# Insecurity of Inductive Reasoning

• November 10th 2008, 09:34 AM
espen180
Insecurity of Inductive Reasoning
I have not encountered this in any math book, but I guess this is where to ask.

A generalisation (more accurately, an inductive generalisation) proceeds from a premise about a sample to a conclusion about the population.

The proportion Q of the sample has attribute A.
Therefore:
The proportion Q of the population has attribute A.

How great the support which the premises provide for the conclusion is dependent on (a) the number of individuals in the sample group compared to the number in the population; and (b) the randomness of the sample.

If we know the proportion $Q$, the sample/population ratio $P$ and the randomness $R$ (where R=0 means 0% randomness and R=1 means 100% randomness), would it be possible to calculate/find a formula for the insecurity $I$ of the inductive reasoning above, if we measure the insecurity in the interval [0,1], where I=0 means 100% secure and I=1 means 0% secure?

The formula for $I$ would most likely have an overall factor $R$, I think.

EDIT:

I just realized that Q and P are the same. That means that the forumula will most likely be $I=R\cdot Q$, right?
• November 10th 2008, 11:16 AM
CaptainBlack
Quote:

Originally Posted by espen180
I have not encountered this in any math book, but I guess this is where to ask.

A generalisation (more accurately, an inductive generalisation) proceeds from a premise about a sample to a conclusion about the population.

The proportion Q of the sample has attribute A.
Therefore:
The proportion Q of the population has attribute A.

How great the support which the premises provide for the conclusion is dependent on (a) the number of individuals in the sample group compared to the number in the population; and (b) the randomness of the sample.

If we know the proportion $Q$, the sample/population ratio $P$ and the randomness $R$ (where R=0 means 0% randomness and R=1 means 100% randomness), would it be possible to calculate/find a formula for the insecurity $I$ of the inductive reasoning above, if we measure the insecurity in the interval [0,1], where I=0 means 100% secure and I=1 means 0% secure?

The formula for $I$ would most likely have an overall factor $R$, I think.

EDIT:

I just realized that Q and P are the same. That means that the forumula will most likely be $I=R\cdot Q$, right?

This is a parody of the way statistics is done. We have a sample with proportion Q, from this we either construct a confidence interval which contains the true proportion with a specified probability (frequentist statistics), or we construct a posterior distribution for the true proportion (Baysian statistics).

From these we may derive a point estimate, but it is associated with an uncertainty interval or equivalent with an interpretation that depends on your statistical school (but usually comparable).

CB