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Math Help - Hilbert System Proof. ( p /\ q ) -> (q /\ p )

  1. #1
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    Hilbert System Proof. ( p /\ q ) -> (q /\ p )

    Hey guys, as the title describes i need some help for this proof!

    I have to use these operations and i "just" need to find the right part of the proof.


    1. {q -> ~p, ~~p} l- _______ ? ? ? Assumption
    2. {q -> ~p, ~~p} l- _______ ? ? ? Thereom 3.25
    3. {q -> ~p, ~~p} l- _______ ? ? ? MP 1,2
    4. {q -> ~p, ~~p} l- _______ ? ? ? Assumption
    5. {q -> ~p, ~~p} l- _______ ? ? ? MP 3,4
    6. {q -> ~p, ~~p} l- _______ ? ? ? Thereom 3.23
    7. {q -> ~p, ~~p} l- _______ ? ? ? MP 5,6
    8. {q -> ~p} l- _______ ? ? ? Dedcution 7
    9. {q -> ~p} l- _______ ? ? ? Contrapositive 8
    10. {q -> ~p} l- _______ ? ? ? Contrapositive 9
    11. l- _______ ? ? ? Deduction 10
    12. l- _______ ? ? ? Thereom 3.25
    13. l- _______ ? ? ? MP 11,12
    14. l- ( p /\ q ) -> (q /\ p ) Definition of /\
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  2. #2
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    Re: Hilbert System Proof. ( p /\ q ) -> (q /\ p )

    Anyone else ? :-D
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  3. #3
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    Re: Hilbert System Proof. ( p /\ q ) -> (q /\ p )

    Quote Originally Posted by Mathic View Post
    Anyone else ? :-D
    Please do not bump!
    You have posted a question that seems to be textbook specific.
    How are we expected what theorem 3.25 is? We have no way to know.
    So why do you expect help?
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  4. #4
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    Re: Hilbert System Proof. ( p /\ q ) -> (q /\ p )

    Plato is right: you need to say what Theorems 3.23 and 3,25 are and what is the definition of /\. I'll venture a guess that p /\ q is ~(p -> ~q), Theorem 3.23 is q -> ~~q and Theorem 3.25 is (p -> q) -> (~q -> ~p). Then the derivation would be as follows.

    1. {q -> ~p, ~~p} l- q -> ~p Assumption
    2. {q -> ~p, ~~p} l- (q -> ~p) -> (~~p -> ~q) Thereom 3.25
    3. {q -> ~p, ~~p} l- ~~p -> ~q MP 1,2
    4. {q -> ~p, ~~p} l- ~~p Assumption
    5. {q -> ~p, ~~p} l- ~q MP 3,4
    6. {q -> ~p, ~~p} l- ~q -> ~~~q Thereom 3.23
    7. {q -> ~p, ~~p} l- ~~~q MP 5,6
    8. {q -> ~p} l- ~~p -> ~~~q Deduction 7
    9. {q -> ~p} l- ~~q -> ~p Contrapositive 8
    10. {q -> ~p} l- p -> ~q Contrapositive 9
    11. l- (q -> ~p) -> (p -> ~q) Deduction 10
    12. l- ((q -> ~p) -> (p -> ~q)) -> (~(p -> ~q) -> ~(q -> ~p)) Thereom 3.25
    13. l- ~(p -> ~q) -> ~(q -> ~p) MP 11,12
    14. l- ( p /\ q ) -> (q /\ p ) Definition of /\
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