I have a couple of questions about the role of the free variables.
The first one is more specific:
1) So a formula to be in a Skolem Normal Form it has to be:
a) In Prenex Normal Form
b) Not to have any ∃ quantifiers.
c) Not to have any free variables
Okay, I know how to achieve a) and b), but I couldn't find information anywhere on how to achieve c)?
2) I am trying to grasp my mind around the idea of the difference between free variables and bound variables. I know the difference mechanically, I know the rules, I understand how to work with them, but I still cannot completely understand the difference in terms of intuition. I will share some questions that will help me understand:
a) If we have the formula: p(x)&∃q(x). The first occurrence of x is free, the second is bound.
- What is different between the two sub formulas in terms of x? My idea is that:
- In both sub formulas if there is no x in the Domain to satisfy the
formula it is false. - If there is at least one x to satisfy both sub formulas: then P(x) is feasible (I am not sure if that is the correct term) and q(x) is true.
- If both subformulas are satisfied for all x then both are true.
- In both sub formulas if there is no x in the Domain to satisfy the
Is that the difference? That if there is at least one x (but not all) that satisfies the formula if we have existence quantifier then the formula is true but if we don't have - it is feasible?
b) What happens when we make a Variable assignment?
- Does the bounded x get an assignment?
- Do both x's get assignment?
Thank you in advance!