Symbolic Logic, D. Jacquette, Wadsworth, 2001, Chpt. 8, IV(14), p.435

The argument:

"There are ghosts if and only if all ghosts or specters are just figments of the imagination. Something is just a figment of the imagination. But not everything fails to be a ghost. Therefore, some ghosts are just figments of the imagination."

The proof:

1. (∃x)Gx ≡ (y)(z)[(Gy ∨Sz) ⊃(∃w)(Iw • Fyw • Fzw)]
2. (∃x)(∃y)(Iy • Fxy)
3. ¬ (x) ¬ Gx
4. ∴ (∃x)[Gx • (∃y)(Iy • Fxy)]
5. ¬ ¬ (∃x) Gx ......... 3 QC
6. (∃x) Gx ......... 5 DN
7. (y)(z)[(Gy ∨Sz) ⊃(∃w)(Iw • Fyw • Fzw)] ......... 6,1 MP
8. Ga ......... 6 EI
9. Ga ∨Sm ......... Add
10. (z)(Ga ∨Sz) ⊃(∃w)(Iw • Faw • Fzw) ......... 7 UI y/a
11. (Ga ∨Sm) ⊃(∃w)(Iw • Faw • Fmw) ......... 10 UI z/m
12. (∃w)(Iw • Faw • Fmw) ......... 9,11 MP
13. Ir • Far • Fmr ......... 12 EI w/r
14. Ir • Far ......... 13 Simp.
15. (∃y)(Iy • Fay) ......... 14 EG
16. Ga • (∃y)(Iy • Fay) ......... 8,15 Conj.
17. (∃x)[Gx • (∃y)(Iy • Fxy)] ......... 16 EG