Problem of induction
Encyclopedia
The problem of induction is the philosophical
question of whether inductive reasoning
leads to knowledge. That is, what is the justification for either:
The problem calls into question all empirical
claims made in everyday life or through the scientific method
and for that reason the philosopher C. D. Broad said that "induction is the glory of science and the scandal of philosophy". Although the problem arguably dates back to the Pyrrhonism
of ancient philosophy
, David Hume
introduced it in the mid-18th century, with the most notable response provided by Karl Popper
two centuries later. (Some feel that the "no-free-lunch theorem for supervised learning" of Wolpert and Macready is a probability-based extension of induction, yet this is misleading, as inductive logic accustomed to probabilistic arguments and the NFL theorem is more a variation of economic rational choice theory)
, 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 8am 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, 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 8am on Monday. In fact, Hume would even argue 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.
skeptic Sextus Empiricus
first questioned the validity of inductive reasoning, positing that a universal rule could not be established from an incomplete set of particular instances. 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
that although Sextus' 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' 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
.
Medieval writers such as al-Ghazali
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
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 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.
. There is a difficulty, however, in discussing induction and Hume. For Hume himself rarely used the term and when he did, he used it to justify some point he was making. He gave no indication that he saw any problem with induction. Induction became associated with Hume only in the early twentieth century. John Maynard Keynes
may have been the first to draw the connection. The connection is now standard, but it must be remembered (as in the following) that what is meant is a connection between Hume's philosophy and what is nowadays called "induction," not what went by that name in Hume's day.
David Hume
described the problem in An Enquiry concerning Human Understanding
, §4, based on his epistemological framework. Here, "reason" refers to deductive reasoning
and "induction" refers to inductive reasoning.
First, Hume ponders the discovery of causal relations
, which form the basis for what he refers to as "matters of fact." He argues that causal relations are found not by reason, but by induction. This is because for any cause, multiple effects are conceivable, and the actual effect cannot be determined by reasoning about the cause; instead, one must observe occurrences of the causal relation to discover that it holds. For example, when one thinks of "a billiard ball moving in a straight line toward another," one can conceive that the first ball bounces back with the second ball remaining at rest, the first ball stops and the second ball moves, or the first ball jumps over the second, etc. There is no reason to conclude any of these possibilities over the others. Only through previous observation can it be predicted, inductively, what will actually happen with the balls. In general, it is not necessary that causal relation in the future resemble causal relations in the past, as it is always conceivable otherwise; for Hume, this is because the negation of the claim does not lead to a contradiction.
Next, Hume ponders the justification of induction. If all matters of fact are based on causal relations, and all causal relations are found by induction, then induction must be shown to be valid somehow. He uses the fact that induction assumes a valid connection between the proposition "I have found that such an object has always been attended with such an effect" and the proposition "I foresee that other objects which are in appearance similar will be attended with similar effects." One connects these two propositions not by reason, but by induction. This claim is supported by the same reasoning as that for causal relations above, and by the observation that even rationally inexperienced or inferior people can infer, for example, that touching fire causes pain. Hume challenges other philosophers to come up with a (deductive) reason for the connection. If that the justification of induction cannot be deductive, then it would beg the question
for induction to be based on an inductive assumption about a connection. Induction, itself, cannot explain the connection.
In this way, the problem of induction is not only concerned with the uncertainty of conclusions derived by induction, but doubts the very principle through which those uncertain conclusions are derived.
presented a different description of the problem of induction in the third chapter of "Fact, Fiction, and Forecast
" entitled "The New Riddle of Induction" (1954). Goodman proposed a new predicate, "grue
". Something is grue if and only if it has been observed to be green before a certain time or blue 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 standard scientific response is to invoke Occam's razor
.
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, or 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 practicable solution to this problem by making the metaphysical
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.
interpreted Hume to say that an inductive inference must be backed, not only by observations, but also by an independent "inductive assumption." Howson combined this idea with Frank P. Ramsey
's view on probabilistic reasoning to conclude that "there is a genuine logic of induction which exhibits inductive reasoning as logically quite sound given suitable premisses, but does not justify those premisses." In other words, there is logic to induction, but it relies on premises. In this sense, the strength of inductive reasoning is comparable to the strength (or lack thereof) of deductive reasoning.
