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I'm currently reading this document by the University of Texas in Austin about Predicate Logic, and got stuck on the following:

Note about nested quantifiers: For predicate P (x, y): ∀x∀yP (x, y) has the same meaning as ∀y∀xP (x, y). ∃x∃yP (x, y) has the same meaning as ∃y∃xP (x, y).

We can not interchange the position of ∀ and ∃ like this!

Example: U = set of married people. True or false?
1. ∀x∃y[x is married to y]
2. ∃y∀x[x is married to y]

I'm doubtful about the answer to this example. Also, some explanation about ordering of ∃ and ∀ operators would be appreciated.

Arpit Quickgun Arora
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2 Answers2

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Another way to consider this is visually. Imagine an $x$-$y$ grid, in which we mark a spot at $(x, y)$ if the proposition is true for that $x$ and that $y$. Then $\forall x \exists y$ amounts to there being at least one spot in every column, and $\exists y \forall x$ amounts there to being at least one row that is completely filled in from left to right.

Brian Tung
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The first says: "For all $x$ it exist a $y$ such that $x$ is married to $y$." So every $x$ is married.

The second one says: "There exist a $y$ such that for all $x$, $y$ is married to $x$. So someone is married to everyone.

Alain Remillard
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