Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument support the conclusion, but do not ensure it. It is used to ascribe properties or relations to types based on limited observations of particular tokens; or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is used, for example, in using specific propositions such as: Reasoning is the act of using reason to derive a conclusion from certain premises. ... In metaphysics (in particular, ontology), the different kinds or ways of being are called categories of being or simply According to the Aristotelian tradition, a being is anything that can be said to be in the various senses of this word. ... A type is a category of being. ... Law (from the Old Norse lagu) in politics and jurisprudence, is a set of rules or norms of conduct which mandate, proscribe or permit specified relationships among people and organizations, intended to provide methods for ensuring the impartial treatment of such people, and provide punishments of/for those who do... A phenomenon (plural: phenomena) is an observable event, especially something special (literally something that can be seen from the Greek word phainomenon = observable). ...

• This ice is cold.
• A billiard ball moves when struck with a cue.

to infer general propositions such as:

• All ice is cold. or: There is no ice in the Sun.
• For every action, there is an equal and opposite reaction.

Examples

Strong:

All observed crows are black.

Therefore all crows are black.

This exemplifies the nature of induction: inducing the universal from the particular. And clearly the conclusion is not certain. Unless we've seen every crow - and how do we know that? - maybe there are some rare blue ones.

Weak:

I always hang pictures on nails.

Therefore all pictures hang from nails.

In this example, the premise is built upon a certainty: "I always hang pictures on nails", but not all people hang pictures on nails and those that do use nails may only do some of the time. There are a number of objects that may be used to hang picture, including, but not limited to: screws, bolts, and clips. The conclusion I draw is an overgeneralization and is, in some instances, false.

Teenagers get lots of speeding tickets.

Therefore all teenagers speed.

In this example, the foundational premise is not built upon a certainty: not every teenager we've observed speeding has received a ticket. It may be in the general nature of teenagers to speed - as it is crows to be black - but the premise is based more on wishful thinking than direct observation.

Validity

Formal logic as most people learn it is deductive rather than inductive. Some philosophers claim to have created systems of inductive logic, but it is controversial whether a logic of induction is even possible. In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same degree of certainty as the initial premises. For example, a conclusion that all swans are white is obviously wrong, but may have been thought correct in Europe until the settlement of Australia. Inductive arguments are never binding but they may be cogent. Inductive reasoning is deductively invalid. (An argument in formal logic is valid if and only if it is not possible for the premises of the argument to be true whilst the conclusion is false.) In traditional Aristotelian logic, deductive reasoning is inference in which the conclusion is of no greater generality than the premises, as opposed to abductive and inductive reasoning, where the conclusion is of greater generality than the premises. ... World map showing Europe Europe is conventionally considered one of the seven continents which, in this case, is more a cultural and political distinction than a physiogeographic one. ... This article discusses validity in logic, for the term in the social sciences see validity (psychometric). ... An argument is cogent if and only if the truth of the arguments premises would render the truth of the conclusion probable (i. ...

In induction there are always many conclusions that can reasonably be related to certain premises. Inductions are open; deductions are closed.

The classic philosophical treatment of the problem of induction, meaning the search for a justification for inductive reasoning, was by the Scotsman David Hume. Hume highlighted the fact that our everyday reasoning depends on patterns of repeated experience rather than deductively valid arguments. For example we believe that bread will nourish us because it has in the past, but it is at least conceivable that bread in the future will poison us. The problem of induction is the philosophical issue involved in deciding the place of induction in determining empirical truth. ... David Hume (April 26, 1711 – August 25, 1776*) was a Scottish philosopher and historian. ...

Someone who insisted on sound deductive justifications for everything would starve to death, said Hume. Instead of unproductive radical skepticism about everything, he advocated a practical skepticism based on common-sense, where the inevitability of induction is accepted. Philosophical skepticism or nihilistic skepticism (UK spelling, scepticism) is the philosophical school of thought in which one critically examines whether the knowledge and perceptions one has are true, and whether or not one can ever be said to have true knowledge. ... Scientific skepticism or rational skepticism (UK spelling, scepticism) sometimes referred to as skeptical inquiry, is a scientific, or practical, epistemological position (or paradigm) in which one questions the veracity of claims unless they can be empirically tested. ...

