flower seed catalog

Help in Logic?? Can You Translate? Sentential Logic? 3?

1. If the seed catalog is correct then if the seeds are planted in April then the flowers bloom in July. The seeds are planted in April. Hence if the flowers do not bloom in July then the seed catalog is not correct.

Hi

1. C = ‘The seed catalogue is correct’, A = ‘The seeds are planted in April’, J = ‘The flowers bloom in July’

C > (A > J), A. Thus, ~J > ~C

2. W = ‘The weather is warm’, C = ‘The sky is clear’, S = ‘We’ll go swimming’, B = ‘We’ll go boating’

(W & C) > (S & B), ~(C > S). Thus, W

This argument is invalid. The conclusion should be ~W.

3. Using the same key:

(W & C) > (S v B), ~(C > S). Thus, ~B > ~W

4. Using the same key:

(W & C) > (S v B), ~(~S > ~C). Thus, W v B

This argument is invalid. The conclusion ought to be ~W v B.

Proofs:

(1) 1. C > (A > J) Premise
(2) 2. A Premise
(3) 3. ~J Assumption for CP
(4) 4. C Assumption for CP
(1,4) 5. A > J 1,4 MP
(1,2,4) 6. J 2,5 MP
(1,2) 7. C > J 4,6 CP
(1,2,3) 8. ~C 3,7 MT
(1,2) 9. ~J > ~C 3,8 CP

(1) 1. (W & C) > (S & B) Premise
(2) 2. ~(C > S) Premise
(3) 3. ~(C & ~S) Assumption for RAA
(4) 4. C Assumption for CP
(5) 5. ~S Assumption for RAA
(4,5) 6. C & ~S 4,5 &I
(3,4,5) 7. (C & ~S) & ~(C & ~S) 3,6 &I
(3,4) 8. ~~S 5,7 RAA
(3,4) 9. S 8 &E
(3) 10. C > S 4,9 CP
(2,3) 11. (C > S) & ~(C > S) 2,10 &I
(2) 12. ~~(C & ~S) 3,11 RAA
(2) 13. C & ~S 12 DNE
(14) 14. W Assumption for CP
(2) 15. C 13 &E
(2,14) 16. W & C 14,15 &I
(1,2,14) 17. S & B 1,16 MP
(1,2,14) 18. S 17 &E
(2) 19. ~S 13 &E
(1,2) 20. W > S 14,18 CP
(1,2) 21. ~W 19,20 MT

(1) 1. (W & C) > (S v B) Premise
(2) 2. ~(C > S) Premise
(3) 3. ~B Assumption
(4) 4. S Assumption
(5) 5. C Assumption
(4,5) 6. C & S 4,5 &I
(4,5) 7. S 6 &E
(4) 8. C > S 5,7 CP
(-) 9. S > (C > S) 4,8 CP
(2) 10. ~S 2,9 MT
(11) 11. S v B Assumption
(3,4) 12. S & ~B 3,4 &I
(3,4) 13. S 12 &E
(2,3,4) 14. S & ~S 10,13 &I
(2,4) 15. ~~B 3,14 RAA
(2,4) 16. B 15 DNE
(17) 17. B Assumption
(2,11) 18. B 11,4,16,17,17 vE
(2,3,11) 19. B & ~B 3,18 &I
(2,3) 20. ~(S v B) 11,19 RAA
(1,2,3) 21. ~(W & C) 1,20 MT
(22) 22. ~C Assumption
(23) 23. ~S Assumption
(5,23) 24. C & ~S 5,23 &I
(5,23) 25. C 24 &E
(5,22,23) 26. C & ~C 22,25 &I
(5,22) 27. ~~S 23,26 RAA
(5,22) 28. S 27 DNE
(22) 29. C > S 5,28 CP
(-) 30. ~C > (C > S) 22,29 CP
(2) 31. ~~C 2,30 MT
(2) 32. C 31 DNE
(33) 33. W Assumption
(2,33) 34. W & C 32,33 &I
(2) 35. W > (W & C) 33,34 CP
(1,2,3) 36. ~W 21,35 MT
(1,2) 37. ~B > ~W 3,36 CP

(1) 1. (W & C) > (S v B) Premise
(2) 2. ~(~S > ~C) Premise
(3) 3. ~(~W v B) Assumption for RAA
(4) 4. ~(~S & C) Assumption for RAA
(5) 5. ~S Assumption for CP
(6) 6. C Assumption for RAA
(5,6) 7. ~S & C 5,6 &I
(4,5,6) 8. (~S & C) & ~(~S & C) 4,7 &I
(4,5) 9. ~C 6,8 RAA
(4) 10. ~S > ~C 5,9 CP
(2,4) 11. (~S > ~C) & ~(~S > ~C) 2,10 &I
(2) 12. ~~(~S & C) 4,11 RAA
(2) 13. ~S & C 12 DNE
(2) 14. C 13 &E
(15) 15. ~W Assumption for CP
(15) 16. ~W v B 15 vI
(-) 17. ~W > (~W v B) 15,16 CP
(3) 18. ~~W 3,17 MT
(3) 19. W 18 DNE
(2,3) 20. W & C 14,19 &I
(1,2,3) 21. S v B 1,20 MP
(2) 22. ~S 13 &E
(23) 23. S Assumption for vE
(24) 24. ~B Assumption for CP
(23,24) 25. S & ~B 23,24 &I
(23,24) 26. S 25 &E
(23) 27. ~B > S 24,26 CP
(2,23) 28. ~~B 22,27 MT
(2,23) 29. B 28 DNE
(30) 30. B Assumption for vE
(1,2,3) 31. B 21,23,29,30,30 vE
(1,2,3) 32. ~W v B 31 vI
(1,2,3) 33. (~W v B) & ~(~W v B) 3,32 &I
(1,2) 34. ~~(~W v B) 3,33 RAA
(1,2) 35. ~W v B 34 DNE

I hope that this helps a little Good luck!

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