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Inductive reasoning is a
method of reasoning Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, la ...
in which a general
principle A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a rule that has to be or usually is to be followed. It can be desirably followed, or it can be an inevitable consequence of something, such as the la ...
is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''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.

# 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 to a conclusion about the
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
. The observation obtained from this sample is projected onto the broader population. : 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 In statistics, sampling bias is a bias in which a sample is collected in such a way that some members of the intended population have a lower or higher sampling probability than others. It results in a biased sample of a population (or non-human f ...
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 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'' and ''sample projections''.

### 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. In other words, the generalization is based on
anecdotal evidence Anecdotal evidence is evidence based only on personal observation, collected in a casual or non-systematic manner. The term is sometimes used in a legal context to describe certain kinds of testimony which are uncorroborated by objective, independ ...
. 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.

## 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. : 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 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''.Introduction to Logic. Harry J. Gensler, Rutledge, 2002. p. 268 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 Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...
. Two dicto simpliciter fallacies can occur in statistical syllogisms: "
accident An accident is an unintended, normally unwanted event that was not directly caused by humans. The term ''accident'' implies that nobody should be blamed, but the event may have been caused by unrecognized or unaddressed risks. Most researche ...
" 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: :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 ''Common Sense'' is a 47-page pamphlet written by Thomas Paine in 1775–1776 advocating independence from Great Britain to people in the Thirteen Colonies. Writing in clear and persuasive prose, Paine collected various moral and political a ...
,
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence f ...
,
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. So ...
,
law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
, and the
humanities Humanities are academic disciplines that study aspects of human society and culture. In the Renaissance, the term contrasted with divinity and referred to what is now called classics, the main area of secular study in universities at the tim ...
, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning. :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, " ere can be no doubt that every resemblance ot known to be irrelevantaffords some degree of probability, beyond what would otherwise exist, in favor of the conclusion." 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. 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

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.''

## 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. The most basic form of enumerative induction reasons from particular instances to all instances, and is thus an unrestricted generalization. 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 The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific ...
. :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. 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. Different evidential tests may also be employed to eliminate possibilities that are entertained. Eliminative induction is crucial to the scientific method and is used to eliminate hypotheses that are inconsistent with observations and experiments. It focuses on possible causes instead of observed actual instances of causal connections.

# History

## Ancient philosophy

For a move from particular to universal,
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
in the 300s BCE used the Greek word ''epagogé'', which
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the est ...
translated into the Latin word ''inductio''.Stefano Gattei, ''Karl Popper's Philosophy of Science: Rationality without Foundations'' (New York:
Routledge Routledge () is a British multinational publisher. It was founded in 1836 by George Routledge, and specialises in providing academic books, journals and online resources in the fields of the humanities, behavioural science, education, law, ...
, 2009), ch. 2 "Science and philosophy"
pp. 28–30

### 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 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 Pyrrhonists were the first Western philosophers to point out the
Problem of induction First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inferen ...
: that induction cannot, according to them, justify the acceptance of universal statements as true.

### Ancient medicine

The
Empiric school The Empiric school of medicine (''Empirics'', ''Empiricists'', or ''Empirici'', el, Ἐμπειρικοί) was a school of medicine founded in Alexandria the middle of the third century BC. The school was a major influence on ancient Greek and ...
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. Epilogism is an inference which moves entirely within the domain of visible and evident things, it tries not to invoke unobservables. The
Dogmatic school The Dogmatic school of medicine (''Dogmatics'', or ''Dogmatici'', el, Δογματικοί) was a school of medicine in ancient Greece and Rome. They were the oldest of the medical sects of antiquity. They derived their name from ''dogma'', a ph ...
of ancient Greek medicine employed ''analogismos'' as a method of inference. This method used analogy to reason from what was observed to unobservable forces.

## Early modern philosophy

In 1620, early modern philosopher
Francis Bacon Francis Bacon, 1st Viscount St Alban (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England. Bacon led the advancement of both ...
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.

