:

:Of a sizeable random sample of voters surveyed, 66% support Measure Z.
:Therefore, approximately 66% of voters support Measure Z.

The measure is highly reliable within a well-defined margin of error provided the sample is large and random. It is readily quantifiable. Compare the preceding argument with the following. "Six of the ten people in my book club are Libertarians. Therefore, about 60% of people are Libertarians." The argument is weak because the sample is non-random and the sample size is very small.

Statistical generalizations are also called ''statistical projections''<ref>Schaum's Outlines, Logic, Second Edition. John Nolt, Dennis Rohatyn, Archille Varzi. McGraw-Hill, 1998. p. 223</ref> and ''sample projections''.<ref>Schaum's Outlines, Logic, p. 230</ref>

==== Anecdotal generalization ====
An anecdotal generalization is a type of inductive argument in which a conclusion about a population is inferred using a non-statistical sample.<ref>{{Cite book|last1=Johnson|first1=Dale D.|url=https://books.google.com/books?id=cMOPtcgQfT8C|title=Trivializing Teacher Education: The Accreditation Squeeze|last2=Johnson|first2=Bonnie|last3=Ness|first3=Daniel|last4=Farenga|first4=Stephen J.|publisher=Rowman & Littlefield|year=2005|isbn=9780742535367|pages=182–83}}</ref> In other words, the generalization is based on [[anecdotal evidence]]. For example:

:So far, this year his son's Little League team has won 6 of 10 games.
:Therefore, by season's end, they will have won about 60% of the games.

This inference is less reliable (and thus more likely to commit the fallacy of hasty generalization) than a statistical generalization, first, because the sample events are non-random, and second because it is not reducible to mathematical expression. Statistically speaking, there is simply no way to know, measure and calculate the circumstances affecting performance that will occur in the future. On a philosophical level, the argument relies on the presupposition that the operation of future events will mirror the past. In other words, it takes for granted a uniformity of nature, an unproven principle that cannot be derived from the empirical data itself. Arguments that tacitly presuppose this uniformity are sometimes called ''Humean'' after the philosopher who was first to subject them to philosophical scrutiny.<ref>Introduction to Logic. Gensler p. 280</ref>

=== Prediction ===
An inductive prediction draws a conclusion about a future, current, or past instance from a sample of other instances. Like an inductive generalization, an inductive prediction relies on a data set consisting of specific instances of a phenomenon. But rather than conclude with a general statement, the inductive prediction concludes with a specific statement about the probability that a single instance will (or will not) have an attribute shared (or not shared) by the other instances.<ref>{{Cite journal|last=Romeyn|first=J. W.|date=2004|title=ypotheses and Inductive Predictions: Including Examples on Crash Data|journal=Synthese|volume=141|issue=3|pages=333–64|doi=10.1023/B:SYNT.0000044993.82886.9e|jstor=20118486|s2cid=121862013|url=https://pure.rug.nl/ws/files/2720641/romeijn_-_hypotheses_and_predictions.pdf}}</ref>

: Proportion Q of observed members of group G have had attribute A.
: Therefore, there is a probability corresponding to Q that other members of group G will have attribute A when next observed.

=== Statistical syllogism ===
{{Main|Statistical syllogism}}

A statistical [[syllogism]] proceeds from a generalization about a group to a conclusion about an individual.

:Proportion Q of the known instances of population P has attribute A.
: Individual I is another member of P.
: Therefore, there is a probability corresponding to Q that I has A.

For example:
:90% of graduates from Excelsior Preparatory school go on to University.
:Bob is a graduate of Excelsior Preparatory school.
:Therefore, Bob will go on to University.

This is a ''statistical syllogism''.<ref name="Logic">Introduction to Logic. Harry J. Gensler, Rutledge, 2002. p. 268</ref> Even though one cannot be sure Bob will attend university, we can be fully assured of the exact probability of this outcome (given no further information). Arguably the argument is too strong and might be accused of "cheating". After all, the probability is given in the premise. Typically, inductive reasoning seeks to ''formulate'' a [[probability]]. Two [[dicto simpliciter]] fallacies can occur in statistical syllogisms: "[[accident (fallacy)|accident]]" and "[[converse accident]]".

