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$\exists x \forall y P(x,y) \equiv \forall y \exists x P(x,y)$

I was told they were not, but I don't see how it can be true.

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    is the order of the quantifier important?2012-10-30
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    p(x,y) = x loves y and p(y,x) = y loves x? no?2012-10-30
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    yes, the order of the quantifier matters, when the quantified variables/arguments are different, like in this case, and when the quantifiers are different, as K. Stm. points out.2012-10-31
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    but the order of the arguments of P(x) doesn't matter?2012-10-31
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    Yes, very much so, order of arguments of a predicate matters, except if P(x, y) is "symmetric" like E(x, y) meaning "x = y". Equality is symmetric: y = x means exactly the same as x = y. But most predicates are not symmetric. L(x, y): x loves y is not the same as L(y, x): y loves x.2012-10-31
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    [Confused between nested quantifiers](http://math.stackexchange.com/questions/64500/confused-between-nested-quantifiers), [Is the order of universal/existential quantifiers important?](http://math.stackexchange.com/questions/201051/is-the-order-of-universal-existential-quantifiers-important)2012-11-04

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