Decide whether this is correct, using the method of resolution. If not, provide a counter example.

Question

I am new to logic and wanted to decide whether the following is correct using the method of resolution: |= p → ¬ (p → (p ∧ (p ∨ q)))

My attempt to this I answered that the conclusion is incorrect, though the premises hold true. Thus, I begin with the negation of the conclusion and solve for validity.

                                   ¬ (p → ¬ (p→(p ∧(p ∨ q)))) ↔

                                   ¬ p ∧ ¬ (¬(p→(p ∧(p ∨ q)))) ↔

                                   ¬ p ∧ (p→( p→(p ∧(p ∨ q)))) ↔

                                   ¬ p ∧ (p→(p ∨ p)) ↔

                                   ¬ p ∧ (p→p) ↔ ¬ p ∨ p 

Therefore, I deduced two clauses from the negated conclusion, ← p and p ←

Resolution gives the clauses in one step. ← p and p ←

Answer 1

Your error is already in the first step. You simplified $\neg(a \rightarrow b)$ to $\neg a \wedge \neg b$ but it should be $\neg(\neg a \vee b) = a \wedge \neg b$.

$$\neg (p \rightarrow \neg (p\rightarrow(p \wedge (p \vee q))))$$ $$p \wedge \neg\neg(p\rightarrow(p \wedge (p \vee q))))$$ $$p \wedge (p\rightarrow(p \wedge (p \vee q))))$$ $$p \wedge (p \vee q)$$ $$p$$