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Monday, August 15, 2011

Inductive reasoning

Induction, also known as inductive reasoning or inductive logic, is a type of reasoning that involves moving from a set of specific facts to a general conclusion. It can also be seen as a form of theory-building, in which specific facts are used to create a theory that explains relationships between the facts and allows prediction of future knowledge. The premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; i.e. they do not ensure its truth. Induction is used to ascribe properties or relations to types based on an observation instance (i.e., on a number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is employed, for example, in using specific propositions such as:

This ice is cold. (or: All ice I have ever touched was cold.)

This billiard ball moves when struck with a cue. (or: Of one hundred billiard balls struck with a cue, all of them moved.)

...to infer general propositions such as:

All ice is cold.

All billiard balls move when struck with a cue.

Another example would be:

3+5=8 and eight is an even number. Therefore, an odd number added to another odd number will result in an even number.

Note that mathematical induction is not a form of inductive reasoning. While mathematical induction may be inspired by the non-base cases, the formulation of a base case firmly establishes it as a form of deductive reasoning.

Strong and Weak Induction

Strong induction

All observed crows are black.


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 falsify the possibility of crows of another colour, the statement (conclusion) may actually be false.

For example, one could examine the bird's genome and learn whether it is capable of producing a differently coloured bird. In doing so, we could discover that albinism is possible, resulting in light-coloured crows. Even if you change the definition of "crow" to require blackness, the original question of the colour possibilities for a bird of that species would stand, only semantically hidden.

A strong induction is thus an argument in which the truth of the premises would make the conclusion probable, but not necessarily guarantee it as being factual.

Weak induction

I always hang pictures on nails.


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 generalisation 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 hung, 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 overgeneralisations.

Many speeding tickets are given to teenagers.


All teenagers drive fast.

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 is false. Moreover, when the link is weak, 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.

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