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Inductive reasoning
Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some
evidence for the truth of the conclusion.[1] It is also described as a method where one's experiences and
observations, including what are learned from others, are synthesized to come up with a general truth.[2]
Many dictionaries define inductive reasoning as the derivation of general principles from specific
observations (arguing from specific to general), although there are many inductive arguments that do
not have that form, such as statistical syllogisms (which argue from general to specific) and arguments
from analogy (which argue from specific to specific).[3]

Inductive reasoning is distinct from deductive reasoning. While the conclusion of a deductive argument
is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence
given.[4]

Contents
Types
Generalization
Statistical generalization
Anecdotal generalization
Prediction
Statistical syllogism
Argument from analogy
Causal inference
Methods
Enumerative induction
Eliminative induction
Comparison with deductive reasoning
History
Ancient philosophy
Early modern philosophy
Late modern philosophy
Contemporary philosophy
Bertrand Russell
Gilbert Harman
Criticism
Biases
Bayesian inference
Inductive inference
See also

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References
Further reading
External links

Types
The three principal types of inductive reasoning are generalization, analogy, and causal inference.[5]
These, however, can still be divided into different classifications. Each of these, while similar, has a
different form.

Generalization

A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample
to a conclusion about the population.[6] The observation obtained from this sample is projected onto the
broader population.[6]

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 5 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 hasty generalization and the biased sample are
generalization fallacies.

Statistical generalization

A statistical inductive generalization, often simply called a statistical generalization, is a type of

inductive argument in which a conclusion about a population is inferred using a 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[7] and sample projections.[8]

Anecdotal generalization

An anecdotal inductive generalization is a type of inductive argument in which a conclusion about a

population is inferred using a non-statistical sample.[9] For example:
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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 than the statistical generalization, first, because the sample events are non-
random, and secondly because it is not reducible to mathematical expression. Statistically speaking,
there is simply no way to know, measure and calculate as to the circumstances affecting performance
that will obtain 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.[10]

Prediction

A prediction draws a conclusion about a future individual from a past sample.

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

A statistical syllogism proceeds from a generalization about a sample 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.[11] Even though one cannot be sure Bob will attend university, we can be
fully assured of the exact probability for 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" and "converse accident".

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:[12]

P and Q are similar in respect to properties a, b, and c.

Object P has been observed to have further property x.
Therefore, Q probably has property x also.

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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.[13]

Mineral A and Mineral B are both igneous rocks often containing veins of quartz and 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, wherein 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 favour of the
conclusion."[14]

Some thinkers contend that analogical induction is a subcategory of inductive generalization because it
assumes a pre-established uniformity governing events. Analogical induction requires an auxiliary
examination of the relevancy of the characteristics cited as common to the pair. In the preceding
example, if an 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 dissimilar. Thus, analogy can mislead if not all relevant comparisons are
made.

Causal inference

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.

Methods
The two principal methods used to reach inductive conclusions are enumerative induction and
eliminative induction.[15][16]

Enumerative induction

Enumerative induction is an inductive method in which a conclusion is constructed based upon the
number of instances that support it. The more supporting instances, the stronger the conclusion.[15][16]

The most basic form of enumerative induction reasons from particular instances to all instances, and is
thus an unrestricted generalization.[17] 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 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.
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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 concluding All is a very bold assertion. A single contrary instance
foils the argument. And last, to quantify the level of probability in any mathematical form is
problematic.[18] By what standard do we measure our Earthly sample of known life against all (possible)
life? For suppose we do discover some new organism—let's say some microorganism floating in the
mesosphere, or better yet, on some asteroid—and it is cellular. Doesn't 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. Eliminative induction is
crucial to the scientific method and is used to eliminate hypotheses that are inconsistent with
observations and experiments.[15][16]

Comparison with deductive reasoning

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 premises are true.[19] 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 "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 "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".[20]
More informally, an inductive argument may be called plausible, probable, likely, reasonable, or
justified, but never certain or necessary. Logic affords no bridge from the probable to the certain.
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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
Argument terminology
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; Charlie can fly." This deductive argument is valid because the logical relations
hold; we are not interested in their factual soundness.

