Why I cannot use universal instantiation rule here?

Question

I have the following premise:

$$\forall x (Fx \vee \neg Fx) \rightarrow Ga$$

Why I cannot use UI and get: $(Fa \vee \neg Fa) \rightarrow Ga$ And what is the difference between the first premise and the below premise:

$$\forall x ((Fx \vee \neg Fx) \rightarrow Ga)$$

Thanks in advance.

Answer 1

You cannot use UI since you can only use UI on statements that are universals. But that first statement is not a universal, but a conditional, whose antecedent is a universal. Maybe it helps you to see that if I add another set of parentheses:

$(\forall x (Fx \lor \neg Fx)) \rightarrow Ga$

Indeed, the difference with the second statement is that the second statement is a universal, and so you can apply UI to that statement to get $(Fa \lor \neg Fa) \rightarrow Ga$

In general, you want rules of inference to apply only to statements as a whole. For example, take Simplification:

$P \land Q$

$\therefore P$

If you try to apply Simplification to the component statement $A \land B$ of the larger statement $(A \land B) \rightarrow C$, the following happens:

$(A \land B) \rightarrow C$

$\therefore A \rightarrow C$

But that is clearly not valid! That's why inference rules can only be applied to statements as a whole, and not to component statements.

Answer 2

And what is the difference between the first premise and the below premise

Suppose you are trying to get a date to a dance.

"If every girl is spoken for, then I am going alone."

"For any girl, if she is spoken for then I am going alone."

Note that in the first case, as long as any girl isn't asked you might have a chance. In the second case, if any girl has made a commitment then you can't get a partner at all. The second statement is equivalent to:

"If any girl is spoken for, then I am going alone."

Quite different from the first statement.