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What does a dot (.) mean in predicates?

$\forall a \in A. \exists d \in D. H(a,d)$

Especially, how is the above different to

$ \exists d \in D. \forall a \in A. H(a,d)$

I've never seen this used in German lecture scripts.

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    $\begingroup$ I've seen used colons too. Often the separator is omitted when followed by another quantor. $\endgroup$ – Bergi Jun 28 '16 at 6:40
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    $\begingroup$ The dots here just separate the $A$ and $D$ from the stuff after them. $\endgroup$ – user253751 Jun 28 '16 at 9:03
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    $\begingroup$ It's used in place of parenthesis around the quantified formula. $$\forall a \in A (\exists d \in D (H(a, d)))\;\text{vs}\;\forall a \in A. \exists d \in D . H(a, d)$$ so the $.$ (or $:$ often) quantifies over everything to the right $\endgroup$ – Bakuriu Jun 28 '16 at 9:48
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    $\begingroup$ Just delete the dots and you will be fine. $\endgroup$ – Andrej Bauer Jun 28 '16 at 11:12
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The dot just means "such that"; it's often omitted.

The difference between the two formulas is the difference between "everybody has a mother" and "there is somebody who is everybody's mother."

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  • $\begingroup$ Thank you, and a comma means "such that", too? $\endgroup$ – alsdkjasdlkja Jun 27 '16 at 22:37
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    $\begingroup$ @nxrd A comma where? If it's after a quantified statement, then probably. But, e.g., the comma in "$H(a,d)$" is just separating the arguments of the relation. $\endgroup$ – David Richerby Jun 27 '16 at 22:57
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    $\begingroup$ I've seen several notations used. The $.$ is particularly popular in logic involving the lambda calculus, which also uses it as a delimiter, e.g. $\lambda x \ldotp x$. $\endgroup$ – jmite Jun 27 '16 at 23:20
  • $\begingroup$ @David Richerby I mean something like $\forall n \in \mathbb{Z}, n \geq z \implies n^2 \geq 4$. I can read and understand that, but somewhere in my brain I am a bit confused by commas and dots $\endgroup$ – alsdkjasdlkja Jun 28 '16 at 3:22
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    $\begingroup$ @nxrd There's nothing to be confused by. It's just punctuation to make things easier to read. It doesn't have any meaning. $\endgroup$ – David Richerby Jun 28 '16 at 7:16
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Well, in $\forall x. P(x)$, the "such that" reading of the dot (or the space, if we don't like to write dots) doesn't make sense grammatically as it does for the existential. One could say, I suppose, "it holds that", or something like that.

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    $\begingroup$ Whoops, I meant this as a comment to David Richerby's answer, sorry... $\endgroup$ – Basil Jun 29 '16 at 22:13

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