Induction (logic)

The inference of general laws from specific observations. In other words: inferring from the specific to the general.

• Ἐπᾰγωγή [Epagogé ]

Whereas a deduction is an inference from a general rule to a specific situation, in an induction a general rule is inferred to from specific observations.

As inductive inferences generally aren’t necessary true but only with some certainty, we normally speak of them not as „valid“ or „invalid“ but rather as „sound“ or „unsound“.

In enumerative induction, a generalizing rule is derived from a large number of observations. This is by far the most common form of induction:

For example:

All people born more than about 120 years ago have died.
It follows that all people die at some point.

This allows to form a general rule like: “all humans are mortal”.

Conclusions of this form often make pragmatic sense, but they can also lead to mistakes, as the following example shows:

Every swan (that I have seen so far) is white.
So all swans are white.

Obviously, this does not take into account that black swans are rare but undoubtedly exist. This can have various causes:

• If one considers too small a sample of the swan population, one may, for example, fall into the pitfall of hasty generalisation – in fact, black swans do exist (albeit very rarely in our part of the world), thereby disproving the rule.

• A hidden sampling bias during observation can can result in certain instances not being considered. In this case, for example, only swan species native to our region were observed, ignoring the black swan species, which is native to Australia.

A method for mathematical proofs that allows to prove the correctness of a statement for an infinitely large set of values.

In principle, the following proofs must be provided:

• There is an initial value 𝑛₀ for which the statement is valid;
• For every value 𝑛 for which the statement is valid, it is also true that for 𝑛+1 the statement is valid.

Figuratively, mathematical induction is often compared to a row of domino stones, where it is sufficient to ensure that each stone will push over the following one, and then to push the first stone so that they all fall.

Because of the strong formal requirements, this form of induction is only suitable for proofs in formal systems, hence the name (see: Mathematical induction)

The term “Mill’s Methods” refers to a collection of five methods of induction described by John Stuart Mill in his work “A System of Logic” (1843).

Discussing these in detail would go beyond the scope of this brief introduction. For readers who wish to explore the subject in greater depth, we refer you to the Wikipedia article: Mill’s Methods.

Another important variant of inductive reasoning is called abduction. In this method, the aim is to find the best possible explanation for the phenomena observed. However, as the aim and methods of abduction differ in some important aspects from “typical” induction, they are explained in a separate article on this site: Abduction.

• Abduction
• Deduction

• Inductive reasoning on Wikipedia