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Is the following proposition true or false?

Every element of the empty set has three toes.

In symbols, the proposition is written as

$$ \forall x \left( x \in \emptyset \to x\text{ has three toes }\right) $$

I think that, due to the fact the set is empty, we cannot test any proposition. Hence, the proposition is false. Is my reasoning correct?

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    $\begingroup$ Can you find an element of the empty set that doesn’t have three toes? If not, then the statement must be true. $\endgroup$
    – Clayton
    Dec 15, 2021 at 16:05
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    $\begingroup$ See Vacuous truth $\endgroup$ Dec 15, 2021 at 16:06
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    $\begingroup$ About the question in the title, the answer is no for "The empty set has at least one element". $\endgroup$
    – lhf
    Dec 15, 2021 at 16:07
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    $\begingroup$ Your subject line is an example of the ambiguous nature of the word "any" in some contexts. "Is any proposition true?" would often mean "Is there any proposition that is true?", so that "any" would in effect mean "some". But I suspect you meant "Is it the case that any proposition at all, no matter which one, is true?", so that "any" would in effect mean "every". English-speaking mathematicians use the word "any" too much. Changing it to "every" would disambiguate the sentence. $\endgroup$ Dec 15, 2021 at 16:20
  • $\begingroup$ I'd think you'd want to change your title to "is every proposition regarding elements_of the empty set true?". $\endgroup$ Dec 16, 2021 at 0:14

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A universal quantification over the empty set is always true. Someone justifies it by saying 'there is no element that makes the consequent false.' That's what we call a vacuous truth.

I want to add some comments about this topic. When I was first taught about the notion of vacuous truth, I thought like 'it must be useful because the proposition is guaranteed to be always true'! But it turned out that a vacuously true proposition does literally nothing in a series of arguments because it says nothing about the truth of the consequent, which is, of course, the matter of our interest. So you can take it as 'the proposition does nothing but at least it does not cause any trouble because it's true anyway.' So we can forget about the use of the proposition and keep deducing another.

Given that $P$ is false, $P \to Q$ is true but the truth of $Q$ is not determined.

The second comment: I thought about a specific case when a vacuous true sentence does something meaningful. Given that

$$ \forall x(x \in A \to Q(x)) \to R $$ If $A = \varnothing$, the consequent $R$ is true.

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Suppose I say: all of my purple hats are purchased online. Now, also suppose I don't have any purple hats. Does that make my statement true or false?

Well, suppose you have three purple hats, and you purchased them all online. That is: all three purple hats you have were purchased online, and you don't have a purple hat that wasn't purchased online: there is no exception to the claim. Or, different yet: the number of purple hats you have (three) equals the number of purple hats that you have that were purchased online (also three).

For me: All zero purple hats I have are purchased online; there is again no exception to this: I have no purple hat that I didn't purchase online. And again: the number of purple hats I have (zero) matches the number of purple hats I have that I purchased online (also zero). So: all purple hats I have are indeed purchased online.

Of course, it is also true that that number of purple hats I have that are not purchased online is also zero, and so all zero hats I have were not purchased online, i.e. it is also true that every purple hat I have is not purchased online.

Indeed, you can fill in anything you want. For example: every purple hat I have has 3 toes!

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Let predicate $t_3 (x)$ denote that $x$ has $3$ toes.

$$ \begin{aligned} \left(\forall x \right) \left( x \in \emptyset \rightarrow t_3 (x) \right) &\equiv \left(\forall x \right) \left( \neg \left( x \in \emptyset \right) \lor t_3 (x) \right) \\ &\equiv \left(\forall x \right) \left( \neg \left( x \in \emptyset \right) \lor \neg \neg t_3 (x) \right) \\ &\equiv \left(\forall x \right) \neg \left( \left( x \in \emptyset \right) \land \neg t_3 (x) \right) \\ &\equiv \neg \left(\exists x \right) \left( \left( x \in \emptyset \right) \land \neg t_3 (x) \right) \end{aligned} $$

Hence, there is no $x$ that belongs to the empty set and that does not have $3$ toes.

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It would be vacuously true. Tautologies appear when your system $\Sigma$, is taken to be empty. More information can be found here Vacuous truth

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Every statement that says "For every member $x$ of the empty set . . ." is true. But not every statement about the empty set is true. "The empty set is infinite." is not true.

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