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Inductive reasoning is inherently uncertain. It only deals in 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 theory with rules for inference such as Bayes' rule. Unlike
deductive reasoning, it does not rely on universals holding over a closed domain of discourse to draw
conclusions, so it can be applicable even in cases of epistemic uncertainty (technical issues with this may
arise however; for example, the second axiom of probability is a closed-world assumption).[21]

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.[22]

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 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 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 extinction of the non-avian
dinosaurs. For example, the release of volcanic gases (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 expect all swans to be white.

Succinctly put: deduction is about certainty/necessity; induction is about probability.[11] 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.

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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 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, discussed below).

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.[23] 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.

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.[24] The ancient Pyrhonists, however, pointed out that
induction cannot justify the acceptance of universal statements as true.[24]

Early modern philosophy

In 1620, early modern philosopher Francis Bacon repudiated the value of mere experience and
enumerative induction alone. 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.

The empiricist David Hume's 1740 stance found enumerative induction to have no rational, let alone
logical, basis but instead induction was 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 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. Hume was 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.

Awakened from "dogmatic slumber" by a German translation of Hume's work, 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 statements
are true by virtue of the arrangement of their terms and meanings, thus analytic statements are
tautologies, merely logical truths, true by necessity. Whereas synthetic statements hold meanings to
refer to states of facts, contingencies. Finding it impossible to know objects as they truly are in
themselves, however, Kant concluded that the philosopher's task should not be to try to peer behind the
veil of appearance to view the noumena, but simply that of handling phenomena.

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Reasoning that the mind must contain its own categories for organizing sense data, making experience of
space and time possible, Kant concluded that the uniformity of nature was an a priori truth.[25] A class
of synthetic statements that was not contingent but true by necessity, was then synthetic a priori. Kant
thus saved both metaphysics and Newton's law of universal gravitation, but as a consequence discarded
scientific realism and developed transcendental idealism. Kant's transcendental idealism gave birth to
the movement of German idealism. Hegel's absolute idealism subsequently flourished across continental
Europe.

Late modern philosophy

Positivism, developed by Saint-Simon and promulgated in the 1830s by his former student Comte, was
the first 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 uniformity of nature,[25] 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
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".[26] 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.[26] Whewell explained:

"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."[26]

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.[26]

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In the 1870s, the originator of pragmatism, 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
abduction or retroduction or hypothesis or presumption.[27] Later philosophers termed Peirce's
abduction, etc., Inference to the Best Explanation (IBE).[28]

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.[29] 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'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".[30] Two decades later, Russell proposed enumerative
induction as an "independent logical principle".[31][32] Russell found:

"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."[32]

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).[28] IBE
is otherwise synonymous with C S Peirce's abduction.[28] Many philosophers of science espousing
scientific realism have maintained that IBE is the way that scientists develop approximately true
scientific theories about nature.[33]
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Criticism
Although philosophers at least as far back as the Pyrrhonist philosopher Sextus Empiricus have pointed
out the unsoundness of inductive reasoning,[34] the classic philosophical critique of the problem of
induction was given by the Scottish philosopher David Hume.[35] 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 .[36]

Hume nevertheless stated that even if induction were proved unreliable, we would still have to rely on it.
So instead of a position of severe skepticism, Hume advocated a practical skepticism based on common
sense, where the inevitability of induction is accepted.[37] 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.[38]

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."[39][40] 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".[40] In Popper's schema,
enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during
a problem shift.[40] An imaginative leap, the tentative solution is improvised, lacking inductive rules to
guide it.[40] The resulting, unrestricted generalization is deductive, an entailed consequence of all
explanatory considerations.[40] Controversy continued, however, with Popper's putative solution not
generally accepted.[41]

More recently, inductive inference has been shown to be capable of arriving at certainty, but only in rare
instances, as in programs of machine learning in artificial intelligence (AI).[42] Popper's stance on
induction being an illusion has been falsified: enumerative induction exists. Even so, inductive reasoning
is overwhelmingly absent from science.[42] Although much-talked of nowadays by philosophers,
abduction, or IBE, lacks rules of inference and the inferences reached by those employing it are arrived
at with human imagination and creativity.[42]

Biases

Inductive reasoning is also known as hypothesis construction because any conclusions made are based
on current knowledge and predictions. As with deductive arguments, biases can distort the proper
application of inductive argument, thereby preventing the reasoner from forming the most logical
conclusion based on the clues. Examples of these biases include the availability heuristic, confirmation
bias, and the predictable-world bias.

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The availability heuristic causes the reasoner to depend primarily upon information that is readily
available to him or her. 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.

