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# Induction (Inductive reasoning)

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Induction is historically the name used to signify a kind of reasoning that proposes to look for general laws from the observation of particular facts, on a probabilistic basis. At present, high school geography curricula involve representative case studies of inductive reasoning. For example, generalizations about the problem of global nutrition are based on the study of a particular case in India or Ethiopia. This pedagogical operation based on the case study was previously implemented by the British.

Note: Although associated in the title of this article with logic, the following presentation corresponds mainly to the Bayesian notion, used consciously or not, of the induction. In Anglo-Saxon mathematics, logic, and computer science, complete induction, referred to as induction (false-friend), refers to recurrence, both in recurrence reasoning and in recursive definitions. The term is often used for the generalizations of recurrence to the right orders and well-founded relationships. In automated reasoning, abduction is a mode of reasoning that aims to emit a hypothesis to explain a fact and it should not be confused with the induction presented here.

### Definition

Induction is the inference leading from several particular affirmations to an affirmation. Aristotle explains that one starts from individual cases to access universal statements, and opposes it to deduction. Cicero explains: “Induction, by making us agree on obvious things, draws from these confessions the means of making us agree on doubtful things, but which have something to do with them.”

### History of the concept

The school of Socrates had adopted a manner of arguing which proceeds by induction; Theophrastus gave preference to the epimereme. The initial idea of ​​this conception of induction was that the repetition of a phenomenon increases the probability of seeing it reproduced. This is, for example, the way the brain reacts in classical conditioning. The accumulation of concordant facts and the absence of counterexamples then makes it possible to increase the level of plausibility of the law until, for the sake of simplification, we choose to consider it as a virtual certainty: thus it is the second principle of thermodynamics. One never reaches “complete” certainty; any appropriate counter-example may put “immediately” in question. Theorems such as that of Cox have given this inductive approach, at first empirical, a firm mathematical basis; they made it possible to calculate the probabilities concerned without any arbitrariness at a given starting position. But the previous definition is quite improper. For example, we could say that “this table is heavy, so this table is heavy” is an example of induction, but in this case, it is not a question of seeking a “general law” from of a particular fact. More recently, “induction” has come to mean a kind of reasoning that does not ensure the truth of its conclusion given the prerequisites. This reasoning is the opposite of deduction, which is a kind of reasoning where the conclusion can not be wrong, given the prerequisites.

### Types

#### Generalization

A generalization (more precisely, inductive generalization) proceeds from a premise on a sample to a conclusion about a population.

The proportion Q of the sample has attribute A.
Therefore :
The proportion Q of the population has attribute A.

Example:

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

#### Statistical syllogism

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

A proportion Q of the population P has the attribute A.
An individual X is a member of P.
Therefore :
There is a probability, which corresponds to Q, that X has A. The proportion in the first premise would be something like “three-fifths of”, “all”, “some”, and so on. Two sophisms dicto simpliciter can occur in statistical syllogisms: “accident” and “conversed accident”.

#### Simple induction

Simple induction proceeds from a principle of a sample group to a conclusion about another person.

The proportion Q of known cases of the population P has the attribute A.
Individual I is another member of P.
Therefore :
There is a probability corresponding to Q that I possesses A.

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

#### Argument of an analogy

The process of analog inference involves noting the shared properties of two or more things, and from this inference base they also share another property:

P and Q are similar with respect to properties a, b, and c .
The object P has been observed to have the property x in addition.
Therefore, Q probably has the property x too.

Analogical reasoning is very common in common sense, in science, philosophy and human sciences, but sometimes it is accepted only as an auxiliary method. A refined approach is case-by-case reasoning.

#### Causal inference

A causal inference draws on a conclusion of causality based on the conditions of the occurrence of an effect. The premise of the correlation between the two things may indicate a causal relationship between them, but other factors must be taken into account to establish the exact form of the cause-and-effect relationship.

#### Prediction

A prediction draws a conclusion on an individual future from a past sample.

The proportion Q of observed members of group G had attribute A.
Therefore :
There is a probability corresponding to Q that other members of group G have attribute A during the next observation.

### Example

For example: If the law of universal gravitation determines that, and how, an apple that is detached from its tree will fall on the ground, the observation of the movement of this same apple makes it possible to establish the general law, but with a probability or a very low certainty. If then we observe that all apples and bodies fall in the same way, if we observe that the bodies in space respect the same law, then the probability of the law will increase to become almost a certainty. In the case of universal gravitation, however, it has been observed that the orbit of Mercury exhibited a precessional effect that was not explained by law. The law of universal gravitation, however, remained universally valid until Albert Einstein developed the theory of general relativity, which explains among other things the precession of the orbit of Mercury. Nevertheless, universal gravitation is still used because it remains valid in common cases, and it is easier to use and understand than the theory of relativity.