[{'from': 'human', 'value': 'Can you present the derivation of the following logical statement?\n((Q ∧ R) → P), (¬(R → S) → (¬W → Q)), (Q ∨ ¬S) ⊢ (R → (P ∨ W))'}, {'from': 'gpt', 'value': "To derive the statement (R → (P ∨ W)) from the premises ((Q ∧ R) → P), (¬(R → S) → (¬W → Q)), and (Q ∨ ¬S), we can use a proof by contradiction. We will assume the opposite of what we are trying to prove (i.e., assume ¬(R → (P ∨ W))), and derive a contradiction.\n\nHere's the derivation:\n\n1. ¬(R → (P ∨ W)) (Assumption)\n2. ¬(¬R ∨ (P ∨ W)) (Implication rule, equivalent to R ∧ ¬(P ∨ W))\n3. R (Disjunctive syllogism, from 2 and ¬(P ∨ W))\n4. ¬(P ∨ W) (From 2 and R)\n5. ¬P ∧ ¬W (De Morgan's law, from 4)\n6. Q ∨ ¬S (Premise)\n7. (Q ∧ R) → P (Premise)\n8. ¬(R → S) → (¬W → Q) (Premise)\n9. ¬(R → S) (Assumption for a proof by contradiction)\n10. ¬¬W → Q (From 8 and 9, by contraposition)\n11. W ∨ Q (From 6 and 10, by disjunctive syllogism)\n12. P ∨ W (From 7, 3, and modus ponens)\n13. P (From 5 and 12, by disjunctive syllogism)\n14. P ∨ W (From 13, by addition)\n15. ¬P ∧ ¬W (From 5)\n16. Contradiction: P ∧ ¬P (From 14 and 15)\n17. Therefore, (R → (P ∨ W)) is true by contradiction (from 1).\n\nThus, we have shown that if ((Q ∧ R) → P), (¬(R → S) → (¬W → Q)), and (Q ∨ ¬S) are true, then (R → (P ∨ W)) must also be true."}, {'from': 'human', 'value': 'b) Use the resolution method of propositional logic to show that\nthe following set of clauses is unsatisfiable:\n{{C, ¬C, A}, {C, ¬A}, {¬B, ¬A}, {¬B, C}, {¬C}, {A, B}}'}, {'from': 'gpt', 'value': 'To show that the set of clauses {{C, ¬C, A}, {C, ¬A}, {¬B, ¬A}, {¬B, C}, {¬C}, {A, B}} is unsatisfiable, we can use the resolution method.\n\nFirst, we write out all possible pairs of clauses:\n\nFrom {C, ¬C, A} and {C, ¬A}: We can resolve on C to obtain {¬A, A}, which can be simplified to the empty clause {}.\nFrom {¬B, ¬A} and {¬B, C}: We can resolve on ¬B to obtain {¬A, C}, which can be simplified to {¬A, C}.\nFrom {¬C} and {¬A, C}: We can resolve on C to obtain {¬A}, which can be simplified to {¬A}.\nFrom {A, B} and {¬A}: We can resolve on A to obtain {B}, which cannot be further simplified.\nAt this point, we have obtained the empty clause {}, which indicates that the set of clauses is unsatisfiable. Therefore, there is no truth assignment that satisfies all the clauses, and the set of clauses is contradictory.'}]