### David Hume

The empiricist
David Hume David Hume (; born David Home; 7 May 1711 NS (26 April 1711 OS) – 25 August 1776) Cranston, Maurice, and Thomas Edmund Jessop. 2020 999br>David Hume" ''Encyclopædia Britannica''. Retrieved 18 May 2020. was a Scottish Enlightenment philo ...
'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 uniformity of nature to produce conclusions that seemed to be certain, the
problem of induction First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inferen ...
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 Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th centu ...
, which was first formulated and advanced by
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
, in 1886, where he referred to it as "reasoning by hypothesis." 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. 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,
Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and aes ...
sought to explain the possibility of
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of consci ...
. In 1781, Kant's '' Critique of Pure Reason'' introduced ''
rationalism In philosophy, rationalism is the epistemological view that "regards reason as the chief source and test of knowledge" or "any view appealing to reason as a source of knowledge or justification".Lacey, A.R. (1996), ''A Dictionary of Philosophy' ...
'' as a path toward knowledge distinct from ''
empiricism In philosophy, empiricism is an epistemological theory that holds that knowledge or justification comes only or primarily from sensory experience. It is one of several views within epistemology, along with rationalism and skepticism. Empi ...
''. Kant sorted statements into two types. Analytic statements are true by virtue of the
arrangement In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orches ...
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. Against both rationalist philosophers like Descartes and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
as well as against empiricist philosophers like Locke and 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 A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfried W ...
''), whereas objects of mere thought (" 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 The theory of sense data is a view in the philosophy of perception, popularly held in the early 20th century by philosophers such as Bertrand Russell, C. D. Broad, H. H. Price, A. J. Ayer, and G. E. Moore. Sense data are taken to be mind-depe ...
, making experience of objects in ''space'' and ''time (
phenomena A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfried W ...
)'' possible, Kant concluded that the uniformity of nature was an ''a priori'' truth. A class of synthetic statements that was not contingent but true by necessity, was then synthetic ''a priori''. Kant thus saved both
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of consci ...
and
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
. On the basis of the argument that what goes beyond our knowledge is "nothing to us," he discarded
scientific realism Scientific realism is the view that the universe described by science is real regardless of how it may be interpreted. Within philosophy of science, this view is often an answer to the question "how is the success of science to be explained?" T ...
. Kant's position that knowledge comes about by a cooperation of perception and our capacity to think (
transcendental idealism Transcendental idealism is a philosophical system founded by German philosopher Immanuel Kant in the 18th century. Kant's epistemological program is found throughout his '' Critique of Pure Reason'' (1781). By ''transcendental'' (a term that dese ...
) gave birth to the movement of
German idealism German idealism was a philosophical movement that emerged in Germany in the late 18th and early 19th centuries. It developed out of the work of Immanuel Kant in the 1780s and 1790s, and was closely linked both with Romanticism and the revolutionary ...
. Hegel's
absolute idealism Absolute idealism is an ontologically monistic philosophy chiefly associated with G. W. F. Hegel and Friedrich Schelling, both of whom were German idealist philosophers in the 19th century. The label has also been attached to others such as ...
subsequently flourished across continental Europe and England.

