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Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence, but not full assurance, for the truth of the conclusion.[1] It is also described as a method where one's experiences and observations, including what are learned from others, are synthesized to come up with a general truth.[2] Many dictionaries define inductive reasoning as the derivation of general principles from specific observations (arguing from specific to general), although there are many inductive arguments that do not have that form.[3]
Inductive reasoning is distinct from deductive reasoning. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.[4]
Types
The three principal types of inductive reasoning are generalization, analogy, and causal inference.[5] These, however, can still be divided into different classifications. Each of these, while similar, has a different form.
Generalization
A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population.[6] The observation obtained from this sample is projected onto the broader population.[6]
- The proportion Q of the sample has attribute A.
- Therefore, the proportion Q of the population has attribute A.
For example, say there are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. An inductive generalization would be that there are 15 bla
Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence, but not full assurance, for the truth of the conclusion.[1] It is also described as a method where one's experiences and observations, including what are learned from others, are synthesized to come up with a general truth.[2] Many dictionaries define inductive reasoning as the derivation of general principles from specific observations (arguing from specific to general), although there are many inductive arguments that do not have that form.[3]
Inductive reasoning is distinct from deductive reasoning. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.[4]
The three principal types of inductive reasoning are generalization, analogy, and causal inference.[5] These, however, can still be divided into different classifications. Each of these, while similar, has a different form.
Generalization
A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population.[6] The observation obtained from this sample is projected onto the broader population.[6]
- The proportion Q of the sample has attribute A.
- Therefore, the proportion Q of the population has attribute A.
For example, say there are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. An inductive generalization would be that there are 15 black and 5 white balls in the urn.
How much the premises support the conclusion depends upon (1) the number in the sample group, (2) the number in the population, and (3) the degree to which the sample represents the population (which may be achieved by taking a random sample). The hasty generalization and the biased sample are generalization fallacies.
Statistical generalization
A statistical generalization is a type of inductive argument in which a conclusion about a population is inferred using a statistically-representative sample. For example:
- Of a sizeable random sample of voters surveyed, 66% support Measure Z.
- Therefore, approximately 66% of voters support Measure Z.
The measure is highly reliable within a well-defined margin of error provided the sample is large and random. It is readily quantifiable. Compare the preceding argument with the following. "Six of the ten people in my book club are Libertarians. Therefore, about 60% of people are Libertarians." The argument is weak because the sample is non-random and the sample size is very small.
Statistical generalizations are also called statistical projections[7] and sample projections.[8]
Anecdotal generalization
An anecdotal generalization is a type of inductive argument in which a conclusion about a population is inferred using a non-statistical sample.[9] In other words, the generalization is based on anecdotal evidence. 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 as to the circumstances affecting performance that will obtain in the future. On a philosophical level, the argument relies on the presupposition that the operation of future events will mirror the past. In other words, it takes for granted a uniformity of nature, an unproven principle that cannot be derived from the empirical data itself. Arguments that tacitly presuppose this uniformity are sometimes called Humean after the philosopher who was first to subject them to philosophical scrutiny.[10]
Prediction
The two principal methods used to reach inductive conclusions are enumerative induction and eliminative induction.The two principal methods used to reach inductive conclusions are enumerative induction and eliminative induction.[16][17]
Enumerative induction
In 1620, early modern philosopher Francis Bacon repudiated the value of mere experience and enumerative induction alone. His method of early modern philosopher Francis Bacon repudiated the value of mere experience and enumerative induction alone. His method of inductivism required that minute and many-varied observations that uncovered the natural world's structure and causal relations needed to be coupled with enumerative induction in order to have knowledge beyond the present scope of experience. Inductivism therefore required enumerative induction as a component.
