The actual full question: If there are free variables in the premises of a natural deduction that doesn't lead to contradiction, then is there a non-empty domain such that its constant terms satisfy the conclusion?
Well, let's consider a set of premises such that it has free variables somewhere. Using the rules of inference, I can get to a conclusion that also has the free variables. Since these free variables are any arbitrary terms, there has to be a non-empty domain that satisfy it. Is it correct? Does the domain have to be previously defined or can it be arbitrary?
Also, is the following inference ok? $$Px \vdash \exists x (Px)$$