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Although this question has sort of been asked before (Why aren'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~

S.C.
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    You might want to check out the idea of a many-sorted language, which I think is what the author is tacitly getting at. The idea is that $3\in 4$ isn't treated as syntactically correct in the first place, and we only give truth values to syntactically correct expressions. – Malice Vidrine Jan 31 '20 at 19:24
  • @MaliceVidrine Interesting. So the statement $\forall n, P(n)$ for $n \in {}$ has been "Agreed upon by mathematicians" as being syntactically correct...and therefore vacuously true (i.e. the community agrees that statements about the null set in this fashion are syntactically correct)? – S.C. Jan 31 '20 at 20:05
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    In a two-sorted language that distinguishes between objects and sets, and only allows the former to be members of the latter, then yes, it's well-formed and vacuously true. Note that working in a two-sorted theory is not a standard way to handle things; in set theory there's only one sort, and $3\in 4$ has a truth value (it's true under the usual definitions), so this issue of undefinedness is an artifact of a particular sort of treatment. – Malice Vidrine Feb 01 '20 at 01:39
  • Your question reminds me of this comment in which I queried whether the statement "it is false that $\frac1x$ is continuous at $0$" is non-meaningful or is true. As it is couched in natural language, it feels like any consensus may not be strong, compared to asking whether a more formal statement is true/false or undefined. – ryang Feb 28 '23 at 15:30

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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."

Dan Christensen
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