More than one quantifiers for one variable: $\forall x\exists x P(x)$

Question

I couldn't find any definiton about this:

$\forall x\exists x P(x)$

Is here the for all or the there exists stronger?

Cheers

Answer 1

The formula as written is quite well formed; it just is not very human readable.

It is best practice to avoid reusing symbols for bound entities when nesting quantifiers.   Thus wise, confusion in the given formula may be averted by a change of symbols for the quantified entities within their own scope.

The entity in the predicate is bound to the scope of the existential quantifier.   As such it does not occur free within the scope of the universal.   We can change symbol for the entity bound within the scope of the existential quantifier, without affecting the symbol of the entity bound in the scope of the universal.   Thusly:

$$\color{red}{\color{black}{ \forall x \color{silver}{\big( \color{black}{\exists x \; P(x)}\big)} \;\equiv\; \forall x \color{silver}{\big(\color{black}{\exists y \; P(y)}\big)} \;\equiv\; \exists y \; P(y) }}$$

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Further   Just to be clear, suppose we had a nested statement with more predicates, such as: $$\forall x\;\Big(Q(x) \wedge \exists x\; \big(P(x)\vee R(x)\big)\Big)$$

To preserve meaning of the statement, we do not touch any of the $x$ symbols that lie outside the scope of the existential quantifier when we change the variable bound within its scope.

$$\forall x\;\Big(Q(x) \wedge \exists y\; \big(P(y)\vee R(y)\big)\Big)$$