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Encyclopedia > Inductive reasoning

Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. This classification has only recently been challenged.

Reasoning is used in determining the validity of inductive concepts, but it is not necessarily involved in the origination of generalizations.

Historically, induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument are believed to support the conclusion but do not ensure it. It is used to ascribe properties or relations to types based on tokens (i.e., on one or a small number of observations or experiences); 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. ... For law within legal systems see law. ... 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.
All billiard balls struck with a cue move.

## Strong and weak induction GA_googleFillSlot("encyclopedia_square");

### Strong induction

All observed crows are black.
therefore
All crows are black.

This exemplifies the nature of induction: inducing the universal from the particular. However, the conclusion is not certain.

Unless we can systematically verify the possibility of crows of another color, the statement may actually be false.

For example, one could examine the bird's genome and learn whether it's capable of producing a differently colored bird without mutation or a long set of breeding changes. In doing so, we could discover that albinism is possible, resulting in light-colored crows. Albinism is a genetic condition resulting in a lack of pigmentation in the eyes, skin and hair. ...

Even if you change the definition of "crow" to require blackness, the original question of the color possibilities for a bird of that species would stand, only semantically hidden. Semantics (Greek semantikos, giving signs, significant, symptomatic, from sema, sign) refers to the aspects of meaning that are expressed in a language, code, or other form of representation. ...

### Weak induction

I always hang pictures on nails.
therefore
All pictures hang from nails.

Assuming the first statement to be true, this example is built on the certainty that "I always hang pictures on nails" leading to the generalization that "All pictures hang from nails". However, the link between the premise and the inductive conclusion is weak. No reason exists to believe that just because one person hangs pictures on nails that there are no other ways for pictures to be, or that other people cannot do other things with pictures. Indeed, not all pictures are hung from nails; moreover, not all pictures are hung. The conclusion cannot be strongly inductively made from the premise. Using other knowledge we can easily see that this example of induction would lead us to a clearly false conclusion. Conclusions drawn in this manner are usually overgeneralizations.

Teenagers are given many speeding tickets.
therefore
All teenagers speed.

In this example, the premise is built upon a certainty; however, it is not one that leads to the conclusion. Not every teenager observed has been given a speeding ticket. In other words, unlike "The sun rises every morning", there are already plenty of examples of teenagers not being given speeding tickets. Therefore the conclusion drawn can easily be true or false (perhaps more easily false than true in this case), and the inductive logic does not give us a strong conclusion. In both of these examples of weak induction, the logical means of connecting the premise and conclusion (with the word "therefore") are faulty, and do not give us a strong inductively reasoned statement.

## 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 false, but may have been thought true in Europe until the settlement of Australia, when Black Swans were discovered. 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 induction there are always many conclusions that can reasonably be related to certain premises. Inductions are open; deductions are closed. It is however possible to derive a true statement using inductive reasoning if you know the conclusion. The only way to have an efficient argument by induction is for the known conclusion to be able to be true only if an unstated external conclusion is true, from which the initial conclusion was built and has certain criteria to be met in order to be true (separate from the stated conclusion). By substitution of one conclusion for the other, you can inductively find out what evidence you need in order for your induction to be true. For example, you have a window that opens only one way, but not the other. Assuming that you know that the only way for that to happen is that the hinges are faulty, inductively you can postulate that the only way for that window to be fixed would be to apply oil (whatever will fix the unstated conclusion). From there on you can successfully build your case. However, if your unstated conclusion is false, which can only be proven by deductive reasoning, then your whole argument by induction collapses. Thus ultimately, pure inductive reasoning does not exist. Deductive reasoning is the kind of reasoning in which the conclusion is necessitated by, or reached from, previously known facts (the premises). ... Binomial name Cygnus atratus Latham, 1790 Subspecies Black Swan New Zealand Swan (extinct) Synonyms Anas atrata Latham, 1790 Chenopis atratus The Black Swan, Cygnus atratus is a large non-migratory waterbird which breeds mainly in the southeast and southwest of Australia. ... In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion. ... An argument is cogent if and only if the truth of the arguments premises would render the truth of the conclusion probable (i. ...

The classic philosophical treatment of the problem of induction, meaning the search for a justification for inductive reasoning, was by the Scottish philosopher 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 done so in the past, but this is not a guarantee that it will always do so. As Hume said, someone who insisted on sound deductive justifications for everything would starve to death. The problem of induction is the philosophical issue involved in deciding the place of induction in determining empirical truth. ... This article is about the Scottish as an ethnic group. ... David Hume (April 26, 1711 â€“ August 25, 1776)[1] was a Scottish philosopher, economist, and historian. ...