## Late modern philosophy

Positivism Positivism is an empiricist philosophical theory that holds that all genuine knowledge is either true by definition or positive—meaning ''a posteriori'' facts derived by reason and logic from sensory experience.John J. Macionis, Linda M ...
, developed by
Henri de Saint-Simon Claude Henri de Rouvroy, comte de Saint-Simon (17 October 1760 – 19 May 1825), often referred to as Henri de Saint-Simon (), was a French political, economic and socialist theorist and businessman whose thought had a substantial influence on p ...
and promulgated in the 1830s by his former student
Auguste Comte Isidore Marie Auguste François Xavier Comte (; 19 January 1798 – 5 September 1857) was a French philosopher and writer who formulated the doctrine of positivism. He is often regarded as the first philosopher of science in the modern sense ...
, was the first late modern philosophy of science. In the aftermath of the
French Revolution The French Revolution ( ) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are consider ...
, fearing society's ruin, Comte opposed
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of consci ...
. Human knowledge had evolved from religion to metaphysics to science, said Comte, which had flowed from
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
to
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxie ...
to
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...
to
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, propertie ...
to
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
to
sociology Sociology is a social science that focuses on society, human social behavior, patterns of social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of empirical investigation ...
—in that order—describing increasingly intricate domains. All of society's knowledge had become scientific, with questions of
theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...
and of
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of consci ...
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 The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific ...
frames predictions, confirms them, and states laws—positive statements—irrefutable by
theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...
or by
metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of consci ...
. Regarding experience as justifying enumerative induction by demonstrating the uniformity of nature,Wesley C Salmon
"The uniformity of Nature"
''Philosophy and Phenomenological Research'', 1953 Sep;14(1):39–48, 9
the British philosopher
John Stuart Mill John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, Member of Parliament (MP) and civil servant. One of the most influential thinkers in the history of classical liberalism, he contributed widely to ...
welcomed Comte's positivism, but thought
scientific laws Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) a ...
susceptible to recall or revision and Mill also withheld from Comte's Religion of Humanity. Comte was confident in treating
scientific law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) a ...
as an irrefutable foundation for all knowledge, and believed that churches, honouring eminent scientists, ought to focus public mindset on ''
altruism Altruism is the principle and moral practice of concern for the welfare and/or happiness of other human beings or animals, resulting in a quality of life both material and spiritual. It is a traditional virtue in many cultures and a core ...
''—a term Comte coined—to apply science for humankind's social welfare via
sociology Sociology is a social science that focuses on society, human social behavior, patterns of social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of empirical investigation ...
, Comte's leading science. During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science,
William Whewell William Whewell ( ; 24 May 17946 March 1866) was an English polymath, scientist, Anglican priest, philosopher, theologian, and historian of science. He was Master of Trinity College, Cambridge. In his time as a student there, he achieved ...
found enumerative induction not nearly as convincing, and, despite the dominance of inductivism, formulated "superinduction".Roberto Torretti, ''The Philosophy of Physics'' (Cambridge:
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pres ...
, 1999)
219–21
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. Whewell explained: These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed ''
consilience In science and history, consilience (also convergence of evidence or concordance of evidence) is the principle that evidence from independent, unrelated sources can "converge" on strong conclusions. That is, when multiple sources of evidence are ...
''—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. In the 1870s, the originator of
pragmatism Pragmatism is a philosophical tradition that considers words and thought as tools and instruments for prediction, problem solving, and action, and rejects the idea that the function of thought is to describe, represent, or mirror reality. Pr ...
,
C S Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
performed vast investigations that clarified the basis of deductive inference as a mathematical proof (as, independently, did
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
). Peirce recognized induction but always insisted on a third type of inference that Peirce variously termed '' abduction'' or ''retroduction'' or ''hypothesis'' or ''presumption''. Later philosophers termed Peirce's abduction, etc., '' Inference to the Best Explanation'' (IBE).

## Contemporary philosophy

### Bertrand Russell

Having highlighted Hume's
problem of induction First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inferen ...
,
John Maynard Keynes John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946), was an English economist whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originally trained in ...
posed ''logical probability'' as its answer, or as near a solution as he could arrive at.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
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". Two decades later, Russell proposed enumerative induction as an "independent logical principle". Russell found:

### 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).Ted Posto
"Foundationalism"
§ b "Theories of proper inference", §§ iii "Liberal inductivism", ''
Internet Encyclopedia of Philosophy The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original papers ...
'', 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 Basic beliefs (also commonly called foundational beliefs or core beliefs) are, under the epistemological view called foundationalism, the axioms of a belief system. Categories of beliefs Foundationalism holds that all beliefs must be justifie ...
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 ' 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 Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. 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".
IBE is otherwise synonymous with
C S Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
's ''abduction''. Many philosophers of science espousing
scientific realism Scientific realism is the view that the universe described by science is real regardless of how it may be interpreted. Within philosophy of science, this view is often an answer to the question "how is the success of science to be explained?" T ...
have maintained that IBE is the way that scientists develop approximately true scientific theories about nature.

