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Math Help - Getting a closed truth tree while it should be open!

  1. #1
    Junior Member
    Joined
    Jul 2008
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    Athens, Greece
    Posts
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    Getting a closed truth tree while it should be open!

    Hello every one!
    It is about a propositional logic problem, I have tried it vey hard maybe... cause I am too tired...

    This is it: a^b-> c |- (a->c)^(b->c) which means that (a->c)^(b->c) is a Logical consequence of a^b-> c. Also means that this trunk of tree:
    a^b-> c -requirement
    ~[(a->c)^(b->c) ] -negative consequence

    should give an open tree because {a=t,b=c=f} and {a=f, b=t, c=f} make true this:
    ((a^b) -> c)^~((a->c)^(b->c)).

    This is the tree:
    a^b->c(1)
    ~((a->c)^(b->c))(2)
    ---------------------
    (3) ~(a^b) c from (1)
    --------------------
    (4) ~(a->c) (5)~(b->c) from (2)
    ----------------------
    ~a ~b from (3)
    -----------------------------------------
    a
    ~c from (4)
    -------------------------------------
    b
    ~c from (5)
    ------------------------------------------
    this tree has all branches closed. Why is this, what I am doing wrong? Any help, any minor idea would be very appreciated!

    Thank you all in advance,
    Melsi
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  2. #2
    MHF Contributor
    Joined
    Oct 2009
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    783
    I assume you are using the method of analytic tableaux. However, I don't see a tree in your writing; I only see a linear sequence. In particular, a ^ b -> c produces two branches and the left one has the following shape.

    Code:
    (1)        ~(a ^ b)
                   |
    (2) ~((a -> c) ^ (b -> c))
                  /\
                 /  \
                /    \
    (3)  ~(a -> c)  ~(b -> c)  from (2)
             /\         /\
            /  \       /  \
    (4)   ~a   ~b    ~a   ~b   from (1)
           |    |     |    |
    (5)    a    a     b    b   from (3)
           |    |     |    |
    (6)   ~c   ~c    ~c   ~c   from (3)
    So, the first and fourth branches are closed, but the second and third are open. Note that they correspond to two truth assignments that make the original two formulas (the premise and the negation of the conclusion) true.
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  3. #3
    Junior Member
    Joined
    Jul 2008
    From
    Athens, Greece
    Posts
    48

    Red face Problem solved

    Hello,

    Thank you very much for your help. I appreceate it a lot and I am greatfull.

    After so much effort and very very deep analization I came to a solution exactly to yours and this is great because I can confirm my solution and have no doubt about it.

    E.g let say A is a sentence to be decomposed and P is the result of decomposation. Your solution says that P should apply to every branch found under A. I had misunderstood the process and thought that P should apply everywhere even to branches that are not under A!!!

    I thought I knew the process very well but this example brought up this problem, I am so glad I made it clear before examination.

    I wish I had found it earlier so you would not spend your time... however thank you very very much again!

    Sincerely,
    Melsi
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