Can you prove that P, Q, or L [Propositional Logics]

[Shaitan00]

Homework Statement

I was given the following text:
If Cleopatra was powerful, then she was venerated but if she was not powerful, then she was not venerated and she was feared. If Cleopatra was either venerated or feared, then she was a queen. Cleopatra was a leader if she was a queen.

P = Cleopatra was Powerful
V = Cleopatra was Venerated
F = Cleopatra was Feared
Q = Cleopatra was a Queen
L = Cleopatra was a Leader

I am being asked if I can prove that Cleopatra was Powerful? A Leader? A Queen? (without using resolution-refutation).

Homework Equations

Propositional clauses:
1. P -> V
2. !P -> (!V and F)
3. (V or F) -> Q
4. Q -> L

CNF Format (shouldn’t be needed but incase):
1. ! P or V
2a. (P or !V)
2b. (P or F)
3a. (!V or Q)
3b. (!F or Q)
4. !Q or L

The Attempt at a Solution

From here I was able, with resolution-refutation, to determine that we cannot prove P but we should be able to prove Q and L… After that I am completely stuck on how to proceed as I am not allowed to prove the question with that approach – only to help me see what answers I should get…

I assume I must either use Forward-Chaining or Backward-Chaining to solve the problems – but no knowledge is given, only implications – so how is one supposed to use either? In all my readings usually we would be given something like F=True (knowledge) or something similar and the chaining would come down to that – but with only implications I can’t see how anything can be proven…

All my attempts (and there have been many) have only added to my confusion.
Any help/hints would be greatly appreciated.
Thanks,

 

[wildman]

Keep in mind that (P or !P) is always true. What does that say about 1, about 2? One has to be true, right? What result do you get if 1 is true? how about 2? How about V and F? Does one of them always have to be true? If so what does that say about Q? how about L?

 

[Shaitan00]

Is it accurate to say the following:

The case contradicting P is if Cleopatra is neither powerful, venerate or feared, then the premises don't imply that Cleaopatra is powerful

Thanks,

 

[wildman]

No.

You don't know if P is true or not. All you know is that if P is true then V is true and if P is false then V is false and F is true. You get that from the two equations: P -> V and the equation !P -> (!V and F). However by the rules of propositional logic you know that either P is true or !P is true. This means that either V is true or F is true.

So since you know that V is true or F is true, what do you know about Q and L?