Inductive reasoning

A generalization (more accurately, an inductive generalization) proceeds from premises about a sample to a conclusion about the population.[4] The observation obtained from this sample is projected onto the broader population.[4]

The proportion Q of the sample has attribute A.

Therefore, the proportion Q of the population has attribute A.

For example, say there are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. An inductive generalization would be that there are 15 black and five white balls in the urn.

How much the premises support the conclusion depends upon (1) the number in the sample group, (2) the number in the population, and (3) the degree to which the sample represents the population (which may be achieved by taking a random sample). The greater the sample size relative to the population and the more closely the sample represents the population, the stronger the generalization is. The hasty generalization and the biased sample are generalization fallacies.

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[5] and sample projections.[6]

An anecdotal generalization is a type of inductive argument in which a conclusion about a population is inferred using a non-statistical sample.[7] 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 the circumstances affecting performance that will occur in the future. On a philosophical level, the argument relies on the presupposition that the operation of future events will mirror the past. In other words, it takes for granted a uniformity of nature, an unproven principle that cannot be derived from the empirical data itself. Arguments that tacitly presuppose this uniformity are sometimes called Humean after the philosopher who was first to subject them to philosophical scrutiny.[8]

An inductive prediction draws a conclusion about a future, current, or past instance from a sample of other instances. Like an inductive generalization, an inductive prediction relies on a data set consisting of specific instances of a phenomenon. But rather than conclude with a general statement, the inductive prediction concludes with a specific statement about the probability that a single instance will (or will not) have an attribute shared (or not shared) by the other instances.[9]

Proportion Q of observed members of group G have had attribute A.

Therefore, there is a probability corresponding to Q that other members of group G will have attribute A when next observed.

A statistical syllogism proceeds from a generalization about a group to a conclusion about an individual.

Proportion Q of the known instances of population P has attribute A.

Individual I is another member of P.

Therefore, there is a probability corresponding to Q that I has A.

For example:

90% of graduates from Excelsior Preparatory school go on to University.

Bob is a graduate of Excelsior Preparatory school.

Therefore, Bob will go on to University.

This is a statistical syllogism.[10] Even though one cannot be sure Bob will attend university, we can be fully assured of the exact probability of this outcome (given no further information). Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".

The process of analogical inference involves noting the shared properties of two or more things and from this basis inferring that they also share some further property:[11]

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

Object P has been observed to have further property x.

Therefore, Q probably has property x also.

Analogical reasoning is very frequent in common sense, science, philosophy, law, and the humanities, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning.[12]

Mineral A and Mineral B are both igneous rocks often containing veins of quartz and are most commonly found in South America in areas of ancient volcanic activity.

Mineral A is also a soft stone suitable for carving into jewelry.

Therefore, mineral B is probably a soft stone suitable for carving into jewelry.

This is analogical induction, according to which things alike in certain ways are more prone to be alike in other ways. This form of induction was explored in detail by philosopher John Stuart Mill in his System of Logic, where he states, "[t]here can be no doubt that every resemblance [not known to be irrelevant] affords some degree of probability, beyond what would otherwise exist, in favor of the conclusion."[13] See Mill's Methods.

Some thinkers contend that analogical induction is a subcategory of inductive generalization because it assumes a pre-established uniformity governing events. Analogical induction requires an auxiliary examination of the relevancy of the characteristics cited as common to the pair. In the preceding example, if a premise were added stating that both stones were mentioned in the records of early Spanish explorers, this common attribute is extraneous to the stones and does not contribute to their probable affinity.

A pitfall of analogy is that features can be cherry-picked: while objects may show striking similarities, two things juxtaposed may respectively possess other characteristics not identified in the analogy that are characteristics sharply dissimilar. Thus, analogy can mislead if not all relevant comparisons are made.

A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.

Enumerative induction is an inductive method in which a generalization is constructed based on the number of instances that support it. The more supporting instances, the stronger the conclusion.[14][15]

The most basic form of enumerative induction reasons from particular instances to all instances and is thus an unrestricted generalization.[16] If one observes 100 swans, and all 100 were white, one might infer a universal categorical proposition of the form All swans are white. As this reasoning form's premises, even if true, do not entail the conclusion's truth, this is a form of inductive inference. The conclusion might be true, and might be thought probably true, yet it can be false. Questions regarding the justification and form of enumerative inductions have been central in philosophy of science, as enumerative induction has a pivotal role in the traditional model of the scientific method.

All life forms so far discovered are composed of cells.

Therefore, all life forms are composed of cells.

