Why is "for all $x\in\varnothing$, $P(x)$" true, but "there exists $x\in\varnothing$ such that $P(x)$" false?

Question

There exists an $X\in A$ such that $P(X)$. When $A$ is the empty set then this statement is false because there is nothing in $A$ that when plugged in for $X$, makes $P(X)$ come out True.

However, when the quantifier is the universal one, meaning "For all $X\in A$" (which is the empty set), then the statement is true !!

How is that ? .. Because by the reasoning in the first statement, there are not any values of $X$ that makes $P(X)$ come out True.

I'm missing something !!

(I'm studying Electrical Engineering and self studying velleman's how to prove it and the quantifiers are already kicking my butt so simplify the answers for me if possible :))

Answer 1

The negation of the statement "for all $x\in X$, $P(x)$ holds" is "there exists $x\in X$ such that $P(x)$ does not hold". Well, then if $X$ is empty, then there is not a single $x\in X$ such that $P(X)$ is not true, and thus it is indeed true that for all $x\in X$, $P(X)$ is true.

The confusing part is that in general the validity of a statement of the form $\forall x\in X: P(X)$ does not imply the existence of a single $x\in X$ for which $P(X)$ holds. It only states that if $x\in X$ then $P(x)$ holds.

You can also think of it this way: Suppose I claim that everything in a box in front of you is pink. Can you sue me if you open it up and see a banana in there? Sure, the banana is yellow, so the judge will rule in your favor. But, can you sue me if you open up the box and see nothing at all in there? Well, no, you have nothing to complain about: everything in the box is indeed pink since nothing in it is not pink.

Answer 2

In classical logic, given a particular structure for some language, if something is not false in the structure, then it's true there. Of course we can easily make things complicated by considering a structure whose universe is made of structures of a particular language, and so on and so forth. So let's stay in the context of one particular structure.

Something is vacuously true if it just cannot be made false. That is to say, there's no counterexample. If it's not false, it has to be true.

So saying "for every $x\in\varnothing$ ..." does not have a counterexample, and therefore it is true; but on the other hand "there exists $x\in\varnothing$ ..." is false because we can prove that there isn't such $x$, and therefore conclude a contradiction.

Answer 3

Let $A$ be the set of monkeys in the room, let $P(x)$ denote the sentence "x is playing chess".

$\forall x\in A, P(x)$ means "every monkey in the room is playing chess", which is true almost all the time, since most rooms are monkey-free.

$\exists x\in A, P(x)$ means "there is a monkey in the room playing chess", which is almost never true.

Answer 4

You are not alone who is puzzled with that. Lewis Carroll (who was a logician by trade) also objected to "vacuous truth". The choice that the modern logic makes is due more to the convenience of drawing implications without worrying if some sets are empty or not at each step, than to the intuition. You want to be able to combine the sentences "Everybody who is not too short can pick an apple on this tree" and "Everybody who is not too tall will be able to enter the vault" into "Everybody who is neither too short, nor too tall will be able to pick the apple and go to the vault with it" without heavy thinking of how the threshold heights in those cases are related to each other. This forces you to assign the "true" value to the last statement even if you need to be at least 6 feet tall to reach the apple and no more than 5 feet tall to enter the vault.

In general, such vacuous usage of "for every" is not uncommon in the real life. When the sheriff in a small town proclaims publicly that "Every criminal will be caught and punished!", the streets are patrolled day and night, and there are no crimes in the town, you don't call him a liar and the police department a worthless waste of taxpayer money, quite the opposite: you suspect that he is so right that even the to-be-criminals believe this statement of his and abstain from misdeeds.

Answer 5

I think I'm getting it. How about the conditional form of this though. For all X(If x belongs to the empty set, then P(X)). This is only false when the hypothesis is True and the conclusion is False. But since the hypothesis is always False since there are not any members of the empty set, therefore the conditional is always True. Great . Now when we change the quantifier into the existential one. Why is it wrong ?! The hypothesis is still wrong because since there aren't any elements in the empty set, then there are not even some elements , Therefore shouldn't it be True as well

Answer 6

Wittgenstein also had problems with it. I'm not sure how well I can explain this, but I'll give it a go.

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Crudely, his claim was that you can't talk about things that don't exist.

He says, in typically abstruse form:

5.521

I dissociate the concept of all from truth-functions.

Frege and Russell introduced generality in association with logical product or sum. This made it difficult to understand the propositions '$(\exists x).fx$' [there exists $x$ and the proposition $f$ holds for it] and $'(x).fx'$ [all $x$ and the proposition $f$ holds for it].

His difficulties are associated with treating "$\exists x$" and "$\forall x$" as valid statements on their own (which may or not be held true these days). The [brackets] are my attempt to interpret the statements in the way he thought was silly. He goes on to say:

5.524

If we are given objects then we are given all objects.

By which he basically means that $\forall$ is redundant and adds nothing to a statement, though he puts it in a very theory laden way. You can always write a statement without it.

But, it seems he would also have a problem with doing this:

$$\forall x : P(x) \rightarrow \nexists x: ¬P(x) $$

or in words, replace "for all $x$, $P$ is true" with "there is no $x$ for which $P$ does not hold". The two statements are mathematically the same (in first order logic). He would, I think, favor

$$P(x)$$

which one might read as "$P$ is true", with the subtext of "for all $x$ that exist, obviously! How could we know if it is true for those things that don't?"