There are other deductive proofs that illustrate the logic of induction, and these are mentioned further below.
's principal positive argument for induction 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 (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 one unrepresentative you are justified in thinking that probably (although not certainly) that it is.
, 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 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 lots of 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
criticises 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 of 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 (but did not) falsify the one theory, while the same kind of experiment, regardless of its outcome, would not falsify the other. So it is rational to choose the well-corroborated theory; it may not be more likely to be true, but at least it is easier to get rid of it if not.
, Neyman–Pearson lemma, de Finetti's theorem
, and Hans Reichenbach
's theories.
Philosopher John Vickers summarizes the ways these mathematical and deductive proofs respond to Hume's Problem of Induction. For one, Hume's algebraic foundations allowed for only rather weak logic, unlike those more complex theorems just mentioned. Moreover, modern mathematical reasoning has incorporated probability in a way that had not been done in Hume's time. At very least these provide good reason to trust inductive methods over simply the wisdom of the crowd
or single testimony
. That is, the method of induction makes good deductive and probabilistic sense, but it is still up to people to determine which starting premises to hold true.
Vickers concludes that we should use induction, not because it yields certainties
, like deduction, but because it is a method that actually seems to correct itself - and is thus more likely to bring us closer to truth than other methods.
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
question of whether inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
leads to knowledge. That is, what is the justification for either:
- generalizing about the properties of a class of objects based on some number of observations of particular instances of that class (for example, the inference that "all swans we have seen are white, and therefore all swans are white," before the discovery of black swanBlack SwanThe Black Swan is a large waterbird, a species of swan, which breeds mainly in the southeast and southwest regions of Australia. The species was hunted to extinction in New Zealand, but later reintroduced. Within Australia they are nomadic, with erratic migration patterns dependent upon climatic...
s) or - presupposing that a sequence of events in the future will occur as it always has in the past (for example, that the laws of physics will hold as they have always been observed to hold). Hume called this the Principle of Uniformity of NatureBackground independenceBackground independence, also called universality, is the concept or assumption, fundamental to all physical sciences, that the nature of reality is consistent throughout all of space and time...
.
The problem calls into question all empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
claims made in everyday life or through the scientific method
Scientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...
and for that reason the philosopher C. D. Broad said that "induction is the glory of science and the scandal of philosophy". Although the problem arguably dates back to the Pyrrhonism
Pyrrhonism
Pyrrhonism, or Pyrrhonian skepticism, was a school of skepticism founded by Aenesidemus in the 1st century BCE and recorded by Sextus Empiricus in the late 2nd century or early 3rd century CE. It was named after Pyrrho, a philosopher who lived from c. 360 to c. 270 BCE, although the relationship...
of ancient philosophy
Ancient philosophy
This page lists some links to ancient philosophy. In Western philosophy, the spread of Christianity through the Roman Empire marked the ending of Hellenistic philosophy and ushered in the beginnings of Medieval philosophy, whereas in Eastern philosophy, the spread of Islam through the Arab Empire...
, David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
introduced it in the mid-18th century, with the most notable response provided by Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
two centuries later. (Some feel that the "no-free-lunch theorem for supervised learning" of Wolpert and Macready is a probability-based extension of induction, yet this is misleading, as inductive logic accustomed to probabilistic arguments and the NFL theorem is more a variation of economic rational choice theory)
Formulation of the problem
In inductive reasoningInductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
, one makes a series of observations and infers
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
a new claim based on them. For instance, from a series of observations that a woman walks her dog by the market at 8am 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, 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 8am on Monday. In fact, Hume would even argue 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.
