2

Then we would have to write out the definitions of the characteristics. For
example, we might give the following definitions for the characteristics we have
just presented:

A whole number is evenif and only if it is divisible by two.

A whole number is oddif and only if it leaves a remainder of one when
divided by two.

A whole number is a multiple of sevenif and only if it is divisible by seven.

A whole number is a perfect squareif and only if it is the product of a whole
number with itself.

Next, we would have to decide a rule for ordering the definitions in the list. That
is, how would we decide which to list first, or which to list second, and so on?

A simple rule for ordering the definitions is based on the number of letters in a
definition. Thus, we count the number of letters in all the definitions and assign
position number 1 to the definition with the smallest number of letters. We assign
position number 2 to the definition with the next smallest number of letters, and
proceed in this way to assign positions to all of the definitions. If two or more
definitions have the same number of letters, we assign their positions on the basis
of the alphabetical order of the letters in each. Therefore, each definition will have
its own position number.

Exercise 1:

Suppose we use the above rule to order the four definitions given
earlier.What are their position numbers?

Exercise 2:

Think of another characteristic of whole numbers and make up a
definition for it. Then assign it a position number in the list of five
definitions and reorder the other definitions as necessary.

Now, we note the following. For a given position number, either it possesses the
characteristic given in the definition to which it corresponds or it does not. For
example, suppose 17 has been assigned to the definition of divisible by five. Then
17 does nothave the characteristic described. On the other hand, if 14 has been
assigned to the definition divisible by seven, then 14 doeshave this characteristic.

Definition:

A position number that does notpossess the characteristic described
in the definition to which it corresponds is called Richardian.

In the above examples, 17 is Richardian and 14 is not Richardian.