Inductive reasoning

== 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] {{Webarchive|url=https://web.archive.org/web/20140407014814/http://plato.stanford.edu/entries/induction-problem/ |date=7 April 2014 }}. 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. 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 thought to be true, then the argument is said to be "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|access-date=24 July 2005|archive-date=19 March 2009|archive-url=https://web.archive.org/web/20090319202920/http://www.dartmouth.edu/~bio125/logic.Giere.pdf|url-status=dead}}</ref> Less formally, the conclusion of 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, and one can regard it as 'true' that the coin is probably 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. For example, after getting 10 heads in a row one might deduce that the coin had met some statistical criterion to be regarded as 'probably two-sided, a conclusion that would not be falsified even if the next toss yielded 'tails'.

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 its conclusions are 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.

The conclusions of inductive reasoning are 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 or empirical 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|access-date=27 November 2015|archive-date=8 December 2015|archive-url=https://web.archive.org/web/20151208082150/http://plato.stanford.edu/entries/kant-reason/#TheReaReaCogRolLim|url-status=live}}</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 had a correct conclusion 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 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'' statistical syllogism was presented by John Vickers:

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

The conclusion fails because the population of swans then known was not actually representative of all swans. A more reasonable conclusion would be: in line with applicable conventions, we might reasonably [[Expectation (epistemic)|expect]] all swans in England to be white, at least in the short-term.

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.