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Possible Duplicate:
Confused between Nested Quantifiers

I asked the question about two sentences. interpreting mixed quantifier

But, I don't know the meaning difference between $∀x∃y(\text{Cube}(x) → (\text{Tet}(y) ∧ \text{LeftOf}(x, y))),$ and $∃y∀x(\text{Cube}(x) → (\text{Tet}(y) ∧ \text{LeftOf}(x, y))),$

"Every cube is to the left of a tetrahedron"
"There is a tetrahedron that is to the right of every cube"

I think these sentences have same meaning.

Is it wrong? please give me your opinion.

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    Related: [this question](http://math.stackexchange.com/q/64500/742).2012-05-16

2 Answers 2

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Let’s look at a simpler example, where we assume that the variables range over real numbers: $\forall y\exists x(x+y=0)$ and $\exists x\forall y(x+y=0)$. The first says that every real number has an additive inverse, which is true. The second says that there is some particular real number $-$ call it $z$, say $-$ such that $z+y=0$ no matter what $y$ is; that’s clearly false.

In general you cannot reverse $\forall$ and $\exists$.

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The two sentences in your question do not have the same meaning: consider an infinite sequence $\text{Cube}\qquad\text{Tetrahedron}\qquad\text{Cube}\qquad\text{Tetrahedron}\qquad\cdots$ Every cube is to the left of a tetrahedron, but there is no tetrahedron that is to the right of every cube.

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    ..I $c$onstructed your example in Tarski's world. But it says that the both sentences are true..... http://thehjh.com/tarski.png2012-05-16