Is $\forall x(x\in\varnothing)$ true or false?

Asked Aug 10 '22 at 11:43

Active Mar 02 '23 at 19:11

Viewed 343 times

4

I know that

$$\forall x(x\in\varnothing)$$

is equivelent with

"There is no element in Universe such that $x$ is not in $\varnothing$."

but, is $\forall x(x\in\varnothing)$ true or false?

Is it an example of vacuously true?

How about the Univers is the empty set?

How about the Univers is not the empty set?

edited Aug 10 '22 at 11:54

asked Aug 10 '22 at 11:43

Stats Cruncher

381

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For all x where? – Michal Adamaszek Aug 10 '22 at 11:51
@MichalAdamaszek Does it matter where x is? Is it possibel that the universe is the empty set? – Stats Cruncher Aug 10 '22 at 11:53
@KevinS I could understand that "there are no members of an empty set". But I don't understand that x∈∅ is false. – Stats Cruncher Aug 10 '22 at 11:59
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Yes, it matters. If the universe is empty it is true. Otherwise false. – Michal Adamaszek Aug 10 '22 at 11:59
If $x$ is an element of a set, this is false. If $x$ is a set, this is true only if $x$ is an empty set. – Vasili Aug 10 '22 at 12:02
@MichalAdamaszek I understand that ∃x(x∈∅) is false, but I don't understand that ∀x(x∈∅) is also false. How come? – Stats Cruncher Aug 10 '22 at 12:03
@Vasili You typed "If x is a set.." Is it what you ment to type? – Stats Cruncher Aug 10 '22 at 12:04
@StatsCruncher: Yes. It's hard to analyze a statement without the context. – Vasili Aug 10 '22 at 12:07
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Suppose $\forall_x$ means for all $x$ in the set of hamsters on Earth. Do all these hamsters belong to the empty set? No. At least the hamster I have at home does not. – Michal Adamaszek Aug 10 '22 at 12:13
@MichalAdamaszek Why is it so difficult to understand? – Stats Cruncher Aug 10 '22 at 12:18
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Let's explicitly include mention the universe $U$ so the statement becomes $\forall x\in U (x\in\varnothing)$. This is the definition of subset so it is equivalent to $U \subseteq \varnothing$. This is true if and only if $U = \varnothing$ because the only subset of the empty set is the empty set. – Jaap Scherphuis Aug 10 '22 at 12:23
@MichalAdamaszek In your example of "hamsters" the Universe is "the set of hamsters on Earth" which is not empty. It seems that tt is easier for me to understand in the aspect of "¬∃x(¬x∈∅)" and the Universe is not empty. – Stats Cruncher Aug 10 '22 at 12:36
@MichalAdamaszek In your example of "hamsters" the Universe is "the set of hamsters on Earth" which is not empty. It seems that it is easier for me to understand your example in the aspect of "¬∃x(¬x∈∅) i.e ¬∃x (x∈U and ¬x∈∅) " and the Universe is not empty. Because ∃x (x∈U and ¬x∈∅) is true. So ¬∃x (x∈U and ¬x∈∅) is false. – Stats Cruncher Aug 10 '22 at 13:00
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I think Jaap's comment should be the answer. – Kevin Aug 10 '22 at 13:05
Vacuous truth arises only when the unfulfillable statement occurs as the restriction in a restricted universal quantification, i.e. as $\phi$ in in $\forall v (\phi \to \psi)$, because then there are no objects that could satisfy $\phi$ but not $\psi$ and thereby falsify the implication. When it is the main predication itself of the universal quantification that fails to be satisfied, then the universal claim is false. – Natalie Clarius Aug 10 '22 at 18:59

5 Answers5

4

$x$ refers to an element of a Set. You need to first define the set to be able to make propositions with it's elements. Your expression is not well defined, meaning, there is not a unique correct answer.

Here is an analogy :

If i asked you how much a car costs. Your answer is probably : I don't know, depends on the car. Also depends on where you buy it. Depends on what you mean by "cost", do you include insurrance ?

Therefore, "How much does a car cost ?" is usually not a well defined expression. Same holds true for your question.

So let's give your question some context. First, what will we call a set ?

You can (naively) define a set by :

Listing all of it's elements.
A property.

Saying : "$\forall x \,: \, (\text{something is true)}$" without specifying a Set, from which to draw $x$ from, wether it be from prior context or explicitly, is in many ways the same as asking that car question.

So let's try to give it context :

Turns out that there are only two possible contexts to interpret your proposition. Or in other words, two ways to make it well-defined. A set is either empty, or it is not. Let's call our Set $X$ :

If $X$ is empty, your proposition becomes : $\forall x \in X : \:(x \in X)$, which is true, by definition of $x$.
If $X$ is not empty : Notice that $\emptyset \subset X$.Also, for any $Y\subset X$, The following proposition : $\forall x \in X : \:(x \in Y)$ is always false, unless $X=Y$. Since by assumption, $X$ is not empty, we have $X \neq Y$ and thus your proposition is false.