=== Argument from analogy ===
{{Main|Argument from analogy}}

The process of analogical inference involves noting the shared properties of two or more things and from this basis inferring that they also share some further property:<ref name="Baronett">{{Cite book|title=Logic|last=Baronett|first=Stan|publisher=Pearson Prentice Hall|year=2008|location=Upper Saddle River, NJ|pages=321–25}}</ref>

:P and Q are similar with respect to properties a, b, and c.
:Object P has been observed to have further property x.
:Therefore, Q probably has property x also.

Analogical reasoning is very frequent in [[common sense]], [[science]], [[philosophy]], [[law]], and the [[humanities]], but sometimes it is accepted only as an auxiliary method. A refined approach is [[case-based reasoning]].<ref>For more information on inferences by analogy, see [http://www.cs.hut.fi/Opinnot/T-93.850/2005/Papers/juthe2005-analogy.pdf Juthe, 2005].</ref>

:Mineral A and Mineral B are both igneous rocks often containing veins of quartz and are most commonly found in South America in areas of ancient volcanic activity.
:Mineral A is also a soft stone suitable for carving into jewelry.
:Therefore, mineral B is probably a soft stone suitable for carving into jewelry.

This is ''analogical induction'', according to which things alike in certain ways are more prone to be alike in other ways. This form of induction was explored in detail by philosopher John Stuart Mill in his ''System of Logic'', where he states, "[t]here can be no doubt that every resemblance [not known to be irrelevant] affords some degree of probability, beyond what would otherwise exist, in favor of the conclusion."<ref>A System of Logic. Mill 1843/1930. p. 333</ref> See [[Mill's Methods]].

Some thinkers contend that analogical induction is a subcategory of inductive generalization because it assumes a pre-established uniformity governing events.{{Citation needed|date=June 2020}} Analogical induction requires an auxiliary examination of the ''relevancy'' of the characteristics cited as common to the pair. In the preceding example, if a premise were added stating that both stones were mentioned in the records of early Spanish explorers, this common attribute is extraneous to the stones and does not contribute to their probable affinity.

A pitfall of analogy is that features can be [[cherry-picked]]: while objects may show striking similarities, two things juxtaposed may respectively possess other characteristics not identified in the analogy that are characteristics sharply ''dis''similar. Thus, analogy can mislead if not all relevant comparisons are made.

=== Causal inference ===
{{Main|Causal reasoning}}
A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.{{cn|date=September 2022}}

== Methods ==
The two principal methods used to reach inductive conclusions are ''enumerative induction'' and ''eliminative induction.''<ref name="dan">{{cite journal |last1=Hunter |first1=Dan |title=No Wilderness of Single Instances: Inductive Inference in Law |journal=Journal of Legal Education |date=September 1998 |volume=48 |issue=3 |pages=370–72 }}</ref><ref name="jm">{{cite book |last1=J.M. |first1=Bochenski |url=https://books.google.com/books?id=fnqhBQAAQBAJ |title=The Methods of Contemporary Thought |publisher=Springer Science & Business Media |year=2012 |isbn=978-9401035781 |editor1-last=Caws |editor1-first=Peter |pages=108–09 |access-date=June 5, 2020}}</ref>

===Enumerative induction===
Enumerative induction is an inductive method in which a conclusion is constructed based on the ''number'' of instances that support it. The more supporting instances, the stronger the conclusion.<ref name="dan" /><ref name="jm" />

The most basic form of enumerative induction reasons from particular instances to all instances, and is thus an unrestricted generalization.<ref>{{cite book|last=Churchill|first=Robert Paul|title=Logic: An Introduction|publisher=St. Martin's Press|year=1990|isbn=978-0-312-02353-9|edition=2nd|location=New York|page=355|oclc=21216829|quote=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.}}</ref>  If one observes 100 swans, and all 100 were white, one might infer a universal [[categorical proposition]] of the form ''All swans are white''. As this [[argument form|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]].

:All life forms so far discovered are composed of cells.
:Therefore, all life forms are composed of cells.

