Sometime after I began studying conditional statements, I started having difficulty understanding vacuous truth. For instance, the fact that for any set $A$ we have $\emptyset\subset A$ is commonly justified by asking (almost) rhetorically, 'Well, can you find an element in the empty set that isn't in $A$?', which undoubtedly is what is required to show that the assertion is false, but that answer always caused me to naively ask, 'Well, can you find elements in the empty set that are in A?'
It wasn't until I finally understood that the statement $\forall x\in A\,P(x)$ does not imply the existence of an $x$ for which $P(x)$ is true. It is only meant to imply that there is no $x$ in $A$ for which $P(x)$ fails (see page 14 here). It then becomes clear to see why the statement $\forall x\in\emptyset\,P(x)$ is always true, for this is the assertion that there is no $x$ in the empty set $\emptyset$ for which $P(x)$ fails, and since there are no elements in the empty set $\emptyset$, the assertion is true, vacuously.
But, recently I learned that the two statements:
(1) $\forall x\in A\,P(x)$
(2) $\forall x(x\in A\implies P(x))$
have the same meaning. For $A=\emptyset$, (1) and (2) become:
(1') $\forall x\in\emptyset\,P(x)$
(2') $\forall x(x\in\emptyset\implies P(x))$.
As seen before, (1') is true. But it isn't clear to me why (2') is true. To explain this, someone might say, 'to claim that (2') is false is to say that $\exists x(x\in\emptyset\,\land\,\lnot P(x))$. Since there are no elements in the empty set, it is impossible to show that (2') is false.' Once again, I naively counter, 'since there are no elements in the empty set, how can we show that (2') is true?'
I think that the source of my unwillingness to be satisfied with explanations that (2') is true is the lack of knowing the precise assertion that a conditional statement is making. As explained in the second paragraph, this certainly was the issue when trying to understand why (1') is true, but once I understood precisely the meaning of the statement $\forall x\in A\,P(x)$, it was easy to see why this was so. I think that if someone could explain precisely what (2') is asserting, specifically by addressing what is meant by a conditional statement, then it would be easy to see why (2') is true.
UPDATE 1:
I should make something clear. When I wrote,
But, recently I learned that the two statements:
…
I meant that I became aware that statements (1) and (2) are equal but did not understand why. This led me to seek out information about universal statements and universal conditional statements. Now, I think I understand why (1) and (2) are equivalent. The universal conditional statement $\forall x\in U(P(x)\implies Q(x))$ means that every object $x$ in $U$ that satisfies $P(x)$ also satisfies $Q(x)$. If I'm not mistaken, this statement is equivalent to $\forall x\in D\,Q(x)$, where $D=\{x\in U:P(x)\}$.
Also, after reading the answer from Graham Kemp, specifically where he writes,
As you've seen, a universal statement is falsified if we can witness a counter example. A vacuous truth is not falsified because no counter examples to any statement exist in the empty set.
I considered (seriously) the negation of the statement $\forall x(x\in\emptyset\implies P(x))$, which is the statement $\exists x(x\in\emptyset\land\lnot P(x))$. The predicate, $x\in\emptyset$, is false for every possible value for x. Therefore, the existential statement can never be true. It follows that the original statement is true since its negation is false. This is (now) absolutely convincing that (2') is true.
However, is this the explanation for the fact that for regular conditional statements, whenever the antecedent is false, the implication is true? Or, does the reasoning for this come from understanding the precise meaning of the conditional of two statements? The reason I ask this is because I read that when dealing with vacuous truth of universal statements, "we need predicate logic to be consistent with propositional logic" (see page 4 here) which suggests that it has already been established that a regular conditional statement is true when its antecedent is false.
UPDATE 2:
This is in response to the last paragraph in UPDATE 1. In order to understand why the conditional of two statements is true when the antecedent is false, it is necessary to understand its precise meaning. The conditional statement, "if $p$, then $q$" is the assertion that "whenever $p$ is true, $q$ is also true." Thus, it is false only when $p$ is true and $q$ is false. In symbols:
$\lnot(p\implies q)\equiv p\land\lnot q$
Since a statement is true if and only if its negation is false, we have:
$p\implies q\equiv\lnot(p\land\lnot q)$
or,
$p\implies q\equiv\lnot p\,\lor q$
Thus, when $p$ is false (which means $\lnot p$ is true), $p\implies q$ is true, vacuously.
UPDATE 3 (Fri Jan 15):
I'm having a bit of difficulty with the fact that any assertion about all of the elements of the empty set $\emptyset$ is vacuously true. For instance, I completely understand and agree with the fact that $\emptyset\subset A$ where $A$ is any nonempty set (i.e., $\forall x\in\emptyset :x\in A$ is a true statement). But isn't the statement $\forall x \in\emptyset : x\notin A$ also true?
I guess my question is, can both of the statements $\forall x\in\emptyset : p(x)$ and $\forall x\in\emptyset : \lnot p(x)$ be true?
RESPONSE TO ZeroXLR:
I guess what I'm seeking is how to confidently explain this to another person.
I agree and understand that the negation of the statement (1) $\forall x\in\emptyset : p(x)$ is not (2) $\forall x\in\emptyset : \lnot p(x)$. The negation of (1) is the statement $\exists x\in\emptyset :\lnot p(x)$, which of course is never true. Hence, (1) is always true and that is how we show that the empty set $\emptyset$ is a subset of any set $A$.
So, would I be correct in saying that since the statements (1) and (2) are both true, all the elements of the empty set satisfy both $p(x)$ and $\lnot p(x)$? And of course, if we take $p(x)$: $x\in A$, then this means that $\emptyset\subset A$.
Best Answer
These statements have the exact same meaning. The first is simply a shortened expression.
$\forall x {\in}\varnothing\; P(x)$ "everything in the empty set satisfies the predicate"
$\forall x\; \big(x\in\varnothing \to P(x)\big)$ "if any thing is in the empty set, then that thing will satisfy the predicate".
As you've seen, a universal statement is falsified if we can witness a counter example. A vacuous truth is not falsified because no counter examples to any statement exist in the empty set.
This can also be explained as: An implication is held to be true if either the antecedent is false or the consequent is true. Because the antecedent ($x\in \varnothing$) is never true, therefore the implication always holds, irrespective of the consequent. Thus no counterexamples to an implication of that form can ever exist.