Suppose that our universe of discourse is $X=\{ x_1, x_2, \cdots , x_n \}$ Then, intuitively, the following seems to be truth.
$$(\exists x\in X) (P(x)) := (P(x_1) \lor P(x_2) \lor \cdots \lor P(x_n))$$
$$(\forall x\in X) (P(x)) := (P(x_1) \wedge P(x_2) \wedge \cdots \wedge P(x_n))$$
I want to make this definition clearer. And from this definition, I want to prove the properties about quantifiers only using the inference rules of propositional logic. (I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.)
If you have an idea, or if you know the references, please help me.