This is ''enumerative induction'', also known as ''simple induction'' or ''simple predictive induction''. It is a subcategory of inductive generalization. In everyday practice, this is perhaps the most common form of induction. For the preceding argument, the conclusion is tempting but makes a prediction well in excess of the evidence. First, it assumes that life forms observed until now can tell us how future cases will be: an appeal to uniformity. Second, the conclusion ''All'' is a bold assertion. A single contrary instance foils the argument. And last, quantifying the level of probability in any mathematical form is problematic.<ref>Schaum's Outlines, Logic, pp. 243–35</ref> By what standard do we measure our Earthly sample of known life against all (possible) life? Suppose we do discover some new organism—such as some microorganism floating in the mesosphere or an asteroid—and it is cellular. Does the addition of this corroborating evidence oblige us to raise our probability assessment for the subject proposition? It is generally deemed reasonable to answer this question "yes," and for a good many this "yes" is not only reasonable but incontrovertible. So then just ''how much'' should this new data change our probability assessment? Here, consensus melts away, and in its place arises a question about whether we can talk of probability coherently at all without numerical quantification.

:All life forms so far discovered have been composed of cells.
:Therefore, the ''next'' life form discovered will be composed of cells.

This is enumerative induction in its ''weak form''. It truncates "all" to a mere single instance and, by making a far weaker claim, considerably strengthens the probability of its conclusion. Otherwise, it has the same shortcomings as the strong form: its sample population is non-random, and quantification methods are elusive.

=== Eliminative induction ===
[[Eliminative induction]], also called variative induction, is an inductive method in which a conclusion is constructed based on the ''variety'' of instances that support it. Unlike enumerative induction, eliminative induction reasons based on the various kinds of instances that support a conclusion, rather than the number of instances that support it. As the variety of instances increases, the more possible conclusions based on those instances can be identified as incompatible and eliminated. This, in turn, increases the strength of any conclusion that remains consistent with the various instances. This type of induction may use different methodologies such as quasi-experimentation, which tests and where possible eliminates rival hypotheses.<ref>{{Cite book|last1=Hoppe|first1=Rob|title=Knowledge, Power, and Participation in Environmental Policy Analysis|last2=Dunn|first2=William N.|year=2001|publisher=Transaction Publishers|isbn=978-1-4128-2721-8|pages=419}}</ref> Different evidential tests may also be employed to eliminate possibilities that are entertained.<ref>{{Cite book|last=Schum|first=David A.|title=The Evidential Foundations of Probabilistic Reasoning|publisher=Northwestern University Press|year=2001|isbn=0-8101-1821-1|location=Evanston, Illinois|pages=32}}</ref>

Eliminative induction is crucial to the scientific method and is used to eliminate hypotheses that are inconsistent with observations and experiments.<ref name="dan" /><ref name="jm" /> It focuses on possible causes instead of observed actual instances of causal connections.<ref>{{Cite book|last1=Hodge|first1=Jonathan|title=The Cambridge Companion to Darwin|url=https://archive.org/details/cambridgecompani00hodg_248|url-access=limited|last2=Hodge|first2=Michael Jonathan Sessions|last3=Radick|first3=Gregory|publisher=Cambridge University Press|year=2003|isbn=0-521-77197-8|location=Cambridge|page=[https://archive.org/details/cambridgecompani00hodg_248/page/n189 174]}}</ref>
{{Expand section|date=June 2020}}

==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 }}</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}}</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"], ''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][https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA216 &#91;216&#93;].</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], [https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA228 228–29].</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 ''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>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].</ref>  Two decades later, [[Bertrand Russell|Russell]] proposed enumerative induction as an "independent logical principle".<ref>Gregory Landini, ''Russell'' (New York: Routledge, 2011), p. [https://books.google.com/books?id=Simvmwn2m2gC&pg=PA230 230].</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"], § 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 ''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"], ''Philosophical Quarterly'', 1996 Jan;'''46'''(182):31–47, [31].</ref>

