TITLE: Quantification over the empty set
QUESTION [5 upvotes]: Possible Duplicate:
Why is predicate “all” as in all(SET) true if the SET is empty? 

In don't quite understand this quantification over the empty set:
$\forall y \in \emptyset: Q(y)$
The book says that this is always TRUE regardless of the value of the predicate $Q(y)$, and it explain that this is because this quantification adds no predicate at all, and therefore can be considered the weakest predicate possible, which is TRUE.
I know that TRUE is the weakest predicate because $ $P$ \Rightarrow$ TRUE is TRUE for every $P$.
I don't see what is the relationship between this weakest predicate and the quantification.

REPLY [4 votes]: When you see a quantification like ‘$\forall \phi x : \psi x$’, this is shorthand for ‘$\forall x : \phi x \to \psi x$’. Since ‘$x \in \emptyset$’ is false for all ‘$x$’, the antecedent of ‘$x \in \emptyset \to \phi x$’ will always be false, thus the entire conditional statement is always true.
Edit:
In the case of existential quantification, the statement ‘$\exists \phi x : \psi x$’ is shorthand for ‘$\exists x : \phi x \land \psi x$’, so in this case, the conjunction ‘$x \in \emptyset \land \phi x$’ will always be false.