This post and video go over the solutions to the proofs assigned as homework in the Proofs with the Rules of Inference post and video. You will probably learn more from watching the video than you would from just looking over the solutions.
1. P // Premise 2. P ⊃ Q // Premise 3. Q ⊃ R // Premise Prove: R 4. Q // 1, 2 Modus Ponens 5. R // 3, 4 Modus Ponens or: 4. P ⊃ R // 2, 3 Hypothetical Syllogism 5. R // 1, 4 Modus Ponens
1. P & ~Q // Premise 2. P ⊃ Q // Premise Prove: R 3. P // 1 Simplification 4. ~Q // 1 Simplification 5. Q // 2, 3 Modus Ponens 6. Q ∨ R // 5 Addition 7. R // 4, 6 Disjunctive Syllogism
1. P ⊃ R // Premise 2. R ⊃ Q // Premise 3. P ∨ T // Premise 4. (P ∨ T) ⊃ (T ⊃ S) // Premise Prove: Q ∨ S 5. T ⊃ S // 3, 4 Modus Ponens 6. P ⊃ Q // 1, 2 Hypothetical Syllogism 7. (P ⊃ Q) & (T ⊃ S) // 5, 6 Conjunction 8. Q ∨ S // 3, 7 Constructive Dilemma
1. P ⊃ (Q ∨ S) // Premise 2. ~Q ⊃ P // Premise 3. Q ⊃ R // Premise 4. ~R ∨ T // Premise 5. ~T // Premise Prove: Q ∨ S 6. ~R // 4, 5 Disjunctive Syllogism 7. ~Q // 3, 6 Modus Tollens 8. P // 2, 7 Modus Ponens 9. Q ∨ S // 1, 8 Modus Ponens
1. P ∨ (T ∨ R) // Premise 2. U ⊃ ~P // Premise 3. (T ⊃ Q) & (R ⊃ S) // Premise 4. W & (U & V) // Premise Prove: Q ∨ S 5. U & V // 4 Simplification 6. U // 5 Simplification 7. ~P // 2, 6 Modus Ponens 8. T ∨ R // 1, 7 Disjunctive Syllogism 9. Q ∨ S // 3, 8 Constructive Dilemma
1. P ⊃ Q // Premise 2. Q ⊃ R // Premise 3. S ⊃ T // Premise 4. T ⊃ U // Premise 5. ~U ∨ ~R // Premise Prove: ~S ∨ ~P 6. P ⊃ R // 1, 2 Hypothetical Syllogism 7. S ⊃ U // 3, 4 Hypothetical Syllogism 8. (S ⊃ U) & (P ⊃ R) // 6, 7 Conjunction 9. ~S ∨ ~P // 5, 8 Destructive Dilemma
1. P ⊃ Q // Premise 2. Q ⊃ R // Premise 3. S ⊃ T // Premise 4. T ⊃ U // Premise 5. ~U ∨ ~R // Premise Prove: ~S ∨ ~P 6. (T ⊃ U) & (Q ⊃ R) // 2, 4 Conjunction 7. ~T ∨ ~Q // 5, 6 Destructive Dilemma 8. (S ⊃ T) & (P ⊃ Q) // 1, 3 Conjunction 9. ~S ∨ ~P // 7, 8 Destructive Dilemma
