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First formulated by
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 ph ...
, 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 inference from the observed to the unobserved is known as "inductive inferences", and Hume, while acknowledging that everyone does and must make such inferences, argued that there is no non-circular way to justify them, thereby undermining one of the Enlightenment pillars of
rationality Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an ab ...
. While David Hume is credited with raising the issue in Western
analytic philosophy Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United ...
in the 18th century, the Pyrrhonist school of
Hellenistic philosophy Hellenistic philosophy is a time-frame for Western philosophy and Ancient Greek philosophy corresponding to the Hellenistic period. It is purely external and encompasses disparate intellectual content. There is no single philosophical school or c ...
and the
Cārvāka Charvaka ( sa, चार्वाक; IAST: ''Cārvāka''), also known as ''Lokāyata'', is an ancient school of Indian materialism. Charvaka holds direct perception, empiricism, and conditional inference as proper sources of knowledge, embra ...
school of ancient
Indian philosophy Indian philosophy refers to philosophical traditions of the Indian subcontinent. A traditional Hindu classification divides āstika and nāstika schools of philosophy, depending on one of three alternate criteria: whether it believes the Veda ...
had expressed skepticism about inductive justification long prior to that. The traditional inductivist view is that all claimed
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
laws, either in everyday life or through 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 scientifi ...
, can be justified through some form of reasoning. The problem is that many philosophers tried to find such a justification but their proposals were not accepted by others. Identifying the inductivist view as the scientific view, C. D. Broad once said that induction is "the glory of science and the scandal of philosophy". In contrast,
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 ...
's
critical rationalism Critical rationalism is an epistemological philosophy advanced by Karl Popper on the basis that, if a statement cannot be logically deduced (from what is known), it might nevertheless be possible to logically falsify it. Following Hume, Popper ...
claimed that inductive justifications are never used in science and proposed instead that science is based on the procedure of conjecturing
hypotheses A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
, deductively calculating consequences, and then empirically attempting to
falsify Falsifiability is a standard of evaluation of scientific theories and hypotheses that was introduced by the philosopher of science Karl Popper in his book '' The Logic of Scientific Discovery'' (1934). He proposed it as the cornerstone of a so ...
them.


Formulation of the problem

In inductive reasoning, one makes a series of observations and infers a new claim based on them. For instance, from a series of observations that a woman walks her dog by the market at 8 am on Monday, it seems valid to infer that next Monday she will do the same, or that, in general, the woman walks her dog by the market every Monday. That next Monday the woman walks by the market merely adds to the series of observations, but it does not prove she will walk by the market every Monday. First of all, it is not certain, regardless of the number of observations, that the woman always walks by the market at 8 am on Monday. In fact,
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 ph ...
even argued that we cannot claim it is "more probable", since this still requires the assumption that the past predicts the future. Second, the observations themselves do not establish the validity of inductive reasoning, except inductively.
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, a ...
illustrated this point in ''
The Problems of Philosophy ''The Problems of Philosophy'' is a 1912 book by the philosopher Bertrand Russell, in which the author attempts to create a brief and accessible guide to the problems of philosophy. He introduces philosophy as a repeating series of (failed) att ...
'':


Ancient and early modern origins


Pyrrhonism

The works of 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 ...
contain the oldest surviving questioning of the validity of inductive reasoning. He wrote: The focus upon the gap between the premises and conclusion present in the above passage appears different from Hume's focus upon the circular reasoning of induction. However, Weintraub claims in ''
The Philosophical Quarterly ''The Philosophical Quarterly'' is a quarterly academic journal of philosophy established in 1950 and published by Wiley-Blackwell on behalf of the Scots Philosophical Club and the University of St Andrews. Since 2014 its publisher is Oxford Acad ...
'' that although Sextus's approach to the problem appears different, Hume's approach was actually an application of another argument raised by Sextus: Although the criterion argument applies to both deduction and induction, Weintraub believes that Sextus's argument "is precisely the strategy Hume invokes against induction: it cannot be justified, because the purported justification, being inductive, is circular." She concludes that "Hume's most important legacy is the supposition that the justification of induction is not analogous to that of deduction." She ends with a discussion of Hume's implicit sanction of the validity of deduction, which Hume describes as intuitive in a manner analogous to modern foundationalism.


