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

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

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).

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

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,

2. Originally Posted by Shaitan00
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).

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

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,
you already asked this here. please wait for your question to be answered there. double posting is against the rules. no matter what forum you post in, people will see it, so don't worry

3. we can prove she was a queen and a leader from the clauses given. we don't know if she was powerful or not. were you given any premises?

we can say she was a queen and a leader because she has to be powerful or not. either way, she is either feared or venerated, and once she is either of those, she is a queen. and once she's a queen, she is a leader.... was that what you were looking for?

4. 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?

5. Originally Posted by 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?
do it by cases

we can't tell if she was powerful or not, so account for both possibilities

case 1: say she was powerful, show that this implies she was a queen and a leader

case 2: say she wasn't powerful, show that this again implies she was a queen and a leader.

6. This might be a stupid question but - is there anyway to prove that we cannot prove that she was powerful (P) other then simply stating it? I mean I completly agree/understand but I am scared that I might be asked to prove how come we cannot prove (P)...

And no - no other information was given (no premises).

7. Originally Posted by Shaitan00
This might be a stupid question but - is there anyway to prove that we cannot prove that she was powerful (P) other then simply stating it? I mean I completly agree/understand but I am scared that I might be asked to prove how come we cannot prove (P)...

And no - no other information was given (no premises).
within the context of this question, i don't see how we can prove P. there is not enough information. we have to be given some kind of premise to tell us that. you may note that (V or !F) => P, but good luck with trying to prove V or !F, also, !V => !P, but again, proving !V is another problem...

8. I have no problems not proving P - actually I don't think it is possible. But I need to show that P is not provable somehow - meaning if this was resolution-refutation I would end with a non-empty set which means not-probable... I was only wondering if there was a way to prove that P was not provable...
(wow that sounds wrong...)

9. 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,

10. I can only prove she was queen and leader

$\displaystyle \begin{array}{llcr} 1 & P -> V && premise \\ 2 & !P -> (!V ~ and ~ F) && premise\\ 3 & (V ~ or ~ F) -> Q && premise\\ 4 & Q -> L && premise\\ \hline\\ 5a & !Q && assume\\ 6a & !(V ~ or ~ F) && MT ~3,5\\ 7a & !V ~ and ~ !F && DeM ~6\\ 8a & !V && Simp ~7\\ 9a & !F && Simp ~7\\ \end{array}$
$\displaystyle \begin{array}{rlcr} 10a & !P && MT ~1,8\\ 11a & !F ~ or ~ V && Add ~9\\ 12a & !(F ~ and ~ !V) ~ \mbox{ } ~ && Dem ~11\\ 13a & P && MT ~2,12 \\ 14a & P ~ and ~ !P && Conj ~10,13\\ 15 & Q && RAA ~5-14\\ 16 & L && MP ~4, 15\\ \end{array}$

11. Originally Posted by 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,
i don't think that statement is correct, neither would i make a statement like that. i'd simply say (if i have to, i don't see why you're pushing this point): "given the current set of statements to use as premises, it is not possible to prove whether or not Cleopatra was powerful. it can be proven however, that she was a queen and a leader, by the following ..."