Inductive reasoning

== Types ==
The types of inductive reasoning include generalization, prediction, statistical syllogism, argument from analogy, and causal inference. There are also differences in how their results are regarded.

=== Inductive generalization ===
A generalization (more accurately, an ''inductive generalization'') proceeds from premises 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, if there are 20 balls—either black or white—in an urn, to estimate their respective numbers, a sample of four balls is drawn, three are black and one is white. An inductive generalization is that there are 15 black and five white balls in the urn.

How much the premises support the conclusion depends upon the number in the sample group, the number in the population, and the degree to which the sample represents the population (which, for a static population, 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 that the selection process was genuinely random and that the numbers of items in the sample having the properties considered are large. 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 a 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=Hypotheses 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|access-date=22 August 2020|archive-date=24 October 2020|archive-url=https://web.archive.org/web/20201024141846/https://pure.rug.nl/ws/files/2720641/romeijn_-_hypotheses_and_predictions.pdf|url-status=live}}</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 probably 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, the exact probability of this outcome is fully assured (given no further information). 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] {{Webarchive|url=https://web.archive.org/web/20090306070520/http://www.cs.hut.fi/Opinnot/T-93.850/2005/Papers/juthe2005-analogy.pdf |date=6 March 2009 }}.</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 possible or probable 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.{{citation needed|date=September 2022}}