Induction is sometimes framed as reasoning about the future from the past, but in its broadest sense it involves reaching conclusions about unobserved things on the basis of what is observed. Inferences about the past from present evidence (e.g. archaeology) count as induction. Induction could also be across space rather than time, e.g. conclusions about the whole universe from what we observe in our galaxy or national economic policy based on local economic performance. Archaeology or archeology (from the Greek words αρχαίος = ancient and λόγος = word/speech/discourse) is the study of human cultures through the recovery, documentation and analysis of material remains and environmental data, including architecture, artifacts, biofacts, human remains, and landscapes. ...

20th Century developments have framed the problem of induction very differently. Rather than a choice about what predictions to make about the future, it can be seen as a choice of what concepts to fit to observation (see the entry for grue) or of what graphs to fit to a set of observed data points. Nelson Goodman posed a “new riddle of induction” by coming up with a property grue to which induction does not apply. Grue is an artificial adjective, coined from green and blue by philosopher Nelson Goodman in one of the seminal works in the philosophy of science, Fact, Fiction, and Forecast. ... Nelson Goodman (7 August 1906, Somerville, Maryland – 25 November 1998) was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, and aesthetics. ... Grue is an artificial adjective, coined from green and blue by philosopher Nelson Goodman in one of the seminal works in the philosophy of science, Fact, Fiction, and Forecast. ...

Types of inductive reasoning

Generalization 

A generalization, or inductive generalization, proceeds from a premise about a sample to a conclusion about the population.

1. A proportion Q of the sample has attribute A.
2. Conclusion: Q of the population has attribute A.

The support which the premises provide for the conclusion is dependent on the number of individuals in the sample group compared to the number in the population, and the randomness of the sample. The hasty generalization and biased sample are fallacies related to generalization. A sample is that part of a population which is actually observed. ... Hasty generalization, also known as fallacy of insufficient statistics, fallacy of insufficient sample, fallacy of the lonely fact, leaping to a conclusion, hasty induction, law of small numbers or secundum quid, is the logical fallacy of reaching an inductive generalization based on too little evidence. ... A biased sample is one that is falsely taken to be typical of a population from which it is drawn. ...

Statistical syllogism 

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

1. A proportion Q of population P has attribute A.
2. An individual I is a member of P.
3. Conclusion: There is a probability which corresponds to Q that I has A.

The proportion in premise 1 can be a word like '3/5 of', 'all' or 'few'. Two dicto simpliciter fallacies can occur in statistical syllogisms. They are "accident" and "converse accident". The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... The logical fallacy of accident, also called destroying the exception or a dicto simpliciter ad dictum secundum quid, is a deductive fallacy occurring in statistical syllogisms (an argument based on a generalization) when an exception to the generalization is ignored. ... The logical fallacy of converse accident (also called reverse accident, destroying the exception or a dicto secundum quid ad dictum simpliciter) is a deductive fallacy that can occur in a statistical syllogism when an exception to a generalization is wrongly called for. ...

Simple Induction 

Simple induction proceeds from a premise about a sample group to a conclusion about another individual.

1. Proportion Q of known instances of population P has attribute A.
2. Individual I is another member of P.
3. Conclusion: There is a probability which corresponds to Q that I has A.

This is actually a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is also the first premise of the statistical syllogism.

Argument from analogy 

An (inductive) analogy proceeds from known similarities between two things to a conclusion about an additional attribute that is common to both things:

1. Thing P is similar to thing Q.
2. P has attribute A.
3. Conclusion: Q has attribute A.

An analogy relies on the inference that the known shared properties (similarities) imply that A is also a shared property. The support which the premises provide for the conclusion is dependent upon the relevance and number of the similarities between P and Q. Analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. ...

