# Relation between Empty Quantifiers and False Statements.

Consider the following statement about $\mathbb{N}$.

$$\alpha: \forall \ prime \ p, \exists \ prime \ p' (p'<p)$$

This is false.

Now, suppose a set of objects $\psi$ is empty. Now, consider the followings statement.

$$\forall A\in \psi \ \alpha$$

Is this true? Why or why not?

More generally, does any false statement combined with a quantifier which ranges over an empty set of objects produces a true statement?

I wonder, what relation does quantifier has with the false statement?

I have some experience with the syntax and semantics of first-order logic, slightly above the beginner level. So, it would really help, if you could explain in simplest possible manner.

This is an instance of the principle of explosion (a.k.a. ex falso sequitur quodlibet). The expression $\forall x \in X~\varphi(x)$ is simply shorthand for $$\forall x~(x \in X \Rightarrow \varphi(x))$$ so if $X$ is empty, then $x \in X$ is false for all $x$, and hence $x \in X \Rightarrow \varphi(x)$ is true by the principle of explosion.

Likewise, $\exists x \in X~\varphi(x)$ is shorthand for $\exists x~(x \in X \wedge \varphi(x))$, meaning that if $X$ is empty then every sentence of the form $\exists x \in X~\varphi(x)$ is false.

Why (or whether) you should accept the principle of explosion is a whole different bag of worms: some logics (including classical logic and intuitionistic logic) accept it as an axiom, whereas some (including minimal logic) do not. This matter has been discussed a lot on this website—if you're interested, you might want to have a look around.

• I see. Thanks for the clear explanation. I did not see that statement $\forall x\in X$ is a short form. – jaspreet Sep 20 '18 at 21:01

It is what is called a 'vacuously true' statement.

Here is another example, to help convince you that such statements are indeed true, rather than false .. or something altogether:

Suppose I say:

'Every time I played the lottery, I won the jackpot'

In fact, I claim that this claim is true.

Wow, how is this true? Am I the luckiest person alive?

No, it is simply because I have never played the lottery!

That is, all zero times that I played the lottery, I did indeed win the jackpot.

Or: if you 'line up' all the times that I played the lottery, and you 'check' each of thosre times, and make sure that indeed I won the jackpot at each of those times, then you will see that indeed all of those times 'check' out.

Put differently yet, there isn't a single time that I played the lottery and I did not win the jackpot. So, there is no exception to my claim that every time I played the lottery, I won the jackpot. So, my claim is true.

• A very nice example, indeed. I have no doubt as to whether this statement is true. In symbolic terms, you are saying $\forall x, p(x)$, where $x$ is an instance of you playing a lottery and $p(x)$ is a predicate which checks if you won. My question is rather more general, I beleive, how this 'empty quantification', when applied to false statements, produces a true statement. Cleve's insight on unpacking the statement $\forall x \in X$ to implicative statement seems to shed much light on the issue. – jaspreet Sep 20 '18 at 21:28