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Thread: Fol 1

  1. #1
    Junior Member
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    Fol 1

    Show that

    $\displaystyle \{\forall x(\alpha \rightarrow \beta), \forall x \alpha \} \models \forall x \beta .$

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    We show that any structure $\displaystyle A$ for the language and any function $\displaystyle s: V \rightarrow |A|$ that satisfies $\displaystyle \{\forall x(\alpha \rightarrow \beta), \forall x \alpha \}$ also satisfies $\displaystyle \forall x \beta$, where $\displaystyle s$ denotes a function from the set V of all variables into the universe $\displaystyle |A|$ of $\displaystyle A$. For any $\displaystyle d \in |A|$, we have $\displaystyle \models_A(\alpha \rightarrow \beta)[s(x|d)]$ and $\displaystyle \models_A \alpha [s(x|d)]$.
    It follows that $\displaystyle \models_A \beta [s(x|d)]$. which in turn implies that $\displaystyle \models_A \forall x \beta [s]$.

    Is this proof O.K or am I missing something?

    Thank you.
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  2. #2
    MHF Contributor
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    Re: Fol 1

    Yes, this proof is OK.
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