For Kev, Understanding Symbolic Logic, V. Klenk, Unit 8, problem 7m

Try Indirect Proof. This helps to unpackage the first two premises. Then the trick is to use addition to set up propositions which will help break down the other two premises.

1. G ⊃(H • I)


2. J ⊃(H • K)


3. [(L ⊃¬ G) • M] ⊃N


4. (M ⊃N) ⊃(L • J)


5. ∴ I ∨K


6. * ¬ (I ∨K) / AIP


7. * ¬ I • ¬ K / 6DeM


8. * ¬ I / 7Simp


9. * ¬ K / 7Simp


10. * ¬ G ∨(H • I) / 1CE


11. * (¬ G ∨H) • (¬ G ∨I) /10Dist


12. * ¬ G ∨I / 11Simp


13. * ¬ G / 8,12DS


14. * ¬ J ∨(H • K) / 2CE


15. * (¬ J ∨H) • (¬ J ∨K) / 14Dist


16. * ¬ J ∨K / 15Simp


17. * ¬ J / 9,16DS


18. * ¬ G ∨¬ L /13Add


19. * ¬ L ∨¬ G / 18Comm


20. * (L ⊃¬ G) / 19CE


21. * ¬ J ∨¬ L / 17Add


22. * ¬ L ∨¬ J / 21Comm


23. * ¬ (L • J) / 22DeM


24. * ¬ (M ⊃N) / 4,23MT


25. * ¬ (¬ M ∨N) / 24CE


26. * M • ¬ N / 25DeM


27. * M / 26Simp


28. * ¬ N / 26Simp


29. * ¬ [(L ⊃¬ G) • M] / 3,28MT


30. * ¬ (L ⊃¬ G) ∨¬ M / 29DeM


31. * ¬ (L ⊃¬ G) / 27,30DS


32. * (L ⊃¬ G) • ¬ (L ⊃¬ G) / 20,31Conj


33. ¬ ¬ (I ∨K) / 6-32IP


34. I ∨K / 33DN