== Comparison with deductive reasoning ==
[[File:Argument terminology used in logic (en).svg|thumb|400px|Argument terminology]]
Inductive reasoning is a form of argument that—in contrast to deductive reasoning—allows for the possibility that a conclusion can be false, even if all of the [[premise]]s are true.<ref>John Vickers. [http://plato.stanford.edu/entries/induction-problem/ The Problem of Induction]. The Stanford Encyclopedia of Philosophy.</ref> This difference between deductive and inductive reasoning is reflected in the terminology used to describe deductive and inductive arguments. In deductive reasoning, an argument is "[[Validity (logic)|valid]]" when, assuming the argument's premises are true, the conclusion ''must'' be true. If the argument is valid and the premises ''are'' true, then the argument is [[Soundness|"sound"]]. In contrast, in inductive reasoning, an argument's premises can never guarantee that the conclusion ''must'' be true; therefore, inductive arguments can never be valid or sound. Instead, an argument is "strong" when, assuming the argument's premises are true, the conclusion is ''probably'' true. If the argument is strong and the premises ''are'' true, then the argument is "cogent".<ref>{{cite web|last=Herms|first=D.|title=Logical Basis of Hypothesis Testing in Scientific Research|url=http://www.dartmouth.edu/~bio125/logic.Giere.pdf}}</ref> Less formally, an inductive argument may be called "probable", "plausible", "likely", "reasonable", or "justified", but never "certain" or "necessary". Logic affords no bridge from the probable to the certain.

The futility of attaining certainty through some critical mass of probability can be illustrated with a coin-toss exercise. Suppose someone tests whether a coin is either a fair one or two-headed. They flip the coin ten times, and ten times it comes up heads. At this point, there is a strong reason to believe it is two-headed. After all, the chance of ten heads in a row is .000976: less than one in one thousand. Then, after 100 flips, every toss has come up heads. Now there is “virtual” certainty that the coin is two-headed. Still, one can neither logically nor empirically rule out that the next toss will produce tails. No matter how many times in a row it comes up heads this remains the case. If one programmed a machine to flip a coin over and over continuously at some point the result would be a string of 100 heads. In the fullness of time, all combinations will appear.

As for the slim prospect of getting ten out of ten heads from a fair coin—the outcome that made the coin appear biased—many may be surprised to learn that the chance of any sequence of heads or tails is equally unlikely (e.g., H-H-T-T-H-T-H-H-H-T) and yet it occurs in ''every'' trial of ten tosses. That means ''all'' results for ten tosses have the same probability as getting ten out of ten heads, which is 0.000976. If one records the heads-tails sequences, for whatever result, that exact sequence had a chance of 0.000976.

An argument is deductive when the conclusion is necessary given the premises. That is, the conclusion must be true if the premises are true.

If a deductive conclusion follows duly from its premises, then it is valid; otherwise, it is invalid (that an argument is invalid is not to say it is false; it may have a true conclusion, just not on account of the premises). An examination of the following examples will show that the relationship between premises and conclusion is such that the truth of the conclusion is already implicit in the premises. Bachelors are unmarried because we ''say'' they are; we have defined them so. Socrates is mortal because we have included him in a set of beings that are mortal. The conclusion for a valid deductive argument is already contained in the premises since its truth is strictly a matter of logical relations. It cannot say more than its premises. Inductive premises, on the other hand, draw their substance from fact and evidence, and the conclusion accordingly makes a factual claim or prediction. Its reliability varies proportionally with the evidence. Induction wants to reveal something ''new'' about the world. One could say that induction wants to say ''more'' than is contained in the premises.

To better see the difference between inductive and deductive arguments, consider that it would not make sense to say: "all rectangles so far examined have four right angles, so the next one I see will have four right angles." This would treat logical relations as something factual and discoverable, and thus variable and uncertain. Likewise, speaking deductively we may permissibly say. "All unicorns can fly; I have a unicorn named Charlie; thus Charlie can fly." This deductive argument is valid because the logical relations hold; we are not interested in their factual soundness.

