Compare the reasoning in (1) with that in (2):

In (1), we are given information about the population of Martians, and on the basis of this information, we arrive at a conclusion about a given sample of Martians. In (2), on the other hand, we are given information about a sample of Martians, and on the basis of this information, we arrive at a conclusion about the entire population of Martians. The latter type of reasoning is is called inductive reasoning. In contrast, the reasoning in (1a) is deductive reasoning. [It would seem natural to take (1b) and (1c) as probabilistic qualitative deduction and probabilistic quantitative deduction, but many logicians would be unhappy about this terminology.]

Given the properties of a population (a set), deductive reasoning allows us to infer the properties of a particular sample (a proper subset) of the population. Given the properties of a sample of a population (a proper subset), inductive reasoning allows us to infer the properties the population (the set) as a whole:

Deduction		Induction
Population		Sample
Sample		Population

In terms of Venn diagrams, the distinction between the two can be diagrammed as follows:

Deduction		Induction

Alternatively, we could say that deductive reasoning allows us to infer the particular from the general, while inductive reasoning allows us to infer the general from the particular.

Deduction		Induction
General		Particular
Particular		General

The reasoning in (8a) illustrates probabilistic quantitative induction (statistical reasoning) while (8b) illustrates probabilistic qualitative induction. (8c) and (8d) both illustrate non-probabilistic qualitative induction, but express caution of two different kinds.Quantitative probabilistic deduction comes under probability theory in mathematics, while quantitative probabilistic induction comes under statistics.

The most conservative rules of inductive inference are probabilistic:

Conservative induction

Given that: Q is true of a large representative random sample of X   it is reasonable to conclude that Q is likely to be true of X.

If we are willing take a risk, we could make the conclusion non-probabilistic as follows:

Radical induction

Given that: Q is true of a sample of X   until we find evidence to the contrary, it is reasonable to conclude that: Q is true of the population of X.

Contributed by K P Mohanan ().