David Hume
Immanuel Kant
Awakened from "dogmatic slumber" by a German translation of Hume's work, Kant sought to explain the po
Awakened from "dogmatic slumber" by a German translation of Hume's work, Kant sought to explain the possibility of metaphysics. In 1781, Kant's Critique of Pure Reason introduced rationalism as a path toward knowledge distinct from empiricism. Kant sorted statements into two types. Analytic statements are true by virtue of the arrangement of their terms and meanings, thus analytic statements are tautologies, merely logical truths, true by necessity. Whereas synthetic statements hold meanings to refer to states of facts, contingencies. Finding it impossible to know objects as they truly are in themselves, however, Kant concluded that the philosopher's task should not be to try to peer behind the veil of appearance to view the noumena, but simply that of handling phenomena.
Reasoning that the mind must contain its own categories for organizing sense data, making experience of space and time possible, Kant concluded that the uniformity of nature was an a prioriReasoning that the mind must contain its own categories for organizing sense data, making experience of space and time possible, Kant concluded that the uniformity of nature was an a priori truth.[29] A class of synthetic statements that was not contingent but true by necessity, was then synthetic a priori. Kant thus saved both metaphysics and Newton's law of universal gravitation, but as a consequence discarded scientific realism and developed transcendental idealism. Kant's transcendental idealism gave birth to the movement of German idealism. Hegel's absolute idealism subsequently flourished across continental Europe.
Positivism, developed by Saint-Simon and promulgated in the 1830s by his former student Comte, was the first late modern philosophy of science. In the aftermath of the French Revolution, fearing society's ruin, Comte opposed metaphysics. Human knowledge had evolved from religion to metaphysics to science, said Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology—in that order—describing increasingly intricate domains. All of society's knowledge had become scientific, with questions of theology and of metaphysics being unanswerable. Comte found enumerative induction reliable as a consequence of its grounding in available experience. He asserted the use of science, rather than metaphysical truth, as the correct method for the improvement of human society.
According to Comte, scientific method frames predictions, confirms them, and states laws—positive statements—irrefutable by theology or by metaphysics. Regarding experience as justify
According to Comte, scientific method frames predictions, confirms them, and states laws—positive statements—irrefutable by theology or by metaphysics. Regarding experience as justifying enumerative induction by demonstrating the uniformity of nature,[29] the British philosopher John Stuart Mill welcomed Comte's positivism, but thought scientific laws susceptible to recall or revision and Mill also withheld from Comte's Religion of Humanity. Comte was confident in treating scientific law as an irrefutable foundation for all knowledge, and believed that churches, honouring eminent scientists, ought to focus public mindset on altruism—a term Comte coined—to apply science for humankind's social welfare via sociology, Comte's leading science.
During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science, William Whewell found enumerative induction not nearly as convincing, and, despite the dominance of inductivism, formulated "superinduction".[30] 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.[30] Whewell explained:
"Although we bind together facts by superinducing upon them a new Conception, this Conception, once introduced and applied, is looked upon as inseparably connected with the facts, and necessarily implied in them. Having once had the phenomena bound together in their minds in virtue of the Conception, men can no longer easily restore them back to detached and incoherent condition in which they were before they were thus combined."[30]
These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed consilience—that is, simultaneously predicting the inductive generalizations in multiple ar
These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed consilience—that is, simultaneously predicting the inductive generalizations in multiple areas—a feat that, according to Whewell, can establish their truth. Perhaps to accommodate the prevailing view of science as inductivist method, Whewell devoted several chapters to "methods of induction" and sometimes used the phrase "logic of induction", despite the fact that induction lacks rules and cannot be trained.[30]
In the 1870
In the 1870s, the originator of pragmatism, C S Peirce performed vast investigations that clarified the basis of deductive inference as a mathematical proof (as, independently, did Gottlob Frege). Peirce recognized induction but always insisted on a third type of inference that Peirce variously termed abduction or retroduction or hypothesis or presumption.[31] Later philosophers termed Peirce's abduction, etc., Inference to the Best Explanation (IBE).[32]
Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer, or as near a solution as he could arrive at.[33] Bertrand Russell found Keynes's Treatise on Probability the best examination of induction, and believed that if read with Jean Nicod's Le Probleme logique de l'induction as well as R B Braithwaite's review of Keynes's work in the October 1925 issue of Mind, that would cover "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics".[34] Two decades later, Russell proposed enumerative induction as an "independent logical principle".[35][36] Russell found:
