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Math Help - Same name predicates

  1. #1
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    Sep 2011
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    Same name predicates

    Hey all,

    I'm doing some homework and one of the exercises got me confused. In particular, it states the following formula:

    $\forall x( A(x) \to \exists y ( B(y) \land \exists y C(x, y) ) )

    I am confused about the double usage of $ \exists y $. Is it allowed to do that twice for the same variable (y) within the same scope? It doesn't make sense to me.

    Thanks in advance!
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  2. #2
    MHF Contributor
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    Re: Same name predicates

    It is allowed, but the two y's have nothing to do with each other. Bound variables can be freely renamed, i.e., \exists y\,F(y) is the same formula as \exists z\,F(z). So \forall x( A(x) \to \exists y ( B(y) \land \exists y C(x, y))) is the same as \forall x( A(x) \to \exists y ( B(y) \land \exists z C(x, z))).

    While having nested bound variables with the same name is legal, unless one wants to make some point about scoping, it is a bad style to use the same name twice, and it is often a typo.
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