Editing Inductive reasoning (section)

== Methods ==
The two principal methods used to reach inductive generalizations 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 generalization 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 probable 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 with or 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 first put forth by [[Francis Bacon]];<ref name=":2">{{Cite book |last1=Goodenough |first1=John B. |last2=Weinstock |first2=Charles B. |last3=Klein |first3=Ari Z. |chapter=Eliminative induction: A basis for arguing system confidence |date=2013 |title=2013 35th International Conference on Software Engineering (ICSE) |chapter-url=https://ieeexplore.ieee.org/document/6606668 |pages=1161–1164 |doi=10.1109/ICSE.2013.6606668 |isbn=978-1-4673-3076-3 |via=IEEE}}</ref> in it a generalization 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. In this context, confidence is the function of how many instances have been identified as incompatible and eliminated. This confidence is expressed as the Baconian probability i|n (read as "i out of n") where n reasons for finding a claim incompatible has been identified and i of these have been eliminated by evidence or argument.<ref name=":2" />

There are three ways of attacking an argument; these ways - known as defeaters in [[defeasible reasoning]] literature - are : rebutting, undermining, and undercutting. Rebutting defeats by offering a counter-example, undermining defeats by questioning the validity of the evidence, and undercutting defeats by pointing out conditions where a conclusion is not true when the inference is. By identifying defeaters and proving them wrong is how this approach builds confidence. <ref name=":2" />

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>
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