Inductive reasoning is inherently [[Uncertainty|uncertain]]. It only deals with the extent to which, given the premises, the conclusion is ''credible'' according to some theory of evidence. Examples include a [[many-valued logic]], [[Dempster–Shafer theory]], or [[Probability|probability theory]] with rules for inference such as [[Bayes' theorem|Bayes' rule]]. Unlike deductive reasoning, it does not rely on universals holding over a [[Closed world assumption|closed domain of discourse]] to draw conclusions, so it can be applicable even in cases of [[Open world assumption|epistemic uncertainty]] (technical issues with this may arise however; for example, the [[Axioms of probability#Second axiom|second axiom of probability]] is a closed-world assumption).<ref>{{cite journal|last=Kosko|first=Bart|year=1990|title=Fuzziness vs. Probability|journal=International Journal of General Systems|volume=17|issue=1|pages=211–40|doi=10.1080/03081079008935108}}</ref>

Another crucial difference between these two types of argument is that deductive certainty is impossible in non-axiomatic systems such as [[reality]], leaving inductive reasoning as the primary route to (probabilistic) knowledge of such systems.<ref>{{cite book|title=Stanford Encyclopedia of Philosophy : Kant's account of reason|publisher=Metaphysics Research Lab, Stanford University|year=2018|chapter=Kant's Account of Reason|chapter-url=http://plato.stanford.edu/entries/kant-reason/#TheReaReaCogRolLim}}</ref>

Given that "if ''A'' is true then that would cause ''B'', ''C'', and ''D'' to be true", an example of deduction would be "''A'' is true therefore we can deduce that ''B'', ''C'', and ''D'' are true". An example of induction would be "''B'', ''C'', and ''D'' are observed to be true therefore ''A'' might be true". ''A'' is a [[Causality|reasonable]] explanation for ''B'', ''C'', and ''D'' being true.

For example:

:A large enough asteroid impact would create a very large crater and cause a severe [[impact winter]] that could drive the non-avian dinosaurs to extinction.
:We observe that there is a [[Chicxulub crater|very large crater]] in the Gulf of Mexico dating to very near the time of the extinction of the non-avian dinosaurs.
:Therefore, it is possible that this impact could explain why the non-avian dinosaurs became extinct.

Note, however, that the asteroid explanation for the mass extinction is not necessarily correct. Other events with the potential to affect global climate also coincide with the [[Cretaceous–Paleogene extinction event|extinction of the non-avian dinosaurs]]. For example, the release of [[volcanic gas]]es (particularly [[sulfur dioxide]]) during the formation of the [[Deccan Traps]] in [[India]].

Another example of an inductive argument:

:All biological life forms that we know of depend on liquid water to exist.
:Therefore, if we discover a new biological life form, it will probably depend on liquid water to exist.

This argument could have been made every time a new biological life form was found, and would have been correct every time; however, it is still possible that in the future a biological life form not requiring liquid water could be discovered.
As a result, the argument may be stated less formally as:

:All biological life forms that we know of depend on liquid water to exist.
:Therefore, all biological life probably depends on liquid water to exist.

A classical example of an ''incorrect'' inductive argument was presented by John Vickers:

:All of the swans we have seen are white.
:Therefore, we ''know'' that all swans are white.

The correct conclusion would be: we [[Expectation (epistemic)|expect]] all swans to be white.

Succinctly put: deduction is about ''certainty/necessity''; induction is about ''probability''.<ref name="Logic" /> Any single assertion will answer to one of these two criteria. Another approach to the analysis of reasoning is that of [[modal logic]], which deals with the distinction between the necessary and the  ''possible'' in a way not concerned with probabilities among things deemed possible.

The philosophical definition of inductive reasoning is more nuanced than a simple progression from particular/individual instances to broader generalizations. Rather, the premises of an inductive [[logical argument]] indicate some degree of support (inductive probability) for the conclusion but do not [[entailment|entail]] it; that is, they suggest truth but do not ensure it. In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms).

Note that the definition of ''inductive'' reasoning described here differs from [[mathematical induction]], which, in fact, is a form of ''deductive'' reasoning. Mathematical induction is used to provide strict proofs of the properties of recursively defined sets.<ref>{{cite book|last1=Chowdhry|first1=K.R.|url=https://books.google.com/books?id=MmVwBgAAQBAJ&q=%22mathematical+induction%22+deduction&pg=PA26|title=Fundamentals of Discrete Mathematical Structures|year=2015|publisher=PHI Learning Pvt. Ltd.|isbn=978-8120350748|edition=3rd|page=26|access-date=1 December 2016}}</ref> The deductive nature of mathematical induction derives from its basis in a non-finite number of cases, in contrast with the finite number of cases involved in an enumerative induction procedure like [[proof by exhaustion]]. Both mathematical induction and proof by exhaustion are examples of [[complete induction]]. Complete induction is a masked type of deductive reasoning.

