Hello, I need help writing proofs for a logic and moral reasoning philosophy class. The proofs are for: 1. Premises: q -> (q &~ q) Conclusion: ~q 2. Premises: k & l
HW 10
Write proofs for the following problems.
(1)
Premises:
a -> b
a -> c
b -> ~c
Conclusion:
~a
| 1. a -> b | A |
| 2. a -> c | |
| 3. b -> ~c | |
| -4. a | SA |
| -5. c | MP 2, 4 |
| -6. b | MP 1, 4 |
| -7. ~c | MP 3, 6 |
| -8. c & ~c | Conj 5, 7 |
| 9. ~a | IP 4-8 |
(2)
Premises:
~(p V q)
(~r) -> (~s)
r -> (q V ~s)
| |
| |
| |
| -4. s | SA |
| -5. ~~s | DN 4 |
| -6. ~~r | MT 2, 5 |
| -7. r | DN 6 |
| -8. q V ~s | MP 3, 7 |
| -9. q | DS 8, 5 |
| -10. p V q | Add 9 |
| -11. (p V q) & ~(p V q) | Conj 10, 1 |
| 12. ~s | IP 4 |
(3)
Premises:
(~q) <-> (~p)
(r <-> s) V ((r & s) V (~r & ~s))
(~r) V p
s V ~q
Conclusion:
| |
| |
| |
| |
| | |
| ME 5 |
| Taut 6 |
| -8. ~r & ~s | SA |
| -9. ~s | Simp 8 |
| -10. s & ~s | Conj 5, 9 |
| 11. ~(~r & ~s) | IP 8-10 |
| 12. r & s | DS 7, 11 |
| 13. r | Simp 12 |
| 14. ~~r | DN 13 |
| 15. p | Ds 3, 14 |
(4)
Premises:
p -> q
q -> r
Conclusion:
p -> (q & r)
| |
| |
| -3. P | SA |
| -4. q | MP 1, 3 |
| -5.r | MP 2, 4 |
| -6. q & r | Conj 4, 5 |
| 7. p -> (q & r) | CP 3-6 |
(5)
Premises:
y -> (z V w)
~w
(~z) V x
Conclusion:
(~x) -> (~y)
| |
| |
| |
| -4. y | SA |
| -5. z V w | MP 1, 4 |
| -6. z | DS 5, 2 |
| -7. ~~z | DN 6 |
| -8. x | DS 3, 7 |
| 9. y -> x | CP 4-8 |
| 10. ~x -> ~y | Trans 9 |
(6)
Premises:
~(a & b)
b V c
Conclusion:
a -> c
| |
| |
| -3. a | SA |
| -4. ~a V ~b | DM 1 |
| -5. ~~a | DM 3 |
| -6. ~b | DS 4, 5 |
| -7. c | DS 2, 6 |
| 8. a ->c | CP 3-7 |
(7)
Premises:
x -> y
((~y) V z) & ((~y) V w)
Conclusion:
x -> z
| |
| |
| -3. x | SA |
| -4. y | MP 1, 3 |
| -5. ~y V z | Simp 2 |
| -6. ~~y | DN 4 |
| -7. z | DS 5, 6 |
| CP 3-7 |
