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My logic class didn't introduce us to multiple quantifiers. I've seen a few variations that seem to have distinct meanings:

$$ \forall x, \forall y(...) $$

$$ \forall x \forall y(...) $$

$$ \left( \forall x \forall y \right) (...) $$

$$ \left( \forall x, \forall y \right) (...) $$

$$ \left( \forall x \right)\left( \forall y \right) (...) $$

Do the meanings of those examples differ?

Thank you.

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2 Answers 2

up vote 6 down vote accepted

No, they are just typographical variants of the same mathematical meaning.

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However, $(\forall x) \phi \to \psi$ could mean $(\forall x . \phi) \to \psi$... –  Zhen Lin Mar 2 at 13:22
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@Zhen Lin - It usually means ... because the usual convention for the omission of parentheses based on "ranking" between opeartors says that quantifiers and negation are ranked last. –  Mauro ALLEGRANZA Mar 2 at 13:47

Just to amplify Henning Makholm's headline answer just a bit ...

  1. Some older texts use $(x)$ [without the rotated 'A'] for the universal quantifier, and some use $(\forall x)$ [with the rotated 'A' and brackets]. In those notations multiple universal quantifiers will look like $(x)(y)\varphi$ or $(\forall x)(\forall y)\varphi$.
  2. The modern habit is to use the rotated 'A' but then not use the unnecessary brackets: thus $\forall x\forall y\varphi$.
  3. In some dialects, we write e.g. instead of $\forall w\forall x\forall y\forall z\varphi$ the simpler $\forall wxyz\varphi$ for brevity.
  4. I can't recall having seen the notation $(\forall x\forall y)\varphi$. Nor can I recall any canonical mathematical logic textbook that uses commas. Certainly these uses are non-standard to my eyes.
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I use $(\forall x)\phi$ in many places because it is more likely to be typeset in an appropriate way. Without the parentheses, a small space is needed between adjacent quantifiers: $\forall x\,\forall y\,\phi$ rather than $\forall x \forall y \phi$. The parentheses avoid the need for explicit spacing, which can be particularly difficult in web typesetting. –  Carl Mummert Mar 2 at 13:49
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It is also common in some contexts (particularly those close to computer science) to use a period to separate the quantifiers and the matrix: $\forall x . \phi$. This is analogous to $\lambda x.f(x)$ I suppose. The $\mu$ operator an be used in the same way: $g(x) = \mu x . f(x,y)=0$ or $g(x) = (\mu x)[f(x,y) = 0]$. –  Carl Mummert Mar 2 at 13:52

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