Indian philosophy

The
Cārvāka Charvaka ( sa, चार्वाक; IAST: ''Cārvāka''), also known as ''Lokāyata'', is an ancient school of Indian materialism. Charvaka holds direct perception, empiricism, and conditional inference as proper sources of knowledge, embra ...
, a materialist and skeptic school of Indian philosophy, used the problem of induction to point out the flaws in using inference as a way to gain valid knowledge. They held that since inference needed an invariable connection between the middle term and the predicate, and further, that since there was no way to establish this invariable connection, that the efficacy of inference as a means of valid knowledge could never be stated. The 9th century Indian skeptic, Jayarasi Bhatta, also made an attack on inference, along with all means of knowledge, and showed by a type of reductio argument that there was no way to conclude universal relations from the observation of particular instances.


Medieval philosophy

Medieval writers such as
al-Ghazali Al-Ghazali ( – 19 December 1111; ), full name (), and known in Persian-speaking countries as Imam Muhammad-i Ghazali (Persian: امام محمد غزالی) or in Medieval Europe by the Latinized as Algazelus or Algazel, was a Persian poly ...
and William of Ockham connected the problem with God's absolute power, asking how we can be certain that the world will continue behaving as expected when God could at any moment miraculously cause the opposite.
Duns Scotus John Duns Scotus ( – 8 November 1308), commonly called Duns Scotus ( ; ; "Duns the Scot"), was a Scottish Catholic priest and Franciscan friar, university professor, philosopher, and theologian. He is one of the four most important ...
, however, argued that inductive inference from a finite number of particulars to a universal generalization was justified by "a proposition reposing in the soul, 'Whatever occurs in a great many instances by a cause that is not free, is the natural effect of that cause. Some 17th-century
Jesuits , image = Ihs-logo.svg , image_size = 175px , caption = ChristogramOfficial seal of the Jesuits , abbreviation = SJ , nickname = Jesuits , formation = , founders = ...
argued that although God could create the end of the world at any moment, it was necessarily a rare event and hence our confidence that it would not happen very soon was largely justified.


