To show that the given well-formed formula cannot be satisfied for any interpretation using the resolution principle, we can apply proof by contradiction. We assume that the formula is satisfiable and try to derive a contradiction using resolution.
Let's first convert the given formula into clause form, which is a set of disjunctions of literals. The formula can be represented as follows:
1. ¬(P → Q) ∨ (¬Q → ¬P)
2. (P → Q) ∨ (¬Q → ¬P) (De Morgan's Law)
3. (¬P ∨ Q) ∨ (¬Q ∨ ¬P) (Implication Elimination)
4. (¬P ∨ Q ∨ ¬Q) ∨ (¬Q ∨ ¬P) (Associativity)
5. (¬P ∨ ¬Q ∨ ¬P) ∨ (¬Q ∨ Q) (Commutation)
6. (¬P ∨ ¬P ∨ ¬Q) ∨ (Q ∨ ¬Q) (Associativity)
7. (¬P ∨ ¬Q) ∨ (Q ∨ ¬Q) (Idempotant)
8. (¬P ∨ ¬Q) ∨ ⊤ (Negation Law)
9. ¬P ∨ ¬Q (Identity Law)
Now, let's assume that there exists an interpretation that satisfies the formula, i.e., there exists a truth assignment that makes the formula true. We will show that this assumption leads to a contradiction.
Assume that there exists an interpretation I such that I(¬P ∨ ¬Q) = true.
To obtain a contradiction, we will apply the resolution principle by deriving the empty clause (⊥) from the set of clauses:
Clause 1: ¬P ∨ ¬Q
Clause 2: ¬P
Applying resolution:
Resolve(Clause 1, Clause 2, P):
¬Q
Now, we have a new clause ¬Q. However, we don't have any other clauses that contain Q to resolve with. Therefore, we cannot derive the empty clause (⊥), which is a contradiction.
Since we cannot derive the empty clause, we conclude that our assumption that the given formula is satisfiable is incorrect. Hence, the well-formed formula cannot be satisfied for any interpretation.
