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Math Help - entailment - interpretation

  1. #1
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    entailment - interpretation

    Hello All

    To prove the following, I have tried resolution with success, but I need to do it using semantics of interpretation; that every interpretation that satisfies the first sentence satisifies the second. I am not sure on how to go about this ....... Your help is extrememly appreciated.

    F(x) iff P(x) or (G(x) and ~N(x))

    G(x) and ~N(x) implies F(x)

    Thanks
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  2. #2
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    Consider any model
    \mathcal{M} \models \forall x (F(x) \leftrightarrow (P(x) \vee (G(x) \wedge \lnot N(x))))

    Using semantics of satisfaction this means, for any valuation  \mu (x) = v
    \mathcal{M}[\mu (x) = v]} \models F(x) \leftrightarrow (P(x) \vee (G(x) \wedge \lnot N(x)))

    Again by semantics of satisfaction, this means that \mathcal{M}[\mu (x) = v] models a truth table for the logical expression shown (of course, rigorously you have to decompose the semantics of all the logical connectives one by one). If the model interpretation is I, then:
     v\in I(F) iff (  v\in I(P) or (  v\in I(G) and not  v\in I(N) ))

    When the truth table of v\in I(...) for the above expression evaluates to true, you can show that
     v\in I(G) and (not  v\in I(N) ) implies  v\in I(F) also evaluates to true.

    This means
    \mathcal{M}[\mu (x) = v]} \models (G(x) \wedge \lnot N(x)) \rightarrow F(x)

    since the valuation for x was arbitrary
    \mathcal{M} \models \forall x ((G(x) \wedge \lnot N(x)) \rightarrow F(x))
    Last edited by snowtea; December 18th 2010 at 08:38 PM.
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