David Hume

David Hume, a Scottish thinker of the Enlightenment era, is the philosopher most often associated with induction. His formulation of the problem of induction can be found in '' An Enquiry concerning Human Understanding'', §4. Here, Hume introduces his famous distinction between "relations of ideas" and "matters of fact." Relations of ideas are propositions which can be derived from deductive logic, which can be found in fields such as geometry and algebra. Matters of fact, meanwhile, are not verified through the workings of deductive logic but by experience. Specifically, matters of fact are established by making an inference about causes and effects from repeatedly observed experience. While relations of ideas are supported by reason alone, matters of fact must rely on the connection of a cause and effect through experience. Causes of effects cannot be linked through a priori reasoning, but by positing a "necessary connection" that depends on the "uniformity of nature." Hume situates his introduction to the problem of induction in ''
A Treatise of Human Nature '' A Treatise of Human Nature: Being an Attempt to Introduce the Experimental Method of Reasoning into Moral Subjects'' (1739–40) is a book by Scottish philosopher David Hume, considered by many to be Hume's most important work and one of th ...
'' within his larger discussion on the nature of causes and effects (Book I, Part III, Section VI). He writes that reasoning alone cannot establish the grounds of causation. Instead, the human mind imputes causation to phenomena after repeatedly observing a connection between two objects. For Hume, establishing the link between causes and effects relies not on reasoning alone, but the observation of "''constant conjunction''" throughout one's sensory experience. From this discussion, Hume goes on to present his formulation of the problem of induction in ''A Treatise of Human Nature'', writing "there can be no ''demonstrative'' arguments to prove, ''that those instances, of which we have had no experience, resemble those, of which we have had experience''." In other words, the problem of induction can be framed in the following way: we cannot apply a conclusion about a particular set of observations to a more general set of observations. While deductive logic allows one to arrive at a conclusion with certainty, inductive logic can only provide a conclusion that is probably true. It is mistaken to frame the difference between deductive and inductive logic as one between general to specific reasoning and specific to general reasoning. This is a common misperception about the difference between inductive and deductive thinking. According to the literal standards of logic, deductive reasoning arrives at certain conclusions while inductive reasoning arrives at probable conclusions. Hume's treatment of induction helps to establish the grounds for probability, as he writes in ''A Treatise of Human Nature'' that "probability is founded on the presumption of a resemblance betwixt those objects, of which we have had experience, and those, of which we have had none" (Book I, Part III, Section VI). Therefore, Hume establishes induction as the very grounds for attributing causation. There might be many effects which stem from a single cause. Over repeated observation, one establishes that a certain set of effects are linked to a certain set of causes. However, the future resemblance of these connections to connections observed in the past depends on induction. Induction allows one to conclude that "Effect A2" was caused by "Cause A2" because a connection between "Effect A1" and "Cause A1" was observed repeatedly in the past. Given that reason alone can not be sufficient to establish the grounds of induction, Hume implies that induction must be accomplished through imagination. One does not make an inductive reference through a priori reasoning, but through an imaginative step automatically taken by the mind. Hume does not challenge that induction is performed by the human mind automatically, but rather hopes to show more clearly how much human inference depends on inductive—not
a priori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
—reasoning. He does not deny future uses of induction, but shows that it is distinct from deductive reasoning, helps to ground causation, and wants to inquire more deeply into its validity. Hume offers no solution to the problem of induction himself. He prompts other thinkers and logicians to argue for the validity of induction as an ongoing dilemma for philosophy. A key issue with establishing the validity of induction is that one is tempted to use an inductive inference as a form of justification itself. This is because people commonly justify the validity of induction by pointing to the many instances in the past when induction proved to be accurate. For example, one might argue that it is valid to use inductive inference in the future because this type of reasoning has yielded accurate results in the past. However, this argument relies on an inductive premise itself—that past observations of induction being valid will mean that future observations of induction will also be valid. Thus, many solutions to the problem of induction tend to be circular.


Nelson Goodman's new riddle of induction

Nelson Goodman Henry Nelson Goodman (7 August 1906 – 25 November 1998) was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, irrealism, and aesthetics. Life and career Goodman was born in Somerville, Ma ...
's '' Fact, Fiction, and Forecast'' presented a different description of the problem of induction in the chapter entitled "The New Riddle of Induction". Goodman proposed the new
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
" grue". Something is grue if and only if it has been (or will be, according to a scientific, general hypothesis) observed to be green before a certain time ''t'', and blue if observed after that time. The "new" problem of induction is, since all emeralds we have ever seen are both green and grue, why do we suppose that after time ''t'' we will find green but not grue emeralds? The problem here raised is that two different inductions will be true and false under the same conditions. In other words: * Given the observations of a lot of green emeralds, someone using a common language will inductively infer that all emeralds are green (therefore, he will believe that any emerald he will ever find will be green, even after time ''t''). * Given the same set of observations of green emeralds, someone using the predicate "grue" will inductively infer that all emeralds, which will be observed after ''t'', will be blue, despite the fact that he observed only green emeralds so far. One could argue, using Occam's Razor, that greenness is more likely than grueness because the concept of grueness is more complex than that of greenness. Goodman, however, points out that the predicate "grue" only appears more complex than the predicate "green" because we have defined grue in terms of blue and green. If we had always been brought up to think in terms of "grue" and "bleen" (where bleen is blue before time ''t'', and green thereafter), we would intuitively consider "green" to be a crazy and complicated predicate. Goodman believed that which scientific hypotheses we favour depend on which predicates are "entrenched" in our language. W. V. O. Quine offers a practical solution to this problem by making the
metaphysical 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 conscio ...
claim that only predicates that identify a " natural kind" (i.e. a real property of real things) can be legitimately used in a scientific hypothesis. R. Bhaskar also offers a practical solution to the problem. He argues that the problem of induction only arises if we deny the possibility of a reason for the predicate, located in the enduring nature of something. For example, we know that all emeralds are green, not because we have only ever seen green emeralds, but because the chemical make-up of emeralds insists that they must be green. If we were to change that structure, they would not be green. For instance, emeralds are a kind of green beryl, made green by trace amounts of chromium and sometimes vanadium. Without these trace elements, the gems would be colourless.