Now, regarding vacuously true. You first need a well defined expression, in order for something to be "vacuously true". And indeed, propositions on the empty set, are "vacuously true". But like i said, they need to be well defined first.

edited Aug 10 '22 at 18:08

answered Aug 10 '22 at 12:30

Pastudent

868

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"If $X$ is empty, $\forall x {\in} X :(x \in X)$ is a tautology (always true)." $\tag{}$ Being "always" true in a given context* (here: an empty universe) does not make a formula a first-order validity, let alone a tautology. – ryang Aug 10 '22 at 17:40
@ryang Yes, you are right ! I think the new formulation is correct. – Pastudent Aug 10 '22 at 18:07
Better, but what does "by definition of $x$" mean? – ryang Aug 10 '22 at 19:01
@ryang By definition of $x$ being an element of the Set. How would you formulate it ? – Pastudent Aug 10 '22 at 19:04

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It is false.

Definition: By axiom, we have $\emptyset$ such that $\neg \exists x(x\in \emptyset)$.

Theorem: $\neg \forall x (x\in \emptyset)$

Proof:

Suppose to the contrary that $\forall x (x\in \emptyset)$.

Substituting $x=\emptyset$, we would then have $\emptyset \in \emptyset$ and a contradiction: $\neg \exists x(x\in \emptyset) \land \exists x(x\in \emptyset)$

Therefore, $\neg \forall x (x\in \emptyset)$.

edited Aug 10 '22 at 16:24

answered Aug 10 '22 at 14:41

Dan Christensen

14,529

What do you mean by ∅∈∅? Did you mean ∅⊆∅ ? – Stats Cruncher Aug 10 '22 at 15:33
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@StatsCruncher No. It was an an application of universal specification. – Dan Christensen Aug 10 '22 at 15:53
I learn ∅∈{∅} before, but I am quire unfamiliar with∅∈∅. – Stats Cruncher Aug 10 '22 at 16:00
This was a proof by contradiction. To prove $\neg \forall x (x\in \emptyset)$, I first assumed (i.e. asked, what if) $ \forall x (x\in \emptyset)$ and then obtained a contradiction: $\neg \exists x(x\in \emptyset) \land \exists x(x\in \emptyset)$. So, that assumption must be false. – Dan Christensen Aug 10 '22 at 16:12
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@StatsCruncher See my edits. – Dan Christensen Aug 10 '22 at 16:25

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$$\forall x(x\in\emptyset)\tag1$$

Translation: Every member of the universe belongs to the empty set.

If the universe is not actually empty, then any member of it belongs to a nonempty set, so sentence $(1)$ is, by counterexample, false.
On the other hand, if the universe is $\emptyset,$ then, given any universal sentence $\forall x\,\psi(x),$ no counterexample can possibly be found to falsify it, so $\forall x\,\psi(x)$ is true and, in particular, sentence $(1)$ is true.

Hence, sentence $(1)$ is true precisely when the universe of discourse is $\emptyset.$ Think of the sentence as characterising an empty universe of discourse.

And, since, in an empty universe, every universal sentence is vacuously true, sentence $(1)$ is vacuously true precisely the universe of discourse is $\emptyset.$

(So, since sentence $(1)$ is false when the discourse domain is, say, $\mathbb R,$ it is not a logical validity.)

edited Mar 02 '23 at 19:11

answered Aug 10 '22 at 18:59

ryang

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No. Vacuous truth arises only when the unfulfillable statement occurs as the restriction in a restricted universal quantification, i.e. as $\psi$ in in $\forall v (\psi \to \phi)$, because then there are no objects that could satisfy $\psi$ but not $\phi$ and thereby falsify the implication.

When it is the main predication itself of the universal quantification that fails to be satisfied, i.e. $\phi$ in $\forall v (\phi)$, then the universal claim is false. It is not the case that all elements of the universe are an element of the empty set, in fact none of them are, so the statement is simply false.

The universe of a first-order-model is, by definition, always non-empty.

answered Aug 10 '22 at 19:04

Natalie Clarius

10,530

@user21820 Done. – Natalie Clarius Aug 23 '22 at 21:14
Thank you very much! – user21820 Aug 24 '22 at 07:37

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The empty set is the set such that $x\notin\emptyset$ for any $x\in X$ where $X$ is some set or another. The confusion arises when you say "for all $x$ in $X$", because if $X$ has no elements, in other words if $X=\emptyset$ then it is true that "for all $x$ in $X$, $x$ is in the empty set" since there is no $x$ in $X$.

It is like me saying that "All of my Aston Martins are covered in gold leaf" a true statement, but meaningless since I don't own an Aston Martin. My friend Al does own an Aston Martin, but it's not covered in gold leaf, so he can't truthfully claim that all of his Aston Martins are covered in gold leaf, whereas I can.

answered Aug 10 '22 at 12:39

Suzu Hirose

11,660