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# Logic

Problem:

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.

Can you prove that Cleopatra was powerful? a leader? a queen?

Do not use proof by resolution-refutation.

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I have broken the text into these propositional statements (~ is a not)

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

1. P -> V
2. ~P -> (~V and F)
3. (V or F) -> Q
4. Q -> L

I started trying to prove that she was powerful and either I am missing something or it cannot be proved?
Can someone give me a hand proving she was powerful, a leader , a queen?

#### Solution Preview

Good start!

Here are the statements that we have:

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

Here is what the problem says:

1. If Cleopatra was powerful, then she was venerated. P --> V
2. If she was not powerful, then she was not venerated and she was feared ~P --> (~V and F)
3. If Cleopatra was either venerated or feared, then she was a queen (V or F) --> Q
4. If she was a queen, then she was a leader. Q --> L

Just to wrap my mind around this, I'm going to set up some truth tables (let me know if you haven't gone over these in your class yet).

1. If Cleopatra was powerful, then she was venerated. P --> V

P &#8594; V

T T T
T F F
F T T
F T F

This will be a true statement in these cases:

She is powerful and she is venerated.
She is not powerful

2. If she was not powerful, then she was ...

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