The confirmation bias is based on the natural tendency to confirm rather than to deny a current
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
upon 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.[43]

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 Bayesian logic.

Inductive inference
Around 1960, Ray Solomonoff founded the theory of universal 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,[44] 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
Abductive reasoning Bayesian probability
Algorithmic probability Counterinduction
Analogy Deductive reasoning
Argument Explanation
Argumentation theory Failure mode and effects analysis
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Falsifiability Machine learning

Grammar induction Mathematical induction
Inductive inference Mill's Methods
Inductive logic programming Minimum description length
Inductive probability Minimum message length
Inductive programming New riddle of induction
Inductive reasoning aptitude Open world assumption
Inductivism Raven paradox
Inquiry Recursive Bayesian estimation
Kolmogorov complexity Retroduction
Lateral thinking Solomonoff's theory of inductive inference
Laurence Jonathan Cohen Statistical inference
Logic Stephen Toulmin
Logical reasoning Marcus Hutter
Logical positivism

References
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CT: Cengage Learning. p. 57. ISBN 978-1-285-19719-7.
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Philosophy, "It is worth noting that some dictionaries and texts define "deduction" as reasoning from
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there are many inductive arguments that do not have that form, for example, 'I saw her kiss him,
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4. Copi, I.M.; Cohen, C.; Flage, D.E. (2006). Essentials of Logic (Second ed.). Upper Saddle River, NJ:
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14. A System of Logic. Mill 1843/1930. p. 333
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15. Hunter, Dan (September 1998). "No Wilderness of Single Instances: Inductive Inference in Law".
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ISBN 978-0-312-02353-9. OCLC 21216829 (https://www.worldcat.org/oclc/21216829). "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."
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19. John Vickers. The Problem of Induction (http://plato.stanford.edu/entries/induction-problem/). The
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21. Kosko, Bart (1990). "Fuzziness vs. Probability". International Journal of General Systems. 17 (1):
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google.com/?id=MmVwBgAAQBAJ&pg=PA26&dq=%22mathematical+induction%22+deduction#v=o
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Ltd. p. 26. ISBN 9788120350748. Retrieved 1 December 2016.
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26. Roberto Torretti, The Philosophy of Physics (Cambridge: Cambridge University Press, 1999), 219–21
(https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA219&dq=Whewell+induction+superinduc
tion)[216] (https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA216&dq=Comte+Mill).
27. Roberto Torretti, The Philosophy of Physics (Cambridge: Cambridge University Press, 1999), pp. 226
(https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA226&dq=Peirce+abduction+deduction+i
nduction), 228–29 (https://books.google.com/books?id=vg_wxiLRvvYC&pg=PA228&dq=Peirce+abdu
ction).
28. Ted Poston "Foundationalism" (http://www.iep.utm.edu/found-ep), § 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 beliefs. 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 '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 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".
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29. David Andrews, Keynes and the British Humanist Tradition: The Moral Purpose of the Market (New
York: Routledge, 2010), pp. 63–65 (https://books.google.com/books?id=1VsYOwRsOVUC&pg=PA63
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30. Bertrand Russell, The Basic Writings of Bertrand Russell (New York: Routledge, 2009), "The validity
of inference"], pp. 157–64, quote on p. 159 (https://books.google.com/books?id=jqun5YJGt-wC&pg=
PA159&dq=Keynes+induction).
31. Gregory Landini, Russell (New York: Routledge, 2011), p. 230 (https://books.google.com/books?id=S
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York: Simon and Schuster, 1945), pp. 673–74.
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Massachusetts, 1933, p. 283.
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19-825060-9. Archived from the original (http://18th.eserver.org/hume-enquiry.html#4) on 31
December 2007. Retrieved 27 December 2007.
36. Vickers, John. "The Problem of Induction" (http://plato.stanford.edu/entries/induction-problem/#2Hum
IndJus) (Section 2). Stanford Encyclopedia of Philosophy. 21 June 2010
37. Vickers, John. "The Problem of Induction" (http://plato.stanford.edu/entries/induction-problem/#IndJu
s) (Section 2.1). Stanford Encyclopedia of Philosophy. 21 June 2010.
38. Russel, Bertrand (1997). The Problems of Philosophy. Oxford: Oxford University Press. p. 66.
ISBN 9780195115529.
39. Popper, Karl R.; Miller, David W. (1983). "A proof of the impossibility of inductive probability". Nature.
302 (5910): 687–88. Bibcode:1983Natur.302..687P (https://ui.adsabs.harvard.edu/abs/1983Natur.30
2..687P). doi:10.1038/302687a0 (https://doi.org/10.1038%2F302687a0).
40. Donald Gillies, "Problem-solving and the problem of induction", in Rethinking Popper (Dordrecht:
Springer, 2009), Zuzana Parusniková & Robert S Cohen, eds, pp. 103–05 (https://books.google.co
m/books?id=R3aywtFIKKsC&pg=PA103#v=twopage).
41. 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 Publishing, 1978), pp. 141–43 (https://books.google.com/books?id=i8kmptHdx3MC&pg=PA14
1&dq=controversy#v=twopage).
42. Donald Gillies, "Problem-solving and the problem of induction", in Rethinking Popper (Dordrecht:
Springer, 2009), Zuzana Parusniková & Robert S Cohen, eds, p. 111 (https://books.google.com/book
s?id=R3aywtFIKKsC&pg=PA111&dq=exceptions): "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 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".
43. Gray, Peter (2011). Psychology (Sixth ed.). New York: Worth. ISBN 978-1-4292-1947-1.