==Problem of induction==
{{Main|The problem of induction}}

Although philosophers at least as far back as the [[Pyrrhonism|Pyrrhonist]] philosopher [[Sextus Empiricus]] have pointed out the unsoundness of inductive reasoning,<ref>Sextus Empiricus, ''Outlines of Pyrrhonism''. Trans. [[Robert Gregg Bury|R.G. Bury]], Harvard University Press, Cambridge, Massachusetts, 1933, p. 283.</ref> the classic philosophical critique of the [[problem of induction]] was given by the Scottish philosopher [[David Hume]].<ref>{{cite book|url=http://18th.eserver.org/hume-enquiry.html#4|title=An Enquiry concerning Human Understanding|author=David Hume|author-link=David Hume|publisher=P.F. Collier & Son|year=1910|isbn=978-0-19-825060-9|access-date=27 December 2007|orig-year=1748|archive-url=https://web.archive.org/web/20071231182807/http://18th.eserver.org/hume-enquiry.html#4|archive-date=31 December 2007|url-status=dead|df=dmy-all}}</ref> Although the use of inductive reasoning demonstrates considerable success, the justification for its application has been questionable. Recognizing this, Hume highlighted the fact that our mind often draws conclusions from relatively limited experiences that appear correct but which are actually far from certain. In deduction, the truth value of the conclusion is based on the truth of the premise. In induction, however, the dependence of the conclusion on the premise is always uncertain. For example, let us assume that all ravens are black. The fact that there are numerous black ravens supports the assumption. Our assumption, however, becomes invalid once it is discovered that there are white ravens. Therefore, the general rule "all ravens are black" is not the kind of statement that can ever be certain. Hume further argued that it is impossible to justify inductive reasoning: this is because it cannot be justified deductively, so our only option is to justify it inductively. Since this argument is circular, with the help of [[Hume's fork]] he concluded that our use of induction is unjustifiable .<ref>Vickers, John. [http://plato.stanford.edu/entries/induction-problem/#2HumIndJus "The Problem of Induction"] (Section 2). ''Stanford Encyclopedia of Philosophy''. 21 June 2010</ref>

Hume nevertheless stated that even if induction were proved unreliable, we would still have to rely on it. So instead of a position of [[Philosophical skepticism|severe skepticism]], Hume advocated a [[Scientific skepticism|practical skepticism]] based on [[common sense]], where the inevitability of induction is accepted.<ref>Vickers, John. [http://plato.stanford.edu/entries/induction-problem/#IndJus "The Problem of Induction"] (Section 2.1). ''Stanford Encyclopedia of Philosophy''. 21 June 2010.</ref> [[Bertrand Russell]] illustrated Hume's skepticism in a story about a chicken, fed every morning without fail, who following the laws of induction concluded that this feeding would always continue, until his throat was eventually cut by the farmer.<ref>{{Cite book|last=Russel|first=Bertrand|title=The Problems of Philosophy|publisher=Oxford University Press|year=1997|isbn=978-0195115529|location=Oxford|pages=66}}</ref>

In 1963, [[Karl Popper]] wrote, "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."<ref>{{cite journal|last1=Popper|first1=Karl R.|last2=Miller|first2=David W.|year=1983|title=A proof of the impossibility of inductive probability|journal=[[Nature (journal)|Nature]]|volume=302|issue=5910|pages=687–88|bibcode=1983Natur.302..687P|doi=10.1038/302687a0|s2cid=4317588}}</ref><ref name="Gillies">Donald Gillies, "Problem-solving and the problem of induction", in ''Rethinking Popper'' (Dordrecht: [[Springer (publisher)|Springer]], 2009), Zuzana Parusniková & Robert S Cohen, eds, [https://books.google.com/books?id=R3aywtFIKKsC&pg=PA103#v=twopage pp. 103–05].</ref>  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]]".<ref name="Gillies" /> In Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during a ''problem shift''.<ref name="Gillies" />  An imaginative leap, the ''tentative solution'' is improvised, lacking inductive rules to guide it.<ref name="Gillies" />  The resulting, unrestricted generalization is deductive, an entailed consequence of all explanatory considerations.<ref name="Gillies" />  Controversy continued, however, with Popper's putative solution not generally accepted.<ref>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|D. Reidel Publishing]], 1978), [https://books.google.com/books?id=i8kmptHdx3MC&pg=PA141#v=twopage pp. 141–43].</ref>

