When should I classify a statement as "undefined" versus "vacuously true/false": implications for induction Although this question has sort of been asked before (Why aren&#39;t vacuous truths just undefined?), I think my question is emphasizing a somewhat different aspect. Rather than knowing "why" vacuous truths aren't just undefined...I want to know when one should treat a statement as undefined, instead of vacuously true (or vacuously false). 

In Tao's Analysis I, there is the following passage: 


  "To summarize so far, among all the objects studied in mathematics, some of the objects happen to be sets; and if $x$ is an object and $A$ is a set, then either $x \in A$ is true or $x \in A$ is false. (If $A$ is not a set, we leave the statement $x \in A$ undefined; for instance, we consider the statement $3 \in 4$ to be neither be true or false, but simply meaningless, since 4 is not
  a set.)"


So while this certainly makes sense...I am reluctant to claim that I comfortably know when to declare something as being vacuous versus undefined. If I were to rephrase this slightly, I could say: Prove to me that $3$ is not an element of $4$. 

...Well, unfortunately, $4$ is not a set so I have no way of disproving your claim. However, this sounds vaguely similar to any sort of argument that invokes the "vacuously true" approach. (And for the situation of a base case of an induction proof, it is quite important to make the distinction between "undefined" and "vacuously true", I would think!) Could anyone offer an explanation that will clear this up for me? 

Cheers~
As I understand, in classical logic, if the antecedent of any implication is false, then that implication is said to to be vacuously true. For any object $x$ and proposition $P$, for example, we have, the implication $x \in \emptyset \implies P$ being (vacuously) true. Likewise, $0=1 \implies P$.

A function $f: A \to B$ is said to be undefined for some object $x$ means only that $x \notin A$. I have never heard of a syntactically correct logical proposition being "undefined" or "meaningless."
As I understand, in classical logic, if the antecedent of any implication is false, then that implication is said to to be vacuously true. For any object $x$ and proposition $P$, for example, we have, the implication $x \in \emptyset \implies P$ being (vacuously) true. Likewise, $0=1 \implies P$.

A function $f: A \to B$ is said to be undefined for some object $x$ means only that $x \notin A$. I have never heard of a syntactically correct logical proposition being "undefined" or "meaningless."