Induction (logic)

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

Other names

• Ἐπᾰγωγή [Epagogé ]

Description

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“.

Enumerative induction

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 a too small part of the swan population is observed, one can make the mistake of hasty generalisation – a single black swan sighting can then disprove the general rule.
• Pre-selection during observation means that certain cases are not considered - in this case, only swan species native to a specific region were observed, not, for example, the Cygnus atratus or “black swan” species native to Australia.

Mathematical induction

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)

Other forms of induction

There are various other forms of inductions. Most famously, John Stuart Mill’s collection of five different methods of induction, known as “Mill’s Methods”.