Hi Everyone! I've been revising for my exams and got stuck at this question:

I would think it's E as I can't really find a model which would satisfy this sentence, though the answer is D. Could you please give me a hint as how I should approach this? Assuming that D is the correct answer I must be missing something.The following sentence of First-Order Logic

$\displaystyle ((\forall X p(X)) \to (\forall X q(X))) \wedge p(a) \wedge \neg q(a)$

is

A. valid and satisfiable

B. valid and unsatisfiable

C. valid but its satisfiability cannot be determined

D. invalid and satisfiable

E. invalid and unsatisfiable

F. invalid but its satisfiability cannot be determined

My thinking: if $\displaystyle p(a)$ is to be true then $\displaystyle \forall X p(X)$ must be true as well. This implies that $\displaystyle \forall X q(X)$ must be true and $\displaystyle \neg p(q)$ will inevitably be false.

EDIT:

OK, if we change $\displaystyle ((\forall X p(X)) \to (\forall X q(X)))$ into $\displaystyle ((\neg \forall X p(X)) \vee (\forall X q(X)))$ I can indeed see that the sentence is satisfiable.

Could someone please tell me why I couldn't see it without transforming the implication into disjunction?