Causal inference 

A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect.

Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.

Prediction 

A prediction draws a conclusion about a future individual from a past sample.

1. Proportion Q of observed members of group G have had attribute A.
2. There is a probability which corresponds to Q that the next observed member of G will have A.

Argument from authority 

An argument from authority draws a conclusion about the truth of a statement based on the proportion of true propositions which a sources says. It has the same form as a prediction.

1. Proportion Q of the claims of authority A have been true.
2. There is a probability which corresponds to Q that this claim of A is true.

Example:

All observed claims from websites about logic are true.

This information came from websites about logic.

Therefore, this information is (probably) true.

Bayesian inference

Of the candidate systems of inductive logic, the most influential is Bayesianism, which uses probability theory as a framework for induction. Bayes theorem is used to calculate how much the strength of one’s belief in a hypothesis should change, given some evidence. Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements; when used with Bayes theorem, it then becomes Bayesian inference. ... The word probability derives from the Latin probare (to prove, or to test). ... Bayes theorem is a result in probability theory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. ...

There is debate around what it is that informs the original degree of belief. Objective Bayesians seek an objective value for the degree of probability of a hypothesis being correct, and so do not avoid the philosophical criticisms of objectivism. Subjective Bayesians hold that the prior probabilities represent subjective degrees of belief, but that repeated application of Bayes’ theorem leads to a high degree of agreement on the posterior probability. They therefore fail to provide an objective standard for choosing between conflicting hypotheses. The theorem can be used to rationally justify belief in some hypothesis, but at the expense of rejecting objectivism. Such a scheme cannot be used, for instance, to objectively decide between conflicting scientific paradigms. Objectivism is the philosophy developed by Russian-born American philosopher and author Ayn Rand. ...

Edwin Jaynes, an outspoken physicist and Bayesian, argued that 'subjective' elements are present in all of inference (e.g. in choosing axioms for deductive inference, in choosing initial degrees of belief or prior probabilities, and in choosing likelihoods), and sought a series of principles for assigning probabilities from qualitative knowledge. Maximum entropy (a generalization of the principle of indifference) and transformation groups are the two resulting tools he produced; both attempt to alleviate the subjectivity of probability assignment in specific situations by converting knowledge of e.g. symmetries of a situation into unambiguous choices for probability distributions. Edwin Thompson Jaynes (July 5th, 1922 – April 30th, 1998) was Wayman Crow Distinguished Professor of Physics at Washington University in St. ... In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. ... The principle of maximum entropy is a method for analyzing the available information in order to determine a unique epistemic probability distribution. ... The principle of indifference is a rule for assigning epistemic probabilities. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

Bayesians feel entitled to call their system an inductive logic because of Cox's Theorem, which derives probability from a set of logical constraints on a system of inductive reasoning.. Coxs theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. ...

See also

Abduction, or abductive reasoning, is the process of reasoning to the best explanations. ... In traditional Aristotelian logic, deductive reasoning is inference in which the conclusion is of no greater generality than the premises, as opposed to abductive and inductive reasoning, where the conclusion is of greater generality than the premises. ... An explanation is a statement which points to causes, context and consequences of some object (or process, state of affairs etc. ... Falsifiability is an important concept in the philosophy of science that amounts to the principle that a proposition or theory cannot be considered scientific if it does not admit the possibility of being shown false. ... Inductive reasoning is a measureable aptitude for how well a person can identify a pattern within a large amount of data. ... It has been suggested that this article or section be merged with statistical inference. ... Wikipedia does not yet have an article with this exact name. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ... Similar to induction, but predicated on a known relationary rule(s) and an observation(s). ...

External links

• Four Varieties of Inductive Argument
• Stanford Encyclopedia of Philosophy entry on Inductive Logic
• Psychology of Inductive Reasoning
• Using Inductive Reasoning to Solve Problems