Relation between Empty Quantifiers and False Statements.

Question

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.

Answer 1

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.

Answer 2

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.