Table 15
| Possible Combination of Statuses | Status 1, Eligible | Status 2, Applied | Status 3, Application Approved | Status 4, Examination Required | Status 5, Insured |
|---|---|---|---|---|---|
| 1 | Yes | Yes | Yes | Yes | No |
| 2 | Yes | Yes | Yes | No | No |
| 3 | Yes | Yes | No | Yes | No |
| 4 | Yes | No | Yes | No | No |
| 5 | No | No | No | No | No |
The question may be asked why employees who are eligible, who have applied for insurance, who have had their applications approved, and who require no medical examination (combination 2) are yet not insured. The answer is that the rules given do not logically lead to this conclusion. As a matter of fact, there might be additional rules, such as: any sick employee must first return to work; or any period from date of approval of application to the first of the following month must first pass.
The first step in putting this problem on the Kalin-Burkhart Logical-Truth Calculator is to rephrase the rules, using the language of the connectives that we have on the machine. The rules rephrased are:
A. If an employee is insured, then he is eligible, he has applied for insurance, and his application has been approved.
if 5, then 1 and 2 and 3
B. If an employee has applied (under these rules) for insurance, then he is eligible.
if 2, then 1
C. If an employee is eligible for insurance, has applied, and requires no medical examination, his application is automatically approved.
if 1 and 2 and not-4, then 3
D. If an employee’s application has been approved, then he has applied.
if 3, then 2
E. If an employee is not eligible, then he does not require a medical examination (under these rules).
if not-1, then not-4
To get the answer we seek, we must add one more rule for this answer only:
F. The employee is not insured.
not-5
We now have a total of 4 + 2 + 4 + 2 + 2 + 1 occurrences of statements, or 15 occurrences. This is beyond the capacity of the existing machine. But fortunately Rule F and Rule A cancel each other; they may both be omitted; and this gives us 10 occurrences instead of 15. In other words, all the possible statuses under “Rule B and Rule C and Rule D and Rule E” will give us the answer we seek.
The rephrasing and reasoning we have done here is perhaps not easy. For example, going from the logical pattern
Only igs may be ows to the logical pattern
If it is an ow, then it is an ig as we did in rephrasing Rule B, deserves rather more thought and discussion than we can give to the subject here. A person who is responsible for preparing problems for the Logical-Truth Calculator should know the algebra of logic.
Choosing an appropriate grouping, we now set on the machine:
{(if 2, then 1) and [IF (1 and 2) and not-4, then 3]} and
[(if 3, then 2) and (IF not-1, then not-4)]
The setting is as shown in Table 16. After this setting, the machine is turned on and set to stop on the “true” cases. The