Predicate Logic question (attempted, assistance required) I'm trying to work out a class exercise and I've got myself stuck. Any help would be appreciated. Thank you.
I am to use use predicate logic reasoning techniques to solve the following problem:
All academics who are computer scientists are programmers or mathematicians. Any logistician is a philosopher. Jack Jones is not a philosopher and he is not a programmer. Prove that if Jack Jones is a logistician, he is not a computer scientist.
Here's what I have done so far.
All academics who are computer scientists are programmers or mathematicians. 
All (∀) academics who are computer scientists (B) are programmers or mathematicians (C).
 ∀x[B(x) -> C(x)]
Any logistician is a philosopher.
Any (∀) logistician (F) is a philosopher (G).
(∃x)[F(x) -> G(x)]
Jack Jones (x) is not a philosopher (H) and he is not a programmer (J).
¬H(x) /\ ¬J(x)
Prove that: if Jack Jones is a logistician, he is not a computer scientist.


*

*If Jack Jones is not a programmer then he cannot be a computer scientist.

*Jack Jones is not a philosopher, however, not all philosophers are logisticians (or not all logisticians are philosophers).

*Meaning Jack Jones can be a logistician without being a philosopher and a computer scientist.

 A: Some of the information is irrelevant, but it can be proven as follows:


*

*Cx = x is a computer scientist

*Px = x is a programmer

*Lx = x is a logistician

*Fx = x is a philosopher

*j = Jack Jones


\begin{array}{l}
& \{1\} & 1. & \forall x[Lx \Rightarrow Fx] & \text{ Prem. }\\
& \{2\} & 2. & \neg Fj \land \neg Pj & \text{ Prem. }\\
& \{3\} & 3. & Lj & \text{ Assum. }\\
& \{1\} & 4. & Lj \Rightarrow Fj & \text{ 1 UE }\\
& \{1,3\} & 5. & Fj & \text{ 3,4 MP }\\
& \{6\} & 6. & Cj & \text{ Assum. }\\
& \{1,3,6\} & 7. & Fj \land Cj & \text{ 5,6 $\land$I }\\
& \{1,3,6\} & 8. & Fj & \text{ 7 $\land$E }\\
& \{1,3\} & 9. & Cj \Rightarrow Fj & \text{ 6,8 CP }\\
& \{2\} & 10. & \neg Fj & \text{ 2 $\land$E }\\
& \{1,2,3\} & 11. & \neg Cj & \text{ 9,10 MT }\\
& \{1,2\} & 12. & Lj \Rightarrow \neg Cj & \text{ 3,11 CP }\\
\end{array}
Explanation:


*

*Premise: For every x, if x is a logistician, x is a philosopher.

*Premise: Jack Jones is neither a philosopher nor a programmer.

*Let's assume that Jack is a logistician.

*From our premise on line 1, if Jack is a logistician, he is a philosopher.

*It follows from our assumption that he is a philosopher.

*Let's assume that Jack is also a computer scientist.

*Given these assumptions, Jack would be both philosopher and a computer scientists. (This step may seem redundant, but it's use by logicians to establish a connection between the propositions.)

*Jack is a philosopher. (Because of line 7, this assertion is now shown to depend on line 1, 3 and 6, whereas before it depended only on line 6. Again, it's used to establish a logical connection.)

*If Jack is a logistician, he is a philosopher. (Because line 7 is shown to depend on line 3, line 7 is considered to be a logical consequence of 3, establishing the implication. The assumption on 3 is discharged.)

*According to the premise on 2, Jack is not a philosopher; 

*Therefore Jack cannot be a logistician. (This is the application of Modus Tollens, together with the implication on line 9. If Jack were a logistician, line 9 says that he would also be a philosopher, contradicting the premise.)

*Based on the assumption on line 3, the conclusion is established: If Jack is a logistician, he is not a computer scientist. (The assumption on 3 is also discharged.) 

