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If you have a formula with existential quantifiers, it is important in which order they appear.

Just to make an easy example:

$\forall$ man $\exists$ woman: the woman is the true love of the man

which is obviously a different statement than

$\exists$ woman $\forall$ man: the women is the true love of the man

The first one means that there a many women - eventually for every man another woman. The second statement means there is (at least) one women that is loved by all men. Good for the woman, eventually bad for the men.

If you have two existential quantifiers or two universal quantifiers, does the order make a difference?

Martin Thoma
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    No, it does not. – Brian M. Scott Sep 23 '12 at 08:06
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    $\forall$ man $\exists$ woman: some woman is the true love of the man -- I think it should be in this way. – Nikita Evseev Sep 23 '12 at 08:10
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    Although, $\forall x \forall (y<x) : (y<x^2)$ certainly is dependent on order, since the other way round makes no sense! (even though it can be written as $\forall x \forall y : (y<x) \to (y<x^2)$ which is the same as $\forall y \forall x : (y<x) \to (y<x^2)$) –  Sep 23 '12 at 09:07
  • Note that the question, when refering to the second example, states that "there is one women that [...]" but this is not true, $\exists$ means that there is at least one, not that there is only one. It could very well be the case that the set of all women s.t. $\forall$ man the women is the true love of the man is equivalent to set of all woman. – Sebastián P. Pincheira Apr 18 '20 at 23:57
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    @RandomUser "there is one women" is the same as "there is at least one women" - but I agree, that it could be missunderstood. I edit it. – Martin Thoma Apr 19 '20 at 06:45

1 Answers1

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Any number of successive quantifiers of the same kind can be replaced by a single quantifier by combining the quantified variables into a tuple; e.g. $\forall x\forall y$ is equivalent to $\forall(x,y)$. The order in the tuple is irrelevant, and thus so is the order of the quantifiers.

joriki
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    The question asks about ∀+∃ whereas you answer, accept and upvote mostly the answer to the different, ∀+∀ question. What is even more surprising is that the duplicate question has the same attitude, http://math.stackexchange.com/questions/491783/intuitive-reason-that-quantifier-order-matters#comment1334623_491792! – Val Jan 09 '14 at 23:32
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    @Val Read the question more carefully - note the bolded part at the bottom. – Noah Schweber Jan 19 '16 at 23:31
  • Why does the order in the tuple not matter? Intuitively, it makes sense because the roles of the distinct variables do not change in the propositional formula (with variables) that comes after the quantifiers, but I was wondering if that is all or if there is a better reason. – Favst Oct 04 '20 at 18:13
  • @Favst: What matters is that all possible values of $x$ can be combined with all possible values of $y$, and the proposition holds for all combinations. It doesn't matter which notation we use for these combinations, whether we call them $(x,y)$ or $(y,x)$, they're the same combinations, the same sets of propositions that need to be satisfied. – joriki Oct 08 '20 at 09:29