Meaning and Argument, Ernest Lepore, Blackwell, 2003, Problem 16.11(2), p. 289

We are only asked to symbolize this argument, but we may as well prove it here. The layering looks daunting.

The father of the father of Annnette flies. Anyone who has a father who flies fears loss. So, the father of Annette fears loss.

Let Fxy - x is a father of y, a - Annette, Vx - x flies, Lx - x fears loss.

1. (∃x){Fxa • (z)(Fza ⊃z = x) • (∃y)[Fyx • (w)(Fwx ⊃w = y) • Vy]}
2. (x)[( ∃y)(Fyx • Vy) ⊃Lx]
3. ∴(∃x)[Fxa • (y)(Fya ⊃y = x) • Lx]
4. Fma • (z)(Fza ⊃z = m) • (∃y)[Fym • (w)(Fwm ⊃w = y) • Vy] ......... 1 EI x/m
5. (∃y)(Fym • Vy) ⊃Lm ......... 2 UI x/m
6. (∃y)[Fym • (w)(Fwm ⊃w = y) • Vy] ......... 4 Simp.
7. Frm • (w)(Fwm ⊃w = r) • Vr ......... 6 EI y/r
8. Frm • Vr ......... 7 Simp.
9. (∃y)(Fym • Vy) ......... 8 EG
10. Lm ......... 9,5 MP
11. (z)(Fza ⊃z = m) ......... 4 Simp.
12. Fya ⊃y = m ........ 11 UI z/y
13. (y)(Fya ⊃y = m) ......... 12 UG
14. Fma ......... 4 Simp.
15. Fma • (y)(Fya ⊃y = m) • Lm ......... 14,10,13 Conj.
16. (∃x)[Fxa • (y)(Fya ⊃y = x) • Lx] ......... 15 EG