This is enumerative induction, also known as simple induction or simple predictive induction. It is a subcategory of inductive generalization. In everyday practice, this is perhaps the most common form of induction. For the preceding argument, the conclusion is tempting but makes a prediction well in excess of the evidence. First, it assumes that life forms observed until now can tell us how future cases will be: an appeal to uniformity. Second, the conclusion All is a bold assertion. A single contrary instance foils the argument. And last, quantifying the level of probability in any mathematical form is problematic.[17] By what standard do we measure our Earthly sample of known life against all (possible) life? Suppose we do discover some new organism—such as some microorganism floating in the mesosphere or an asteroid—and it is cellular. Does the addition of this corroborating evidence oblige us to raise our probability assessment for the subject proposition? It is generally deemed reasonable to answer this question "yes," and for a good many this "yes" is not only reasonable but incontrovertible. So then just how much should this new data change our probability assessment? Here, consensus melts away, and in its place arises a question about whether we can talk of probability coherently at all without numerical quantification.

All life forms so far discovered have been composed of cells.

Therefore, the next life form discovered will be composed of cells.

This is enumerative induction in its weak form. It truncates "all" to a mere single instance and, by making a far weaker claim, considerably strengthens the probability of its conclusion. Otherwise, it has the same shortcomings as the strong form: its sample population is non-random, and quantification methods are elusive.

Eliminative induction, also called variative induction, is an inductive method in which a generalization is constructed based on the variety of instances that support it. Unlike enumerative induction, eliminative induction reasons based on the various kinds of instances that support a conclusion, rather than the number of instances that support it. As the variety of instances increases, the more possible conclusions based on those instances can be identified as incompatible and eliminated. This, in turn, increases the strength of any conclusion that remains consistent with the various instances. This type of induction may use different methodologies such as quasi-experimentation, which tests and, where possible, eliminates rival hypotheses.[18] Different evidential tests may also be employed to eliminate possibilities that are entertained.[19]

Eliminative induction is crucial to the scientific method and is used to eliminate hypotheses that are inconsistent with observations and experiments.[14][15] It focuses on possible causes instead of observed actual instances of causal connections.[20]

For a move from particular to universal, Aristotle in the 300s BCE used the Greek word epagogé, which Cicero translated into the Latin word inductio.[21]

Aristotle's Posterior Analytics covers the methods of inductive proof in natural philosophy and in the social sciences. The first book of Posterior Analytics describes the nature and science of demonstration and its elements: including definition, division, intuitive reason of first principles, particular and universal demonstration, affirmative and negative demonstration, the difference between science and opinion, etc.

The ancient Pyrrhonists were the first Western philosophers to point out the Problem of induction: that induction cannot, according to them, justify the acceptance of universal statements as true.[21]

The Empiric school of ancient Greek medicine employed epilogism as a method of inference. 'Epilogism' is a theory-free method that looks at history through the accumulation of facts without major generalization and with consideration of the consequences of making causal claims.[22] Epilogism is an inference which moves entirely within the domain of visible and evident things, it tries not to invoke unobservables.

The Dogmatic school of ancient Greek medicine employed analogismos as a method of inference.[23] This method used analogy to reason from what was observed to unobservable forces.

In 1620, early modern philosopher Francis Bacon repudiated the value of mere experience and enumerative induction alone. His method of inductivism required that minute and many-varied observations that uncovered the natural world's structure and causal relations needed to be coupled with enumerative induction in order to have knowledge beyond the present scope of experience. Inductivism therefore required enumerative induction as a component.

The empiricist David Hume's 1740 stance found enumerative induction to have no rational, let alone logical, basis; instead, induction was the product of instinct rather than reason, a custom of the mind and an everyday requirement to live. While observations, such as the motion of the sun, could be coupled with the principle of the uniformity of nature to produce conclusions that seemed to be certain, the problem of induction arose from the fact that the uniformity of nature was not a logically valid principle, therefore it could not be defended as deductively rational, but also could not be defended as inductively rational by appealing to the fact that the uniformity of nature has accurately described the past and therefore, will likely accurately describe the future because that is an inductive argument and therefore circular since induction is what needs to be justified.