Notable interpretations


Hume

Although induction is not made by reason, Hume observes that we nonetheless perform it and improve from it. He proposes a descriptive explanation for the nature of induction in §5 of the ''Enquiry'', titled "Skeptical solution of these doubts". It is by custom or habit that one draws the inductive connection described above, and "without the influence of custom we would be entirely ignorant of every matter of fact beyond what is immediately present to the memory and senses". The result of custom is belief, which is instinctual and much stronger than imagination alone.


David Stove and Donald Williams

David Stove David Charles Stove (15 September 1927 – 2 June 1994) was an Australian philosopher. Philosophy His work in philosophy of science included criticisms of David Hume's Inductive scepticism. He offered a positive response to the problem of i ...
's argument for induction, based on the statistical syllogism, was presented in the ''Rationality of Induction'' and was developed from an argument put forward by one of Stove's heroes, the late
Donald Cary Williams Donald Cary Williams (28 May 1899 – 16 January 1983), usually cited as D. C. Williams, was an American philosopher and a professor at both the University of California Los Angeles (from 1930 to 1938) and at Harvard University (from 1939 to 1967 ...
(formerly Professor at Harvard) in his book ''The Ground of Induction''. Stove argued that it is a statistical truth that the great majority of the possible subsets of specified size (as long as this size is not too small) are similar to the larger population to which they belong. For example, the majority of the subsets which contain 3000 ravens which you can form from the raven population are similar to the population itself (and this applies no matter how large the raven population is, as long as it is not infinite). Consequently, Stove argued that if you find yourself with such a subset then the chances are that this subset is one of the ones that are similar to the population, and so you are justified in concluding that it is likely that this subset "matches" the population reasonably closely. The situation would be analogous to drawing a ball out of a barrel of balls, 99% of which are red. In such a case you have a 99% chance of drawing a red ball. Similarly, when getting a sample of ravens the probability is very high that the sample is one of the matching or "representative" ones. So as long as you have no reason to think that your sample is an unrepresentative one, you are justified in thinking that probably (although not certainly) that it is.


Biting the bullet: Keith Campbell and Claudio Costa

An intuitive answer to Hume would be to say that a world inaccessible to any inductive procedure would simply not be conceivable. This intuition was taken into account by Keith Campbell by considering that, to be built, a concept must be reapplied, which demands a certain continuity in its object of application and consequently some openness to induction. Recently, Claudio Costa has noted that a future can only be a future of its own past if it holds some identity with it. Moreover, the nearer a future is to the point of junction with its past, the greater are the similarities tendentially involved. Consequently, – ''contra'' Hume – some form of principle of homogeneity (causal or structural) between future and past must be warranted, which would make some inductive procedure always possible.


