TITLE: Why is predicate "all" as in all(SET) true if the SET is empty?
QUESTION [18 upvotes]: Can anyone explain why the predicate all is true for an empty set? If the set is empty, there are no elements in it, so there is not really any elements to apply the predicate on? So it feels to me it should be false rather than true.

REPLY [4 votes]: Another approach: the 'vacuous truth' for $\forall$ is roughly the logical equivalent of an empty product being defined as 1 or an empty sum being defined as 0.  Just as we want $\sum_{i=1}^{n+1} a_i = a_{n+1} + \sum_{i=1}^{n} a_i$ (and want this to hold in every case, even the 'base case' where $n=0$) and want $\prod_{i=1}^{n+1} a_i = a_{n+1}\cdot \prod_{i=1}^{n} a_i$, so too we want $\forall x\in (S\cup \{z\})\ P(x) \Longleftrightarrow \bigl(\ (\forall x\in S\ P(x))\ \wedge P(z)\bigr)$ to hold even in the 'base case' where $S$ is empty.  You should be able to convince yourself (through some relatively straightforward logical manipulation) that this is requires defining $\forall x\in\emptyset \ P(x)$ to be true for all predicates $P()$.