[[Donald A. Gillies]] argues that [[Rule of inference|rules of inferences]] related to inductive reasoning are overwhelmingly absent from science, and describes most scientific inferences as "involv[ing] conjectures thought up by human ingenuity and creativity, and by no means inferred in any mechanical fashion, or according to precisely specified rules."<ref name="Gillies-p111" /> Gillies also provides a rare counterexample "in the machine learning programs of [[Artificial Intelligence|AI]]."<ref name="Gillies-p111">Donald Gillies, "Problem-solving and the problem of induction", in ''Rethinking Popper'' (Dordrecht: [[Springer (publisher)|Springer]], 2009), Zuzana Parusniková & Robert S Cohen, eds, [https://books.google.com/books?id=R3aywtFIKKsC&pg=PA111 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 [[Artificial Intelligence|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 be 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".</ref>

===Biases===
Inductive reasoning is also known as hypothesis construction because any conclusions made are based on current knowledge and predictions.{{Citation needed|date=March 2012}} As with deductive arguments, biases can distort the proper application of inductive argument, thereby preventing the reasoner from forming the most [[Logical consequence|logical conclusion]] based on the clues. Examples of these biases include the [[availability heuristic]], [[confirmation bias]], and the  [[Gambler's fallacy|predictable-world bias]].

The availability heuristic causes the reasoner to depend primarily upon information that is readily available. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of people who died from various causes, most respondents choose the causes that have been most prevalent in the media such as terrorism, murders, and airplane accidents, rather than causes such as disease and traffic accidents, which have been technically "less accessible" to the individual since they are not emphasized as heavily in the world around them.

Confirmation bias is based on the natural tendency to confirm rather than deny a hypothesis. Research has demonstrated that people are inclined to seek solutions to problems that are more consistent with known hypotheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seek answers that fit established hypotheses, thus confirming these hypotheses. For example, if it is hypothesized that Sally is a sociable individual, subjects will naturally seek to confirm the premise by asking questions that would produce answers confirming that Sally is, in fact, a sociable individual.

The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist, either at all or at a particular level of abstraction. Gambling, for example, is one of the most popular examples of predictable-world bias. Gamblers often begin to think that they see simple and obvious patterns in the outcomes and therefore believe that they are able to predict outcomes based on what they have witnessed. In reality, however, the outcomes of these games are difficult to predict and highly complex in nature. In general, people tend to seek some type of simplistic order to explain or justify their beliefs and experiences, and it is often difficult for them to realise that their perceptions of order may be entirely different from the truth.<ref>{{cite book|title=Psychology|last=Gray|first=Peter|publisher=Worth|year=2011|isbn=978-1-4292-1947-1|edition=Sixth|location=New York}}</ref>

== Bayesian inference ==
As a logic of induction rather than a theory of belief, [[Bayesian inference]] does not determine which beliefs are ''a priori'' rational, but rather determines how we should rationally change the beliefs we have when presented with evidence. We begin by committing to a [[prior probability]] for a hypothesis based on logic or previous experience and, when faced with evidence, we adjust the strength of our belief in that hypothesis in a precise manner using [[conditional probability|Bayesian logic]].

==Inductive inference==
Around 1960, [[Ray Solomonoff]] founded the theory of universal [[Solomonoff's theory of inductive inference|inductive inference]], a theory of prediction based on observations, for example, predicting the next symbol based upon a given series of symbols. This is a formal inductive framework that combines [[algorithmic information theory]] with the Bayesian framework. Universal inductive inference is based on solid philosophical foundations,<ref>{{cite journal |first1=Samuel |last1=Rathmanner |author2-link=Marcus Hutter |first2=Marcus |last2=Hutter |title=A Philosophical Treatise of Universal Induction |journal=Entropy |volume=13 |issue=6 |pages=1076–136 |year=2011 |doi=10.3390/e13061076 |bibcode=2011Entrp..13.1076R |arxiv=1105.5721 |s2cid=2499910 |doi-access=free }}</ref> and can be considered as a mathematically formalized [[Occam's razor]]. Fundamental ingredients of the theory are the concepts of [[algorithmic probability]] and [[Kolmogorov complexity]].