Ancient and early modern origins
PyrrhonianPyrrhonism
Pyrrhonism, or Pyrrhonian skepticism, was a school of skepticism founded by Aenesidemus in the 1st century BCE and recorded by Sextus Empiricus in the late 2nd century or early 3rd century CE. It was named after Pyrrho, a philosopher who lived from c. 360 to c. 270 BCE, although the relationship...
skeptic Sextus Empiricus
Sextus Empiricus
Sextus Empiricus , was a physician and philosopher, and has been variously reported to have lived in Alexandria, Rome, or Athens. His philosophical work is the most complete surviving account of ancient Greek and Roman skepticism....
first questioned the validity of inductive reasoning, positing that a universal rule could not be established from an incomplete set of particular instances. 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
Circular reasoning
Circular reasoning, or in other words, paradoxical thinking, is a type of formal logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in one of the premises. For example:"Only an untrustworthy person would run for office...
of induction. However, Weintraub claims in The Philosophical Quarterly
The Philosophical Quarterly
The Philosophical Quarterly is a quarterly academic journal of philosophy established in 1950. It is published by Wiley-Blackwell on behalf of the Scots Philosophical Club and the University of St Andrews. The current editor-in-chief is Tim Mulgan. Every year the journal holds an Essay...
that although Sextus' 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' 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
Foundationalism
Foundationalism is any theory in epistemology that holds that beliefs are justified based on what are called basic beliefs . This position is intended to resolve the infinite regress problem in epistemology...
.
Medieval writers such as al-Ghazali
Al-Ghazali
Abu Hāmed Mohammad ibn Mohammad al-Ghazzālī , known as Algazel to the western medieval world, born and died in Tus, in the Khorasan province of Persia was a Persian Muslim theologian, jurist, philosopher, and mystic....
and William of Ockham
William of Ockham
William of Ockham was an English Franciscan friar and scholastic philosopher, who is believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of...
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
Duns Scotus
Blessed John Duns Scotus, O.F.M. was one of the more important theologians and philosophers of the High Middle Ages. He was nicknamed Doctor Subtilis for his penetrating and subtle manner of thought....
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 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
Few philosophers are as associated with induction as David HumeDavid Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
. There is a difficulty, however, in discussing induction and Hume. For Hume himself rarely used the term and when he did, he used it to justify some point he was making. He gave no indication that he saw any problem with induction. Induction became associated with Hume only in the early twentieth century. John Maynard Keynes
John Maynard Keynes
John Maynard Keynes, Baron Keynes of Tilton, CB FBA , was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments...
may have been the first to draw the connection. The connection is now standard, but it must be remembered (as in the following) that what is meant is a connection between Hume's philosophy and what is nowadays called "induction," not what went by that name in Hume's day.
David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
described the problem in An Enquiry concerning Human Understanding
An Enquiry Concerning Human Understanding
An Enquiry Concerning Human Understanding is a book by the Scottish empiricist philosopher David Hume, published in 1748. It was a revision of an earlier effort, Hume's A Treatise of Human Nature, published anonymously in London in 1739–40...
, §4, based on his epistemological framework. Here, "reason" refers to deductive reasoning
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...
and "induction" refers to inductive reasoning.
First, Hume ponders the discovery of causal relations
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
, which form the basis for what he refers to as "matters of fact." He argues that causal relations are found not by reason, but by induction. This is because for any cause, multiple effects are conceivable, and the actual effect cannot be determined by reasoning about the cause; instead, one must observe occurrences of the causal relation to discover that it holds. For example, when one thinks of "a billiard ball moving in a straight line toward another," one can conceive that the first ball bounces back with the second ball remaining at rest, the first ball stops and the second ball moves, or the first ball jumps over the second, etc. There is no reason to conclude any of these possibilities over the others. Only through previous observation can it be predicted, inductively, what will actually happen with the balls. In general, it is not necessary that causal relation in the future resemble causal relations in the past, as it is always conceivable otherwise; for Hume, this is because the negation of the claim does not lead to a contradiction.
