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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.

  • 3
    The difference between "for every x there is a y such that..." and "there is a y such that for every x ..." is the difference between "Everyone is loved by someone" and "Someone loves everyone."2012-05-16
  • 3
    A better translation of the first sentence is "Every cube is to the left of **some** tetrahedron", to make it clear that it is not necessarily the *same* tetrahedron that works for every cube; whereas in the second sentence, the *same* tetrahedron must act as witness for every cube.2012-05-16
  • 0
    Related: [this question](http://math.stackexchange.com/q/64500/742).2012-05-16

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