Exercise 1.13.5: Determine and prove whether an argument inEnglish is valid or invalid.
Prove whether each argument is valid or invalid. First find theform of the argument by defining predicates and expressing thehypotheses and the conclusion using the predicates. If the argumentis valid, then use the rules of inference to prove that the form isvalid. If the argument is invalid, give values for the predicatesyou defined for a small domain that demonstrate the argument isinvalid.
QUESTION A AND B ALREADY SOLVED. PLEASE solve part C, D, E usingsame method, and give some explantion for your answer. THANKS!
The domain for each problem is the set of students in aclass.
(a)
Every student on the honor roll received an A.
No student who got a detention received an A.
No student who got a detention is on the honor roll.
- H(x): x is on the honor roll
- A(x): x received an A.
- D(x): x got a detention.
∀x (H(x) → A(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ H(x))
Valid.
| 1. | ∀x (H(x) → A(x)) | Hypothesis |
| 2. | c is an arbitrary element | Element definition |
| 3. | H(c) → A(c) | Universal instantiation, 1, 2 |
| 4. | ¬∃x (D(x) ∧ A(x)) | Hypothesis |
| 5. | ∀x ¬(D(x) ∧ A(x)) | De Morgan's law, 4 |
| 6. | ¬(D(c) ∧ A(c)) | Universal instantiation, 2, 5 |
| 7. | ¬D(c) ∨ ¬A(c) | De Morgan's law, 6 |
| 8. | ¬A(c) ∨ ¬D(c) | Commutative law, 7 |
| 9. | ¬H(c) ∨ A(c) | Conditional identity, 3 |
| 10. | A(c) ∨ ¬H(c) | Commutative law, 9 |
| 11. | ¬D(c) ∨ ¬H(c) | Resolution, 8, 10 |
| 12. | ¬(D(c) ∧ H(c)) | De Morgan's law, 11 |
| 13. | ∀x ¬(D(x) ∧ H(x)) | Universal generalization, 2, 12 |
| 14. | ¬∃x (D(x) ∧ H(x)) | De Morgan's law, 13 |
(b)
No student who got an A missed class.
No student who got a detention received an A.
No student who got a detention missed class.
- M(x): x missed class
- A(x): x received an A.
- D(x): x got a detention.
¬∃x (A(x) ∧ M(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ M(x))
The argument is not valid. Consider a class that consists of asingle student named Frank. If M(Frank) = D(Frank) = T and A(Frank)= F, then the hypotheses are all true and the conclusion is false.In other words, Frank got a detention, missed class, and did notget an A.
(c)
Every student who missed class got a detention.
Penelope is a student in the class.
Penelope got a detention.
Penelope missed class.
(d)
Every student who missed class got a detention.
Penelope is a student in the class.
Penelope did not miss class.
Penelope did not get a detention.
(e)
Every student who missed class or got a detention did not get anA.
Penelope is a student in the class.
Penelope got an A.
Penelope did not get a detention.
