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Math Help - Need logic help

  1. #1
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    Unhappy Need logic help

    Consider a first-order language with ternary predicate T and equality predicate =. We define the formulas F1, F2, F3 as follows.
    F1: ∀x∀y∃zT(x,y,z).
    F2: ∀x∀y∀z∀w ((T(x,y,z) ∧ T(y,x,w)) → =(z,w))
    F3: ∀x∀y∀z (T(x,y,z) → T(y,x,z)))

    a) Prove F1 ∧ F2 logically implies F3
    b) Prove F1 ∧ F3 does not logically imply F2

    How do I start this, I am very lost here...
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  2. #2
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    Re: Need logic help

    Assume that F1 and F2 are true in some interpretation. To prove that F3 holds in that interpretation, fix some x, y, z and assume T(x,y,z). Does there exist some w such that T(x,y,w)? What can you say about it in view of F2?

    For b), let T(x,y,z) mean "point z is located between points x and y."
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  3. #3
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    Re: Need logic help

    For b)
    Let domain be {0,1}
    T(x,y,z): z is between x and y
    Let x=0 y=3 z=1 w=2
    Then F1 and F3 does not logically imply F2

    Hows this? I am just trying to show a counter example, I hope this is enough to prove it though.
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  4. #4
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    Re: Need logic help

    Quote Originally Posted by Sneaky View Post
    For b)
    Let domain be {0,1}
    T(x,y,z): z is between x and y
    Let x=0 y=3
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  5. #5
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    Re: Need logic help

    For b)
    Let domain be {0,1,2,3}
    T(x,y,z): z is between x and y
    Let x=0 y=3 z=1 w=2
    Then F1 and F3 does not logically imply F2

    Oops I missed that...
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  6. #6
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    Re: Need logic help

    OK. As I said in another thread, you should say that in the described interpretation, F1 and F3 are true and F2 is false, not that "then" F1 and F3 does not logically imply F2.
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  7. #7
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    Re: Need logic help

    For b)
    Let domain be {0,1,2,3}
    T(x,y,z): z is between x and y
    Let x=0 y=3 z=1 w=2
    Then F1 is true and F3 is true and F2 is false.
    Therefore F1 and F3 does not logically imply F2.

    edit: Do you mean I shouldn't have that last line "Therefore F1 and F3 does not logically imply F2." at all?
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