# 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
premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
with rules for inference such as
Bayes' rule In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
. 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). Another crucial difference between these two types of argument is that deductive certainty is impossible in non-axiomatic systems such as
reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary. The term is also used to refer to the ontological status of things, indicating their existence. In physical terms, r ...
, leaving inductive reasoning as the primary route to (probabilistic) knowledge of such systems. 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 An impact winter is a hypothesized period of prolonged cold weather due to the impact of a large asteroid or comet on the Earth's surface. If an asteroid were to strike land or a shallow body of water, it would eject an enormous amount of dust, ...
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 Sulfur dioxide (IUPAC-recommended spelling) or sulphur dioxide (traditional Commonwealth English) is the chemical compound with the formula . It is a toxic gas responsible for the odor of burnt matches. It is released naturally by volcanic a ...
) during the formation of the
Deccan Traps The Deccan Traps is a large igneous province of west-central India (17–24°N, 73–74°E). It is one of the largest volcanic features on Earth, taking the form of a large shield volcano. It consists of numerous layers of solidified flood ...
in
India India, officially the Republic of India ( Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on th ...
. 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''. Any single assertion will answer to one of these two criteria. Another approach to the analysis of reasoning is that of
modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend othe ...
, 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 An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
indicate some degree of support (inductive probability) for the conclusion but do not
entail In English common law, fee tail or entail is a form of trust established by deed or settlement which restricts the sale or inheritance of an estate in real property and prevents the property from being sold, devised by will, or otherwise aliena ...
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 Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
, which, in fact, is a form of ''deductive'' reasoning. Mathematical induction is used to provide strict proofs of the properties of recursively defined sets. 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

Although philosophers at least as far back as the Pyrrhonist philosopher
Sextus Empiricus Sextus Empiricus ( grc-gre, Σέξτος Ἐμπειρικός, ; ) was a Greek Pyrrhonist philosopher and Empiric school physician. His philosophical works are the most complete surviving account of ancient Greek and Roman Pyrrhonism, and beca ...
have pointed out the unsoundness of inductive reasoning, the classic philosophical critique of the
problem of induction First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inferen ...
was given by the Scottish philosopher
David Hume David Hume (; born David Home; 7 May 1711 NS (26 April 1711 OS) – 25 August 1776) Cranston, Maurice, and Thomas Edmund Jessop. 2020 999br>David Hume" ''Encyclopædia Britannica''. Retrieved 18 May 2020. was a Scottish Enlightenment philo ...
. 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 . 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 ''Common Sense'' is a 47-page pamphlet written by Thomas Paine in 1775–1776 advocating independence from Great Britain to people in the Thirteen Colonies. Writing in clear and persuasive prose, Paine collected various moral and political a ...
, where the inevitability of induction is accepted.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
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. In 1963,
Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the c ...
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."Donald Gillies, "Problem-solving and the problem of induction", in ''Rethinking Popper'' (Dordrecht:
Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...
, 2009), Zuzana Parusniková & Robert S Cohen, eds
pp. 103–05
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 First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inferen ...
". In Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during a ''problem shift''. An imaginative leap, the ''tentative solution'' is improvised, lacking inductive rules to guide it. The resulting, unrestricted generalization is deductive, an entailed consequence of all explanatory considerations. Controversy continued, however, with Popper's putative solution not generally accepted. Donald A. Gillies argues that rules of inferences related to inductive reasoning are overwhelmingly absent from science, and describes most scientific inferences as "involv ngconjectures thought up by human ingenuity and creativity, and by no means inferred in any mechanical fashion, or according to precisely specified rules." Gillies also provides a rare counterexample "in the machine learning programs of AI."Donald Gillies, "Problem-solving and the problem of induction", in ''Rethinking Popper'' (Dordrecht:
Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...
, 2009), Zuzana Parusniková & Robert S Cohen, eds
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 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".

