The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 4th edition, Exercise 10.5E, 5(d)

Show that the following set of sentences is inconsistent. And to do so, we show that we can derive a sentence and its negation within the set of primary assumptions.

1. (∃x)(y)[Hxy ⊃ (w)Jww] / Assumption
2. (∃x) ¬ Jxx • ¬ (∃x) ¬ Hxm / Assumption
3. (∃x) ¬ Jxx / 2Simp.
4. ¬ Jaa / 3EI x/a
5. (∃w) ¬ Jww / 4EG
6. ¬ (w)Jww / 5CQ
7. (y)[Hcm ⊃ (w)Jww] / 1EI x/c
8. Hcm ⊃ (w)Jww / UI y/s
9. ¬ Hcm / 6,8MT
10. (∃x) ¬ Hxm / 8EG
11. ¬ (x)Hxm / 10CQ
12. ¬ (∃x) ¬ Hxm / 2Simp.
13. ¬ ¬ (x) Hxm / 12 CQ
14. (x)Hxm / 13DN
15. ¬ (x) Hxm / 10CQ
16. ¬ (x) Hxm • (x)Hxm / 14,15Conj.