Next, Hume ponders the justification of induction. If all matters of fact are based on causal relations, and all causal relations are found by induction, then induction must be shown to be valid somehow. He uses the fact that induction assumes a valid connection between the proposition "I have found that such an object has always been attended with such an effect" and the proposition "I foresee that other objects which are in appearance similar will be attended with similar effects." One connects these two propositions not by reason, but by induction. This claim is supported by the same reasoning as that for causal relations above, and by the observation that even rationally inexperienced or inferior people can infer, for example, that touching fire causes pain. Hume challenges other philosophers to come up with a (deductive) reason for the connection. If that the justification of induction cannot be deductive, then it would beg the question
Beg the Question
Beg the Question is a graphic novel by Bob Fingerman. It chronicles the trials and tribulations of Rob, a squeamish freelance cartoonist/pornographer, and Sylvia, a beauty salon manager with loftier aspirations, with a supporting cast featuring Jack, an unhappily celibate literary stalker; Max, a...
for induction to be based on an inductive assumption about a connection. Induction, itself, cannot explain the connection.
In this way, the problem of induction is not only concerned with the uncertainty of conclusions derived by induction, but doubts the very principle through which those uncertain conclusions are derived.
Nelson Goodman's New Problem of Induction
Nelson GoodmanNelson Goodman
Henry Nelson Goodman was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, irrealism and aesthetics.-Career:...
presented a different description of the problem of induction in the third chapter of "Fact, Fiction, and Forecast
Fact, Fiction, and Forecast
Fact, Fiction, and Forecast is a book by Nelson Goodman in which he explores some problems regarding scientific law and counterfactual conditionals and presents his New Riddle of Induction...
" entitled "The New Riddle of Induction" (1954). Goodman proposed a new predicate, "grue
Grue (color)
Grue and bleen are artificial predicates, coined as two portmanteaux of "green" and "blue" by philosopher Nelson Goodman in his book Fact, Fiction, and Forecast...
". Something is grue if and only if it has been observed to be green before a certain time or blue 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 standard scientific response is to invoke Occam's razor
Occam's razor
Occam's razor, also known as Ockham's razor, and sometimes expressed in Latin as lex parsimoniae , is a principle that generally recommends from among competing hypotheses selecting the one that makes the fewest new assumptions.-Overview:The principle is often summarized as "simpler explanations...
.
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, or 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
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
offers a practicable solution to this problem by making the metaphysical
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
claim that only predicates that identify a "natural kind
Natural kind
In philosophy, a natural kind is a "natural" grouping, not an artificial one. Or, it is something that a set of things has in common which distinguishes it from other things as a real set rather than as a group of things arbitrarily lumped together by a person or group of people.If any natural...
" (i.e. a real property of real things) can be legitimately used in a scientific hypothesis.
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.Colin Howson
Colin HowsonColin Howson
Professor Colin Howson is a British philosopher who is a Professor of Philosophy at the University of Toronto, where he joined the faculty on July 1, 2008. Previously, he was Professor of Logic at the London School of Economics. He completed a PhD on the philosophy of probability in 1981...
interpreted Hume to say that an inductive inference must be backed, not only by observations, but also by an independent "inductive assumption." Howson combined this idea with Frank P. Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
's view on probabilistic reasoning to conclude that "there is a genuine logic of induction which exhibits inductive reasoning as logically quite sound given suitable premisses, but does not justify those premisses." In other words, there is logic to induction, but it relies on premises. In this sense, the strength of inductive reasoning is comparable to the strength (or lack thereof) of deductive reasoning.
There are other deductive proofs that illustrate the logic of induction, and these are mentioned further below.
David Stove and Donald Williams
David StoveDavid Stove
David Charles Stove , was an Australian philosopher of science.His work in philosophy of science included detailed criticisms of David Hume's inductive skepticism, as well as what he regarded as the irrationalism of his disciplinary contemporaries Karl Popper, Thomas Kuhn, Imre Lakatos, and Paul...
's principal positive argument for induction 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 (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 one unrepresentative you are justified in thinking that probably (although not certainly) that it is.
Karl Popper
Karl PopperKarl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
, a philosopher of science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
, 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 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 lots of 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 C. Salmon
Wesley C. Salmon was a metaphysician and contemporary philosopher of science concerned primarily with the topics of causation and explanation....
criticises 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
David Miller (philosopher)
David W. Miller is a philosopher and prominent exponent of critical rationalism. He taught in the Department of Philosophy at the University of Warwick in Coventry, UK....
has criticized this kind of criticism of 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 (but did not) falsify the one theory, while the same kind of experiment, regardless of its outcome, would not falsify the other. So it is rational to choose the well-corroborated theory; it may not be more likely to be true, but at least it is easier to get rid of it if not.
