I have a question about vacuous true and it always make me confused. If I want to prove that the empty set is the subset of all the set A, the proof is as following: if x is in empty set, then x is in A. since x is in empty set is always false,, so the conditional statement is always true~ my question is why x is in empty set is always false, what if x=unicorn, since unicorn is in empty set so x is in empty set is true, right? the question goes to why I can not plug a non-exist thing( like unicorn)into the variable x?
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You want to prove something like the following:
If $x \in \emptyset$, then $x \in A$.
If you don't like the vacuous nature of the hypothesis, consider the contrapositive (which is logically equivalent to the original statement):
If $x \notin A$, then $x \notin \emptyset$.
Select any $x$ that does not belong to $A$. Surely it also does not belong to the empty set, since the empty set contains no elements at all. (Notice the vacuous hypothesis in the original statement corresponds to a tautological conclusion in the contrapositive.) Having proved the contrapositive, we can also conclude the original statement.
@AustinMorh makes the key point. But a footnote about unicorns. We must distinguish two different claims.
Suppose $U$ is the predicate satisfied by all and only unicorns, then
indeed comes out true for sensible domains (since nothing satisfies the antecedent of the conditional). 'All unicorns are members of the empty set' is another vacuous truth! [But note, we are not "plugging a non-existent thing into a variable": the variables still run over what there is in the domain.]
Suppose however $u$ is now a constant, purportedly naming a unicorn. In standard logic where empty names are not allowed, then $u$ is illegitimate. But take a free logic which does allow empty names. Then atomic wffs involving empty names are not true [on some accounts, they are false, on others they are neither true nor false]. So, since there are no unicorns to be named and $u$ is an empty name, no atomic wff $Fu$ is true, and in particular
is not true.
So what are we to make of
If x is intended to be name-like, then "x is in empty set" is not true. The truth in the vicinity is something else, i.e. $\forall x(Ux \to x \in \emptyset)$