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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The set of unicorns is the empty set. The set of unicorns does not contain any unicorns, since there are no unicorns. – André Nicolas Apr 20 '13 at 5:53
@AndréNicolas I would say that the intersection of the set of unicorns and the set of animals of this planet is empty xD – MphLee Apr 20 '13 at 8:02
@MphLee Indeed, up to some degree, unicorns do exist... the intersection of the set of unicorns and the set of fairy tale animals is definitely non-empty. – A.P. Apr 20 '13 at 9:33

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.

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Hi Austin thank you so much I got what you mean, but what make me confused is that why x is an element of empty set is always false, why x can not be something like unicorns? if x is unicorn, then unicorn is in empty set is true(it is another vacuous true), so for – dingding Apr 20 '13 at 19:28
Hi Austin thank you so much I got what you mean, but what make me confused is that why x is an element of empty set is always false, why x can not be something like unicorns? if x is unicorn, then unicorn is in empty set is true(it is another vacuous true), so x is an element of empty set is not always false in this sense~ thanks again – dingding Apr 20 '13 at 19:46
@dingding Think of a set as a bucket with objects in it. The empty set is the bucket that has nothing in it (the empty bucket). If you reach inside, you absolutely will not find a unicorn in there, so "unicorn belongs to the empty bucket" is a false statement. In fact, no matter what $x$ is, "$x$ belongs to the empty bucket" is a false statement, since the empty bucket has no objects at all inside it. – Austin Mohr Apr 20 '13 at 19:54

@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

$\forall x(Ux \to x \in \emptyset)$

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

$u \in \emptyset$

is not true.

So what are we to make of

if x=unicorn, since unicorn is in empty set so x is in empty set is true?

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. $\forall x(Ux \to x \in \emptyset)$

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Thanks Peter by . '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.] – dingding Apr 20 '13 at 10:52
Thanks Peter by . '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.] do you mean that:x still run over all the exist element, which means actually we never plug a non-exist unicorn into x? So, since there are no unicorns to be named and u is an empty name, no atomic wff Fu is true, and in particular u∈∅ I do not understand this part, do you also want to make it clear that x can not be plugged by some non-esist thing by saying these? thanks – dingding Apr 20 '13 at 10:59
Thanks Peter by . '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.] do you mean that:x still run over all the exist element, which means actually we never plug a non-exist unicorn into x? So, since there are no unicorns to be named and u is an empty name, no atomic wff Fu is true, and in particular u∈∅ I do not understand this part, do you also want to make it clear that x can not be plugged by some non-esist thing by saying these? thanks a lot – dingding Apr 20 '13 at 19:27
Hi Dr.Smith Thanks so much for your help, at the time I ask this question I can not fully understand your answer,but now I guess I understand most of it, do you mean that," we should never plug a constant non-exist thing into our variable?" since empty name is not allowed in logic something also something like u(a constant unicorn) is in empty set is always false not as what I imagine is always true, what is always true is "all the unicorns are in empty set" – dingding Jul 20 '13 at 3:10