Karl Popper

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 ...
, a philosopher of science, sought to solve the problem of induction. He argued that science does not use induction, and induction is in fact a myth. Instead, knowledge is created by
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
and criticism. The main role of observations and experiments in science, he argued, is in attempts to criticize and refute existing theories. According to Popper, the problem of induction as usually conceived is asking the wrong question: it is asking how to justify theories given they cannot be justified by induction. Popper argued that justification is not needed at all, and seeking justification "begs for an authoritarian answer". Instead, Popper said, what should be done is to look to find and correct errors. Popper regarded theories that have survived criticism as better corroborated in proportion to the amount and stringency of the criticism, but, in sharp contrast to the inductivist theories of knowledge, emphatically as less likely to be true. Popper held that seeking for theories with a high probability of being true was a false goal that is in conflict with the search for knowledge. Science should seek for theories that are most probably false on the one hand (which is the same as saying that they are highly falsifiable and so there are many ways that they could turn out to be wrong), but still all actual attempts to falsify them have failed so far (that they are highly corroborated).
Wesley C. Salmon Wesley Charles Salmon (August 9, 1925 – April 22, 2001) was an American philosopher of science renowned for his work on the nature of scientific explanation. He also worked on confirmation theory, trying to explicate how probability theory via ...
criticizes Popper on the grounds that predictions need to be made both for practical purposes and in order to test theories. That means Popperians need to make a selection from the number of unfalsified theories available to them, which is generally more than one. Popperians would wish to choose well-corroborated theories, in their sense of corroboration, but face a dilemma: either they are making the essentially inductive claim that a theory's having survived criticism in the past means it will be a reliable predictor in the future; or Popperian corroboration is no indicator of predictive power at all, so there is no rational motivation for their preferred selection principle. David Miller has criticized this kind of criticism by Salmon and others because it makes inductivist assumptions. Popper does not say that corroboration is an indicator of predictive power. The predictive power is in the theory itself, not in its corroboration. The rational motivation for choosing a well-corroborated theory is that it is simply easier to falsify: Well-corroborated means that at least one kind of experiment (already conducted at least once) could have falsified (but did not actually falsify) the one theory, while the same kind of experiment, regardless of its outcome, could not have falsified the other. So it is rational to choose the well-corroborated theory: It may not be more likely to be true, but if it is actually false, it is easier to get rid of when confronted with the conflicting evidence that will eventually turn up. Accordingly, it is wrong to consider corroboration as a reason, a justification for believing in a theory or as an argument in favor of a theory to convince someone who objects to it.Thomas Bullemore
"Some Remarks on the Pragmatic Problem of Induction"
Academia.edu.


See also

* ''A priori'' and ''a posteriori'' * Abductive reasoning *
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 ...
* Consilience * Hasty generalization * Inductive logic programming *
Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
*
Solomonoff's theory of inductive inference Solomonoff's theory of inductive inference is a mathematical proof that if a universe is generated by an algorithm, then observations of that universe, encoded as a dataset, are best predicted by the smallest executable archive of that dataset. T ...
*
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 ...


Notes


References

* * * *


External links

* * * * * (1973) by
David Stove David Charles Stove (15 September 1927 – 2 June 1994) was an Australian philosopher. Philosophy His work in philosophy of science included criticisms of David Hume's Inductive scepticism. He offered a positive response to the problem of i ...

Discovering Karl Popper
by
Peter Singer Peter Albert David Singer (born 6 July 1946) is an Australian moral philosopher, currently the Ira W. DeCamp Professor of Bioethics at Princeton University. He specialises in applied ethics and approaches ethical issues from a secular ...

The Warrant of Induction
by D. H. Mellor
Hume and the Problem of Induction


by James N. Anderson

by Thomas Bullemore

by Nicholas Maxwell. This book sets out to solve the problem of induction and related problems associated with scientific progress.
The problem of induction and metaphysical assumptions concerning the comprehensibility and knowability of the universe
by Nicholas Maxwell {{DEFAULTSORT:Problem of Induction David Hume Karl Popper Induction Philosophy of science Philosophical logic Inductive reasoning Concepts in epistemology Pyrrhonism Skepticism