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.
Best Answer
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.