Instead of approaching everything with unproductive skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted. This article or section may contain original research or unverified claims. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This article or section does not cite its references or sources. ...

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 has been observed. Inferences about the past from present evidence – for instance, as in archaeology, count as induction. Induction could also be across space rather than time, for instance as in physical cosmology where conclusions about the whole universe are drawn from the limited perspective we are able to observe (see cosmic variance); or in economics, where national economic policy is derived from local economic performance. Archaeology, archeology, or archÃ¦ology (from the Greek words Î±ÏÏ‡Î±Î¯Î¿Ï‚ = ancient and Î»ÏŒÎ³Î¿Ï‚ = 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. ... Physical cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ... Cosmic variance is the idea that we are only able to observe one universe at one particular time, so it is difficult to make statistical statements about cosmology on the scale of the entire universe. ... This article or section does not cite its references or sources. ...

Twentieth-century philosophy has approached induction very differently. Rather than a choice about what predictions to make about the future, induction can be seen as a choice of what concepts to fit to observations or of how to graph or represent a set of observed data. Nelson Goodman posed a "new riddle of induction" by inventing the property "grue" to which induction does not apply. Nelson Goodman (7 August 1906, Somerville, Maryland â€“ 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

Sources for the examples that follow are: (1), (2), (3).

### Generalization

A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population: A sample is that part of a population which is actually observed. ...

The proportion Q of the sample has attribute A.
therefore
The proportion Q of the population has attribute A.

How great the support which the premises provide for the conclusion is dependent on (a) the number of individuals in the sample group compared to the number in the population; and (b) the randomness of the sample. The hasty generalization and biased sample are fallacies related to generalization. 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, unrepresentative sample 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:

A proportion Q of population P has attribute A.
An individual I is a member of P.
therefore
There is a probability which corresponds to Q that I has A.

The proportion in the first premise would be something like "3/5ths of", "all", "few", etc. Two dicto simpliciter fallacies can occur in statistical syllogisms: "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.

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.

This is 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 common to both things: Analogy is either the cognitive process of transferring or giving information from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. ...

P is similar to Q.
P has attribute A.
therefore
Q has attribute A.

An analogy relies on the inference that the properties known to be shared (the 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. The fallacy related to this process is false analogy. False analogy is a logical fallacy applying to inductive arguments. ...

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

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.

### Argument from authority

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

Proportion Q of the claims of authority A have been true.
therefore
There is a probability corresponding to Q that this claim of A is true.

For instance:

All observed claims from websites about logic are true.
Information X came from a website about logic.
therefore
Information X is likely to be true.

## Bayesian inference

Of the candidate systems for an inductive logic, the most influential is Bayesianism. This uses probability theory as the framework for induction. Given new evidence, Bayes' theorem is used to evaluate how much the strength of a belief in a hypothesis should change. 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. ... Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ... Bayess theorem (also known as Bayess rule or Bayess law) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. ...

There is debate around what 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 prior probabilities represent subjective degrees of belief, but that the 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 produce a rational justification for a belief in some hypothesis, but at the expense of rejecting objectivism. Such a scheme cannot be used, for instance, to decide objectively between conflicting scientific paradigms. Objectivity, as a concept of philosophy, is dependent upon the presupposition distinguishing references in the field of epistemology regarding the ontological status of a possible objective reality, and the state of being objective in regard to references towards whatever is considered as objective reality. ...

Edwin Jaynes, an outspoken physicist and Bayesian, argued that "subjective" elements are present in all inference, for instance in choosing axioms for deductive inference; in choosing initial degrees of belief or prior probabilities; or in choosing likelihoods. He thus sought principles for assigning probabilities from qualitative knowledge. Maximum entropy – a generalization of the principle of indifference – and transformation groups are the two tools he produced. Both attempt to alleviate the subjectivity of probability assignment in specific situations by converting knowledge of features such as a situation's symmetry into unambiguous choices for probability distributions. Edwin Thompson Jaynes (July 5, 1922 â€“ April 30, 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. ...

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

Results from FactBites:

 Inductive reasoning - Wikipedia, the free encyclopedia (1934 words) Inductive reasoning is the complement of deductive reasoning. 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. Inductive arguments are never binding but they may be cogent.
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