THE SIMPLEST LOGICAL PATTERNS

Many simple logical patterns are so familiar that we often use them without being conscious of doing so. The simple logical patterns are marked by words like “and,” “or,” “else,” “not,” “if,” “then,” “only.” In the same way, simple arithmetical patterns are marked by words like “plus,” “minus,” “times,” “divided by.”

Let us see what some of these simple logical patterns are. Suppose that we take two statements about which we have no factual information that might interfere with logical supposing:

In practice, we might be concerned with such statements when writing the rules governing a plan of insurance for a group of employees. Here, we shall play a game:

(1) We shall make up some new statements from statements 1 and 2, using the words “and,” “or,” “else,” “not,” “if,” “then,” “only.”

(2) We shall examine the logical patterns that we can make.

(3) We shall see what we can find out about their logical truth.

Suppose we make up the following statements:

3. John Doe is not eligible for insurance.

4. John Doe does not require a medical examination.

5. John Doe is eligible for insurance and requires a medical examination.

6. John Doe is eligible for insurance, and John Doe is eligible for insurance.

7. John Doe is eligible for insurance, or John Doe requires a medical examination.

8. If John Doe is eligible for insurance, then he requires a medical examination.

9. John Doe requires a medical examination if and only if he is eligible for insurance.

10. John Doe is eligible for insurance or else he requires a medical examination.

Now clearly it is troublesome to repeat quantities of words when we are interested only in the way that “and,” “or,” “else,” “not,” “if,” “then,” “only” occur. So, let us use just 1 and 2 for the two original statements, remembering that “1 and 2” means here “statement 1 AND statement 2” and does not mean 1 plus 2. Then we have:

3:not-1
4:not-2
5:1 and 2
6:1 and 1
7:1 or 2
8:if 1, then 2
9:1 if and only if 2
10:1 or else 2

Here then are some simple logical patterns that we can make.