Since Hume first wrote about the dilemma between the invalidity of deductive arguments and the circularity of inductive arguments in support of the uniformity of nature, this supposed dichotomy between merely two modes of inference, deduction and induction, has been contested with the discovery of a third mode of inference known as abduction, or abductive reasoning, which was first formulated and advanced by Charles Sanders Peirce, in 1886, where he referred to it as "reasoning by hypothesis."[24] Inference to the best explanation is often yet arguably treated as synonymous to abduction as it was first identified by Gilbert Harman in 1965 where he referred to it as "abductive reasoning," yet his definition of abduction slightly differs from Pierce's definition.[25] Regardless, if abduction is in fact a third mode of inference rationally independent from the other two, then either the uniformity of nature can be rationally justified through abduction, or Hume's dilemma is more of a trilemma. Hume was also skeptical of the application of enumerative induction and reason to reach certainty about unobservables and especially the inference of causality from the fact that modifying an aspect of a relationship prevents or produces a particular outcome.

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. Against both rationalist philosophers like Descartes and Leibniz as well as against empiricist philosophers like Locke and Hume, Kant's Critique of Pure Reason is a sustained argument that in order to have knowledge we need both a contribution of our mind (concepts) as well as a contribution of our senses (intuitions). Knowledge proper is for Kant thus restricted to what we can possibly perceive (phenomena), whereas objects of mere thought ("things in themselves") are in principle unknowable due to the impossibility of ever perceiving them.

Reasoning that the mind must contain its own categories for organizing sense data, making experience of objects in space and time (phenomena) possible, Kant concluded that the uniformity of nature was an a priori truth.[26] 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. On the basis of the argument that what goes beyond our knowledge is "nothing to us,"[27] he discarded scientific realism. Kant's position that knowledge comes about by a cooperation of perception and our capacity to think (transcendental idealism) gave birth to the movement of German idealism. Hegel's absolute idealism subsequently flourished across continental Europe and England.

Positivism, developed by Henri de Saint-Simon and promulgated in the 1830s by his former student Auguste Comte, was the first late modern philosophy of science. In the aftermath of the French Revolution, fearing society's ruin, Comte opposed metaphysics. Human knowledge had evolved from religion to metaphysics to science, said Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology—in that order—describing increasingly intricate domains. All of society's knowledge had become scientific, with questions of theology and of metaphysics being unanswerable. Comte found enumerative induction reliable as a consequence of its grounding in available experience. He asserted the use of science, rather than metaphysical truth, as the correct method for the improvement of human society.

According to Comte, scientific method frames predictions, confirms them, and states laws—positive statements—irrefutable by theology or by metaphysics. Regarding experience as justifying enumerative induction by demonstrating the uniformity of nature,[26] 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".[28] 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.[28] 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."[28]

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.[28]

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.[29] Later philosophers termed Peirce's abduction, etc., Inference to the Best Explanation (IBE).[30]

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.[31] 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".[32] Two decades later, Russell proposed enumerative induction as an "independent logical principle".[33][34] Russell found:

"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must, therefore, be, or be deduced from, an independent principle not based on experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are forbidden. These, however, are not questions directly raised by Hume's arguments. What these arguments prove—and I do not think the proof can be controverted—is that induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible."[34]

In a 1965 paper, Gilbert Harman explained that enumerative induction is not an autonomous phenomenon, but is simply a disguised consequence of Inference to the Best Explanation (IBE).[30] IBE is otherwise synonymous with C S Peirce's abduction.[30] Many philosophers of science espousing scientific realism have maintained that IBE is the way that scientists develop approximately true scientific theories about nature.[35]

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

The availability heuristic causes the reasoner to depend primarily upon information that is readily available. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of people who died from various causes, most respondents choose the causes that have been most prevalent in the media such as terrorism, murders, and airplane accidents, rather than causes such as disease and traffic accidents, which have been technically "less accessible" to the individual since they are not emphasized as heavily in the world around them.

Confirmation bias is based on the natural tendency to confirm rather than deny a hypothesis. Research has demonstrated that people are inclined to seek solutions to problems that are more consistent with known hypotheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seek answers that fit established hypotheses, thus confirming these hypotheses. For example, if it is hypothesized that Sally is a sociable individual, subjects will naturally seek to confirm the premise by asking questions that would produce answers confirming that Sally is, in fact, a sociable individual.

The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist, either at all or at a particular level of abstraction. Gambling, for example, is one of the most popular examples of predictable-world bias. Gamblers often begin to think that they see simple and obvious patterns in the outcomes and therefore believe that they are able to predict outcomes based on what they have witnessed. In reality, however, the outcomes of these games are difficult to predict and highly complex in nature. In general, people tend to seek some type of simplistic order to explain or justify their beliefs and experiences, and it is often difficult for them to realise that their perceptions of order may be entirely different from the truth.[50]