## 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 The availability heuristic, also known as availability bias, is a mental shortcut that relies on immediate examples that come to a given person's mind when evaluating a specific topic, concept, method, or decision. This heuristic, operating on the ...
, confirmation bias, and the 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.

# Bayesian inference

As a logic of induction rather than a theory of belief,
Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
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 In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
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 Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' re ...
, 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 Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as ...
with the Bayesian framework. Universal inductive inference is based on solid philosophical foundations, and can be considered as a mathematically formalized
Occam's razor Occam's razor, Ockham's razor, or Ocham's razor ( la, novacula Occami), also known as the principle of parsimony or the law of parsimony ( la, lex parsimoniae), is the problem-solving principle that "entities should not be multiplied beyond neces ...
. Fundamental ingredients of the theory are the concepts of algorithmic probability and
Kolmogorov complexity In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that prod ...
.

*
Analogy Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject (t ...
*
Argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
* Argumentation theory *
Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification ...
* Counterinduction *
Explanation An explanation is a set of statements usually constructed to describe a set of facts which clarifies the causes, context, and consequences of those facts. It may establish rules or laws, and may clarify the existing rules or laws in relatio ...
* Failure mode and effects analysis * Falsifiability *
Grammar induction Grammar induction (or grammatical inference) is the process in machine learning of learning a formal grammar (usually as a collection of ''re-write rules'' or '' productions'' or alternatively as a finite state machine or automaton of some kind) fr ...
* Inductive logic programming * Inductive probability * Inductive programming *
Inductive reasoning aptitude Inductive reasoning aptitude (also called differentiation or inductive learning ability) measures how well a person can identify a pattern within a large amount of data. It involves applying the rules of logic when inferring general principles fr ...
* Inductivism *
Inquiry An inquiry (also spelled as enquiry in British English) is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ...
*
Intuitive statistics Intuitive statistics, or folk statistics, refers to the cognitive phenomenon where organisms use data to make generalizations and predictions about the world. This can be a small amount of sample data or training instances, which in turn contribute ...
* Lateral thinking * Laurence Jonathan Cohen *
Logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
* Logical reasoning *
Logical positivism Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, is a movement in Western philosophy whose central thesis was the verification principle (also known as the verifiability criterion o ...
* Minimum description length * Minimum message length * New riddle of induction *
Open world assumption In a formal system of logic used for knowledge representation, the open-world assumption is the assumption that the truth value of a statement may be true irrespective of whether or not it is ''known'' to be true. It is the opposite of the clos ...
* Raven paradox * Recursive Bayesian estimation *
Statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...
* Marcus Hutter *
Stephen Toulmin Stephen Edelston Toulmin (; 25 March 1922 – 4 December 2009) was a British philosopher, author, and educator. Influenced by Ludwig Wittgenstein, Toulmin devoted his works to the analysis of moral reasoning. Throughout his writings, he sought t ...

# References

* * * *

* * * *
''Four Varieties of Inductive Argument''
from the Department of Philosophy,
University of North Carolina at Greensboro The University of North Carolina at Greensboro (UNCG or UNC Greensboro) is a public research university in Greensboro, North Carolina. It is part of the University of North Carolina system. UNCG, like all members of the UNC system, is a stand-al ...
. *  , a psychological review by Evan Heit of the University of California, Merced.
''The Mind, Limber''
An article which employs the film
The Big Lebowski ''The Big Lebowski'' () is a 1998 crime comedy film written, produced, and directed by Joel and Ethan Coen. It stars Jeff Bridges as Jeffrey "The Dude" Lebowski, a Los Angeles slacker and avid bowler. He is assaulted as a result of mistaken ...
to explain the value of inductive reasoning.
The Pragmatic Problem of Induction
by Thomas Bullemore
Arguments against Popper's Falsificationism
{{DEFAULTSORT:Inductive Reasoning Arguments Causal inference Concepts in epistemology Concepts in logic Concepts in metaphysics Concepts in the philosophy of science Critical thinking Epistemology of science Intellectual history Logic Philosophy of statistics Problem solving skills Reasoning