In Kleene's "Mathematical Logic" and "Introduction to Metamathematics" for a classical predicate calculus the following two rules of inference are chosen.
If $A(x) \Rightarrow C$ then $(\exists xA(x)) \Rightarrow (C)$ and
if $C \Rightarrow A(x)$ then $(C) \Rightarrow (\forall xA(x))$ where $C$ does not contain variable $x$ free.
I tried motivating these choices but unfortunately I could not. Because it is a classical predicate calculus I tried considering truth table semantics to somehow see why these results should be valid, but what I found (not sure if correct) is that the following results are semantically valid as well.
If $A(x) \Rightarrow C$ then $(\forall x(A(x)) \Rightarrow (C)$ and also if $C \Rightarrow A(x)$ then $(C) \Rightarrow (\exists x A(x))$ where $C$ again does not contain variable $x$ free.
If this is indeed true then I am confused as one sees that $\exists$ and $\forall$ act in inference rules in exactly the same way while intuitively I would think that these two logical symbols should act differently.
I would appreciate your advice or thoughts about this.
Best Answer
An intuitive answer, in a world of match sticks and fire and no magic, i.e. one can not make fire out of nothing.
Consider a variable $x$ to mean a match stick. Let $A(x)$ mean "Stick $x$ smokes" and $C$ mean "There is fire".
The rule "if $A(x)\Rightarrow C$ then $(\exists x A(x))\Rightarrow C$" states the very obvious: "if stick x smokes then there's fire" means "if there is some stick $x$ that smokes, then there's fire". This sentence holds in any universe of match sticks.
Now, consider your rule "if $A(x)\Rightarrow C$ then $(\forall x A(x))\Rightarrow C$". It states the following: "if stick x smokes, then there's fire" means also that "if all sticks $x$ are smoking, then there's fire". This derivation is valid only in models that contain at least one object. Because in an empty universe that has no match sticks, all match sticks (which are actually none) are smoking. However, there can be no fire without match sticks!