==See also==
{{Portal|Philosophy|Psychology}}
{{div col|colwidth=20em}}
* [[Analogy]]
* [[Argument]]
* [[Argumentation theory]]
* [[Bayesian probability]]
* [[Counterinduction]]
* [[Explanation]]
* [[Failure mode and effects analysis]]
* [[Falsifiability]]
* [[Grammar induction]]
* [[Inductive logic programming]]
* [[Inductive probability]]
* [[Inductive programming]]
* [[Inductive reasoning aptitude]]
* [[Inductivism]]
* [[Inquiry]]
* [[Intuitive statistics]]
* [[Lateral thinking]]
* [[Laurence Jonathan Cohen]]
* [[Logic]]
* [[Logical reasoning]]
* [[Logical positivism]]
* [[Minimum description length]]
* [[Minimum message length]]
* [[New riddle of induction]]
* [[Open world assumption]]
* [[Raven paradox]]
* [[Recursive Bayesian estimation]]
* [[Statistical inference]]


* [[Marcus Hutter]]
* [[Stephen Toulmin]]
{{div col end}}

==References==
{{Reflist}}

==Further reading==
* {{cite web |last=Herms |first=D. |title=Logical Basis of Hypothesis Testing in Scientific Research |url=http://www.dartmouth.edu/~bio125/logic.Giere.pdf }}
* {{cite web |last=Kemerling |first=G. |title=Causal Reasoning |date=27 October 2001 |url=http://www.philosophypages.com/lg/e14.htm}}
* {{cite book |last1=Holland |first1=J.H. |last2=Holyoak |first2=K.J. |last3=Nisbett |first3=R.E. |last4=Thagard |first4=P.R. |year=1989 |title=Induction: Processes of Inference, Learning, and Discovery |location=Cambridge, MA |publisher=[[MIT Press]] |isbn=978-0-262-58096-0}}
* {{cite book |last1=Holyoak |first1=K. |last2=Morrison |first2=R. |year=2005 |title=The Cambridge Handbook of Thinking and Reasoning |location=New York |publisher=[[Cambridge University Press]] |isbn=978-0-521-82417-0|url=https://books.google.com/books?id=znbkHaC8QeMC}}

==External links==

{{Wiktionary|inductive reasoning}}
{{Americana Poster|year=1920|Induction (logic)|Inductive reasoning}}
* {{cite IEP |url-id=conf-ind |title=Confirmation and Induction}}
* {{cite SEP |url-id=logic-inductive |title=Inductive Logic}}
* {{PhilPapers|category|induction}}
* {{InPho|taxonomy|2256}}
* [https://web.archive.org/web/20070927003210/http://www.uncg.edu/phi/phi115/induc4.htm ''Four Varieties of Inductive Argument''] from the Department of Philosophy, [[University of North Carolina at Greensboro]].
* {{cite web |url= http://faculty.ucmerced.edu/sites/default/files/eheit/files/heit2000.pdf |title= Properties of Inductive Reasoning |access-date= 16 July 2013 |archive-date= 8 August 2017 |archive-url= https://web.archive.org/web/20170808171702/http://faculty.ucmerced.edu/sites/default/files/eheit/files/heit2000.pdf |url-status= dead }}&nbsp;{{small|(166&nbsp;[[Kibibyte|KiB]])}}, a psychological review by Evan Heit of the [[University of California, Merced]].
* [http://dudespaper.com/the-mind-limber.html ''The Mind, Limber''] An article which employs the film [[The Big Lebowski]] to explain the value of inductive reasoning.
* [https://www.academia.edu/4154895/Some_Remarks_on_the_Pragmatic_Problem_of_Induction.html The Pragmatic Problem of Induction], by Thomas Bullemore
* [http://sresearch.scienceontheweb.net/philosophy.php Arguments against Popper's Falsificationism]

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