Same name predicates

**infnty** · October 13th 2011, 08:40 AM

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!

**emakarov** · October 13th 2011, 01:17 PM

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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