Applying Universal Instantiation to ∀xP(x) → ∀xQ(x)

Question

I understand if i have: ∀x(P(x) → Q(x))

Applying Universal Instantiation i have:

P(c) → Q(c)

as ∀x is bound to both P(x) and Q(x).

I'm struggling to understand how i'd apply Universal Instantiation to:

∀xP(x) → ∀xQ(x)

Obviously it cannot be P(c) → Q(c), but i'm unsure how i'd represent it. Some lines of thought i had were, using a second arbitrary constant say d:

P(c) → Q(d)

However this seems wrong to me, as c and d are both arbitrary constants so represent the same thing. Another thing i was thinking was something like:

(P(c)) → (Q(c))

But again i haven't been able to find any examples of this online.

Any help would be much appreciated.

Thanks.

Answer 1

You can't apply Universal Instatiation to $\forall x \ P(x) \to \forall x \ Q(x)$

You can only apply Universal iIstantiation to a universal statement, and $\forall x \ P(x) \to \forall x \ Q(x)$ is not a universal statement ... it is a conditional statement

In general, you can never apply inference rules to a component statement of a larger statement.