"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inference
"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must, therefore, be, or be deduced from, an independent principle not based on experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are forbidden. These, however, are not questions directly raised by Hume's arguments. What these arguments prove—and I do not think the proof can be controverted—is that induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible."[36]
Gilbert HarmanIn 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).[32] IBE is otherwise synonymous with C S Peirce's abduction.[32] Many philosophers of science espousing scientific realism have maintained that IBE is the way that scientists develop approximately true scientific theories about nature.[37]
Criticism
Pyrrhonist philosopher Sextus Empiricus have pointed out the unsoundness of inductive reasoning,[38] the classic philosophical critique of the problem of induction was given by the Scottish philosopher David Hume.[39] 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 .[40]
Hume nevertheless stated that even if induction were proved unreliable, we would still have to rely on it. So instead of a position of severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.Hume nevertheless stated that even if induction were proved unreliable, we would still have to rely on it. So instead of a position of severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.[41] Bertrand Russell illustrated Hume's skepticism in a story about a chicken, fed every morning without fail, who following the laws of induction concluded that this feeding would always continue, until his throat was eventually cut by the farmer.[42]
In 1963, Karl Popper wrote, "Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure."[43][44] 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".[44] In Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during a problem shift.[44] An imaginative leap, the tentative solution is improvised, lacking inductive rules to guide it.[44] The resulting, unrestricted generalization is deductive, an entailed consequence of all explanatory considerations.[44] Controversy continued, however, with Popper's putative solution not generally accepted.[45]
More recently, inductive inference has been shown to be capable of arriving at certainty, but only in rare instances, as in programs of machine learning in artificial intelligence (AI).[46][failed verification] Popper's stance on induction being an illusion has been falsified: enumerative induction exists. Even so, inductive reasoning is overwhelmingly absent from science.[46] Although much-talked of nowadays by philosophers, abduction, or IBE, lacks rules of inference and the inferences reached by those employing it are arrived at with human imagination and creativity.[46]
Inductive reasoning is also known as hypothesis construction because any conclusions made are based on current knowledge and predictions.[citation needed] As with deductive arguments, biases can distort the proper application of inductive argument, thereby preventing the reasoner from forming the most logical conclusion based on the clues. Examples of these biases include the availability heuristic, confirmation bias, and the predictable-world bias.
The availability heuristic causes the reasoner to depend primarily upon information that is readily available to him or her. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of pe
The availability heuristic causes the reasoner to depend primarily upon information that is readily available to him or her. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of people who died from various causes, most respondents choose the causes that have been most prevalent in the media such as terrorism, murders, and airplane accidents, rather than causes such as disease and traffic accidents, which have been technically "less accessible" to the individual since they are not emphasized as heavily in the world around them.
The confirmation bias is based on the natural tendency to confirm rather than to deny a current hypothesis. Research has demonstrated that people are inclined to seek solutions to problems that are more consistent with known hypotheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seek answers that fit established hypotheses, thus confirming these hypotheses. For example, if it is hypothesized that Sally is a sociable individual, subjects will naturally seek to confirm the premise by asking questions that would produce answers confirming that Sally is, in fact, a sociable individual.
The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist, either at all or at a particular level of abstraction. Gambling, for example, is one of the most popular examples of predictable-world bias. Gamblers often begin to think that they see simple and obvious patterns in the outcomes and therefore believe that they are able to predict outcomes based upon what they have witnessed. In reality, however, the outcomes of these games are difficult to predict and highly complex in nature. In general, people tend to seek some type of simplistic order to explain or justify their beliefs and experiences, and it is often difficult for them to realise that their perceptions of order may be entirely different from the truth.[47]
As a logic of induction rather than a theory of belief, Bayesian inference does not determine which beliefs are a priori rational, but rather determines how we should rationally change the beliefs we have when presented with evidence. We begin by committing to a prior probability for a hypothesis based on logic or previous experience and, when faced with evidence, we adjust the strength of our belief in that hypothesis in a precise manner using Bayesian logic.
