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,

 

[HallsofIvy]

You say "if Cleopatra was powerful then she was venerated" and "if Cleopatra was not powerful then whe was feared.

Okay, so whatever was the case about being powerful, Cleopatra was either venerated or feared.

But you also say "If Cleopatra was either venerated or feared, then she was a queen."

Okay, so it follows that Cleopatra was a queen but none of the others.

 

[Shaitan00]

I think I get it - you can't prove that she was Powerful neither can you prove that she wasn't correct? I think that is where most of my confusion came from - thinking that I had to either prove she was or wasn't when I only need to state that I can't prove it either way...

Do you know if there is a way to prove this using the P,V,F,Q,L clauses somehow?

Also - from your deducation - if she is a Queen (as you stated) then doesn't that also imply that she was a Leader given clause #4 (If she is a Queen then she is a leader = Q->L)? So that would prove Q and L - but not P (which is what I expected). Does that sound right?

 

[HallsofIvy]

Yes, you are right- I missed that. You can prove, from what you are given that Cleopatra was a queen and a leader.

 

[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,