Steve would like to determine the relative salaries of three coworkers using two facts:
- First, he knows that if Fred is not the highest paid of three, then Janice is.
- Second, he knows that if Janice is not the lowest paid,, then Maggie is paid the most.
Is it possible to determine the relative salaries of Fred, Maggie and Janice from what Steve knows? If so, who is paid the most and who the least?
Fact 1 says that if Fred is not the highest paid, then Janice is. This results in two situations:
- Fred is not the highest paid
- Janice is the highest paid.
- Then by fact 2, Janice is not the lowest pad hence Maggie is the most paid.
- Which contradicts with established truth that Janice is highest paid.
- Hence, this case is incorrect and results in second condition.
- Fred is the highest paid
- Now Maggie can't be the most paid, because Fred is the most paid. Hence, the second statement is false.
- Which means that Janice is lowest paid.
This establishes the relative order Janice < Maggie < Fred
Solution using propositions to Find the Relative Salary
In the previous puzzle "Identify the Murder"
, we solved the puzzle with reasoning without formal methodologies. Here we will try to find our way through the propositional calculus and arrive to a situation.
Let us consider three statements:
- F denotes that, "Fred is highest paid."
- J denotes that, "Janice is lowest paid."
- M denotes that,"Maggie is the most paid."
Then the first fact says that, if NOT F then NOT J AND NOT M. This can be expressed using propositions as ¬F → (¬J ∧ ¬M).
The second fact says that, if NOT J then M. This can be expressed using propositions as ¬J → M.
We can use these two propositions to check for consistency. For this purpose we can use a truth table as shown below:
The last two columns of the truth table shows the two facts we know. Also, both the facts are true only in the last three rows. Let us now examine the last three rows of the table separately.
Looking at the first three columns, shows the following:
- Row 1 & Row 3 has F and M true, which means Fred is highest paid and Maggie is most paid. But both cannot be highest paid, hence these two rows do not satisfy the propositional consistency.
- Whereas Row 2 is not contradicting the proposition.
So, the solution to this problem would be when F is true, J is true and M is false. Which means
- Fred is highest paid
- Janice is lowest paid and
- Maggie is not the most paid.
Hence the relative order of their salaries would be Janice < Maggie < Fred.