Inductive inferenceAround 1960, Ray Solomonoff founded the theory of universal inductive inference, a theory of prediction based on observations, for example, predicting the next symbol based upon a given series of symbols. This is a formal inductive framework that combines algorithmic information theory with the Bayesian framework. Universal inductive inference is based on solid philosophical foundations,[48] and can be considered as a mathematically formalized Occam's razor. Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity.
See also
Criticism
Hume nevertheless stated that even if induction were proved unreliable, we would still have to rely on it. So instead of a position of severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.Hume nevertheless stated that even if induction were proved unreliable, we would still have to rely on it. So instead of a position of severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.[41] Bertrand Russell illustrated Hume's skepticism in a story about a chicken, fed every morning without fail, who following the laws of induction concluded that this feeding would always continue, until his throat was eventually cut by the farmer.[42]
In 1963, Karl Popper wrote, "Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure."[43][44] 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".[44] In Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during a problem shift.[44] An imaginative leap, the tentative solution is improvised, lacking inductive rules to guide it.[44] The resulting, unrestricted generalization is deductive, an entailed consequence of all explanatory considerations.[44] Controversy continued, however, with Popper's putative solution not generally accepted.[45]
More recently, inductive inference has been shown to be capable of arriving at certainty, but only in rare instances, as in programs of machine learning in artificial intelligence (AI).[46][failed verification] Popper's stance on induction being an illusion has been falsified: enumerative induction exists. Even so, inductive reasoning is overwhelmingly absent from science.[46] Although much-talked of nowadays by philosophers, abduction, or IBE, lacks rules of inference and the inferences reached by those employing it are arrived at with human imagination and creativity.[46]
Inductive reasoning is also known as hypothesis construction because any conclusions made are based on current knowledge and predictions.[citation needed] As with deductive arguments, biases can distort the proper application of inductive argument, thereby preventing the reasoner from forming the most logical conclusion based on the clues. Examples of these biases include the availability heuristic, confirmation bias, and the predictable-world bias.
The availability heuristic causes the reasoner to depend primarily upon information that is readily available to him or her. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of pe
The availability heuristic causes the reasoner to depend primarily upon information that is readily available to him or her. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of people who died from various causes, most respondents choose the causes that have been most prevalent in the media such as terrorism, murders, and airplane accidents, rather than causes such as disease and traffic accidents, which have been technically "less accessible" to the individual since they are not emphasized as heavily in the world around them.
The confirmation bias is based on the natural tendency to confirm rather than to deny a current hypothesis. Research has demonstrated that people are inclined to seek solutions to problems that are more consistent with known hypotheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seek answers that fit established hypotheses, thus confirming these hypotheses. For example, if it is hypothesized that Sally is a sociable individual, subjects will naturally seek to confirm the premise by asking questions that would produce answers confirming that Sally is, in fact, a sociable individual.
The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist, either at all or at a particular level of abstraction. Gambling, for example, is one of the most popular examples of predictable-world bias. Gamblers often begin to think that they see simple and obvious patterns in the outcomes and therefore believe that they are able to predict outcomes based upon what they have witnessed. In reality, however, the outcomes of these games are difficult to predict and highly complex in nature. In general, people tend to seek some type of simplistic order to explain or justify their beliefs and experiences, and it is often difficult for them to realise that their perceptions of order may be entirely different from the truth.[47]
As a logic of induction rather than a theory of belief, Bayesian inference does not determine which beliefs are a priori rational, but rather determines how we should rationally change the beliefs we have when presented with evidence. We begin by committing to a prior probability for a hypothesis based on logic or previous experience and, when faced with evidence, we adjust the strength of our belief in that hypothesis in a precise manner using Bayesian logic.