Symbolic Logic, D. Jacquette, Wadsworth, 2001, Chpt. 8, IV(17), p. 436

The argument:

"If Annie is elected, then everything we worked to achieve in the last 4 years will be destroyed. Either Annie or Sara will be elected. But it's definitely not going to be Sara. Whatever is destroyed must be rebuilt from scratch. Hence, everything we worked to achieve in the last 4 years must be rebuilt from scratch."

The proof:

1. Ea ⊃(x)(Wx ⊃Dx)
2. Ea ∨Es
3. ¬ Es
4. (x)(Dx ⊃ Rx)
5. ∴(x)(Wx ⊃ Rx)
6. Ea ......... 3,2 DS
7. (x)(Wx ⊃Dx) ......... 6,1 MP
8. Wx ⊃Dx ......... 7 UI x/x
9. Dx ⊃ Rx ......... 4 UI x/x
10. Wx ⊃ Rx ......... 8,9 HS
11. (x)(Wx ⊃ Rx) ......... 10 UG

Symbolic Logic, D. Jacquette, Wadsworth, 2001, Chpt. 8, IV(16), p. 435

The argument:

"All schnauzers are prize dogs and champion hunters if they are properly trained. Ralph is a schnauzer. Therefore, there is something that if properly trained is a champion hunter."

The proof:

1. (x)[(Sx • Tx) ⊃(Px • Dx • Cx • Hx)]
2. Sr
3. ∴ (∃x)[Tx⊃(Cx • Hx)]
4. * ¬ (∃x)[Tx⊃(Cx • Hx)] ......... AIP
5. * (x) ¬ [Tx⊃(Cx • Hx)] ......... 4 QC
6. * (x) ¬ [ ¬ Tx ∨(Cx • Hx)] ......... 5 MI
7. * (x)[Tx • ¬ (Cx • Hx)] ......... 6 DeM
8. * Tr • ¬ (Cr • Hr) ......... 7 UI
9. * Tr ......... 8 Simp.
10. * Sr • Tr ......... 2,9 Conj.
11. * (Sr • Tr) ⊃(Pr • Dr • Cr • Hr) ......... 1 UI
12. * Pr • Dr • Cr • Hr ......... 10,11 MP
13. * ¬ (Cr • Hr) ......... 8 Simp.
14. * ¬ Cr ∨ ¬ Hr ......... 13 DeM.
15. * Cr ......... 12 Simp.
16. * ¬ Hr ......... 15,14 DS
17. * Hr ......... 12 Simp.
18. * Hr • ¬ Hr ......... 16,17 Conj.
19. ¬ ¬ (∃x)[Tx⊃(Cx • Hx)] ......... 4-18 IP
20. (∃x)[Tx⊃(Cx • Hx)] ......... 19 DN

Symbolic Logic, D. Jacquette, Wadsworth, 2001, Chpt. 8, IV(15), p. 435

The argument:

"Something is costly and something is free. Something is prohibitively expensive only if nothing is free. As a result, nothing is prohibitively expensive."

The proof:

1. (∃x)Cx • (∃x)Fx
2. (∃x)(Px• Cx) ⊃¬ (∃x)Fx
3. ∴¬ (∃x)(Px• Cx)
4. (∃x)Fx ......... 1 Simp.
5. ¬ (∃x)(Px• Cx) ......... 4,2 MT

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