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44. Rathmanner, Samuel; Hutter, Marcus (2011). "A Philosophical Treatise of Universal Induction".
Entropy. 13 (6): 1076–136. arXiv:1105.5721 (https://arxiv.org/abs/1105.5721).
Bibcode:2011Entrp..13.1076R (https://ui.adsabs.harvard.edu/abs/2011Entrp..13.1076R).
doi:10.3390/e13061076 (https://doi.org/10.3390%2Fe13061076).

Further reading
Cushan, Anna-Marie (1983/2014). Investigation into Facts and Values: Groundwork for a theory of
moral conflict resolution. [Thesis, Melbourne University], Ondwelle Publications (online): Melbourne.
[1] (http://www.ondwelle.com/ValueJudgements.pdf)
Herms, D. "Logical Basis of Hypothesis Testing in Scientific Research" (http://www.dartmouth.edu/~bi
o125/logic.Giere.pdf) (PDF).
Kemerling, G. (27 October 2001). "Causal Reasoning"
(http://www.philosophypages.com/lg/e14.htm).
Holland, J.H.; Holyoak, K.J.; Nisbett, R.E.; Thagard, P.R. (1989). Induction: Processes of Inference,
Learning, and Discovery. Cambridge, MA: MIT Press. ISBN 978-0-262-58096-0.
Holyoak, K.; Morrison, R. (2005). The Cambridge Handbook of Thinking and Reasoning (https://book
s.google.com/books?id=znbkHaC8QeMC&printsec=frontcover&dq=isbn:9780521824170&hl=en&sa
=X&ved=0ahUKEwjt5bj01NDjAhWKXM0KHXMgCVkQ6AEIKjAA#v=snippet&q=inductive%20OR%2
0induction&f=false). New York: Cambridge University Press. ISBN 978-0-521-82417-0.

External links
"Confirmation and Induction" (http://www.iep.utm.edu/conf-ind). Internet Encyclopedia of Philosophy.
Zalta, Edward N. (ed.). "Inductive Logic" (https://plato.stanford.edu/entries/logic-inductive/). Stanford
Encyclopedia of Philosophy.
Inductive reasoning (https://philpapers.org/browse/induction) at PhilPapers
Inductive reasoning (https://inpho.cogs.indiana.edu/taxonomy/2256) at the Indiana Philosophy
Ontology Project
Four Varieties of Inductive Argument (https://web.archive.org/web/20070927003210/http://www.unc
g.edu/phi/phi115/induc4.htm) from the Department of Philosophy, University of North Carolina at
Greensboro.
"Properties of Inductive Reasoning" (http://faculty.ucmerced.edu/sites/default/files/eheit/files/heit200
0.pdf) (PDF). (166 KiB), a psychological review by Evan Heit of the University of California, Merced.
The Mind, Limber (http://dudespaper.com/the-mind-limber.html) An article which employs the film
The Big Lebowski to explain the value of inductive reasoning.
The Pragmatic Problem of Induction (https://www.academia.edu/4154895/Some_Remarks_on_the_
Pragmatic_Problem_of_Induction.html), by Thomas Bullemore

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