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Inductive Reasoning

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

(1)
  1. All Martians have green skin. Therefore, it is reasonable to conclude that the Martians in this spaceship have green skin.
  2. Most Martians have green skin. Therefore, it is reasonable to conclude that Xilo the Martian most likely has green skin.
  3. 99.7% of Martians have green skin. Therefore, it is reasonable to conclude that there is a 0.997 probability of Xilo the Martian having green skin.
(2)
  1. 995 Martians in our representative random sample of 1,000 Martians have purple eyes. Therefore, it is reasonable to conclude that 99.5% of Martians have purple eyes.
  2. Every Martian in our large representative random sample of Martians has purple eyes. Therefore, it is reasonable to conclude that all Martians are likely to have purple eyes.
  3. Every Martian in our large representative random sample of Martians has purple eyes. Therefore, it is reasonable to conclude that all Martians have purple eyes.
  4. Every Martian in our sample of Martians has purple eyes. Therefore, until we find Martians with non-purple eyes, it is reasonable to conclude that all Martians have purple eyes.

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 ().