The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, Ex. 10E, 5(f), p. 563

The following set of sentences is inconsistent. Show that it is so. Strategy: we show that the original assumptions lead to a contradiction on line 16.

1. (x)[(Sx • Bxx) ⊃ Kax] / Assumption
2. (x)(Hx ⊃Bxx) / Assumption
3. (∃x)(Sx • Hx) / Assumption
4. (x) ¬ (Kax • Hx) / Assumption
5. Sm • Hm / 3EI x/m
6. Hm / 5Simp.
7. Hm ⊃Bmm / 2UI x/m
8. Bmm / 6,7MP
9. Sm / 5Simp.
10. Sm • Bmm / 8,9Conj.
11. (Sm • Bmm) ⊃ Kam / 1UI x/m
12. Kam / 10,11MP
13. ¬ (Kam • Hm) / 4UI x/m
14. ¬ Kam ∨ ¬ Hm / 13DeM
15. ¬ Hm / 12,14DS
16. Hm • ¬ Hm / 6,15Conj.