Other proofs
There are some deductive and mathematical ways to justify some of the logic behind induction. Besides those responses mentioned above, of further relevance to deductively proving aspects of the practice of inference are: law of large numbersLaw of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
, Neyman–Pearson lemma, de Finetti's theorem
De Finetti's theorem
In probability theory, de Finetti's theorem explains why exchangeable observations are conditionally independent given some latent variable to which an epistemic probability distribution would then be assigned...
, and Hans Reichenbach
Hans Reichenbach
Hans Reichenbach was a leading philosopher of science, educator and proponent of logical empiricism...
's theories.
Philosopher John Vickers summarizes the ways these mathematical and deductive proofs respond to Hume's Problem of Induction. For one, Hume's algebraic foundations allowed for only rather weak logic, unlike those more complex theorems just mentioned. Moreover, modern mathematical reasoning has incorporated probability in a way that had not been done in Hume's time. At very least these provide good reason to trust inductive methods over simply the wisdom of the crowd
Wisdom of the crowd
The wisdom of the crowd refers to the process of taking into account the collective opinion of a group of individuals rather than a single expert to answer a question. This process, while not new to the information age, has been pushed into the mainstream spotlight by social information sites such...
or single testimony
Testimony
In law and in religion, testimony is a solemn attestation as to the truth of a matter. All testimonies should be well thought out and truthful. It was the custom in Ancient Rome for the men to place their right hand on a Bible when taking an oath...
. That is, the method of induction makes good deductive and probabilistic sense, but it is still up to people to determine which starting premises to hold true.
Vickers concludes that we should use induction, not because it yields certainties
Certainty
Certainty can be defined as either:# perfect knowledge that has total security from error, or# the mental state of being without doubtObjectively defined, certainty is total continuity and validity of all foundational inquiry, to the highest degree of precision. Something is certain only if no...
, like deduction, but because it is a method that actually seems to correct itself - and is thus more likely to bring us closer to truth than other methods.
See also
- ApophasisApophasisApophasis refers, in general, to "mention by not mentioning". Apophasis covers a wide variety of figures of speech.-Apophasis:...
- Nassim TalebNassim TalebNassim Nicholas Taleb is a Lebanese American essayist whose work focuses on problems of randomness and probability. His 2007 book The Black Swan was described in a review by Sunday Times as one of the twelve most influential books since World War II....
- Abductive ReasoningAbductive reasoningAbduction is a kind of logical inference described by Charles Sanders Peirce as "guessing". The term refers to the process of arriving at an explanatory hypothesis. Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true...
- Probability TheoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
- Hasty GeneralizationHasty generalizationHasty generalization is a logical fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence essentially making a hasty conclusion without considering all of the variables...
- Bayesian InferenceBayesian inferenceIn statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...
- A Priori and A Posteriori Truths
External links
- The Problem of Induction - Stanford Encyclopedia of Philosophy
- David Hume: Metaphysics and Epistemology at the Internet Encyclopedia of PhilosophyInternet Encyclopedia of PhilosophyThe Internet Encyclopedia of Philosophy is a free online encyclopedia on philosophical topics and philosophers founded by James Fieser in 1995. The current general editors are James Fieser and Bradley Dowden...
- Probability and Hume's Inductive Scepticism (1973) by David StoveDavid StoveDavid Charles Stove , was an Australian philosopher of science.His work in philosophy of science included detailed criticisms of David Hume's inductive skepticism, as well as what he regarded as the irrationalism of his disciplinary contemporaries Karl Popper, Thomas Kuhn, Imre Lakatos, and Paul...
- Discovering Karl Popper by Peter SingerPeter SingerPeter Albert David Singer is an Australian philosopher who is the Ira W. DeCamp Professor of Bioethics at Princeton University and Laureate Professor at the Centre for Applied Philosophy and Public Ethics at the University of Melbourne...
- The Warrant of Induction by D. H. Mellor
- Hume and the Problem of Induction
- Secular Responses to the Problem of Induction, by James N. Anderson