Boolean representation of logical statements of more than two extreme conditions I had a question to determine the highest and and lowest and the middle paid of three employees. I tried to solve the problem using logical values T or F, then getting the connection between the truth tables, but I got stuck since the first statement concerns the highest and the second concerns the lowest paid and there is a hole about the middle one that I could't represent, using truth values since not highest may be lowest or middle and vice versa. The question is below from Rosen discrete math book.

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 the 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? Explain your reasoning.

 A: OK, claim $1$: if Fred is not the highest paid, then Janice is. This means Maggie is not the highest paid (because then claim $1$ becomes $True\implies False$ which cannot be).
Claim $2$: If Janice is not the lowest paid, then Maggie is paid the most. However, we know Maggie is not paid the most, so Janice is the lowest paid (because $A\implies B$ is the same as $\neg B\implies \neg A$, so we know if Maggie is not paid the most, then Janice is the lowest paid, and we know Maggie is not paid the most. Therefore, Janice is the lowest paid).
Now, since Janice is the lowest paid, we know (because Maggie is not the highest paid) that Fred must be the highest paid. So its Fred>Maggie>Janice.

Brute force:
Claim one says: "If $F$ is not the highest, then $J$ is the highest.
Claim two says: "If $J$ is not the lowest, then $M$ is the highest.
There are only $6$ possible ways this goes:

*

*F>M>J  -  This option is possible


*F>J>M  - This option is not possible because it conflicts with claim $2$.


*M>F>J  - This option conflicts with claim $1$


*M>J>F  - This option conflicts with claim $1$


*J>M>F  - This option conflicts with claim $2$


*J>F>M  - This option conflicts with claim $2$
So the only possible option is F>M>J.
A: 
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 the three, then Janice is. 

$$(F{\lt}J\vee F{\lt}M)\to (F{<}M{<}J\vee M{<}F{<}J)\tag 1$$
Clearly that means that: $$F{<}J\to (F{<}M{<}J\vee M{<}F{<}J) \tag{1.1}$$ ... and, less obviously, that: $$F{<}M\to (F{<}M{<}J)\tag {1.2}$$

Second, he knows that if Janice is not the lowest paid, then Maggie is paid the most. 

$$(F{<}J\vee M{<}J)\to (F{<}J{<}M\vee J{<}F{<}M) \tag 2$$
Likewise, this infers that: $$F{<}J\to F{<}J{<}M \tag{2.1}$$ ... and also that: $$M{<}J\to \bot\tag{2.2}$$

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? Explain your reasoning.

Fred cannot be paid less than Jannice (1.1 and 2.1 contradict), and Fred cannot be paid less than Maggie (1.2 and 2.1 contradict), and Maggie cannot be paid less than Jannice (2.2).   Which means Fred is paid the most, and Jannice the least. $$J{<}M{<}F\tag{done}$$
