A Concise Introduction to Logic, Patrick J. Hurley, Wardsworth, 2006, 9th ed., 8.6, III, 3, p. 436

Translate then derive the conclusion.

A horse is an animal. Therefore, whoever owns a horse owns an animal. (Hx: x is a horse; Ax: x is an animal; Oxy: x owns y)

 

The difficulty here is to translate the conclusion correctly. Before we do that, it helps to paraphrase it:

 

Therefore, anyone who owns a horse owns an animal.

Therefore, if someone owns a horse, then he owns an animal.

 

or, less elegantly but, as is almost always the case, more helpfully:

 

Therefore, if for any person there is a horse that he owns, then there is an animal that he owns.

 

The last translation in particular gives us a hint as to how to translate it into formal logic. Only ‘he’ is anaphoric – which will make the overall sentence structure a conditional – not the ‘horse’. This is not a donkey type sentence. Hence, two existential quantifiers – one for the antecedent, one for the consequent of the conditional.

1.     (x)(Hx ⊃ Ax) ∴ (z)[(∃x)(Hx • Ozx) ⊃ (∃y)(Ay • Ozy)] 2.     ~ (z)[(∃x)(Hx • Ozx) ⊃ (∃y)(Ay • Ozy)] 3.     ~ (z)[~ (∃x)(Hx • Ozx) v (∃y)(Ay • Ozy)] 4.     ~ (z)[(x)(Hx ⊃ ~ Ozx) v (∃y)(Ay • Ozy)] 5.     (∃z) ~ [(x)(Hx ⊃ ~ Ozx) v (∃y)(Ay • Ozy)] 6.     (∃z)[~ (x)(Hx ⊃ ~ Ozx) • ~ (∃y)(Ay • Ozy)] 7.     (∃z)[(∃x)~ (Hx ⊃ ~ Ozx) • ~ (∃y)(Ay • Ozy)] 8.     (∃z)[(∃x)~ (Hx ⊃ ~ Ozx) • (y)~ (Ay • Ozy)] 9.     (∃z)[(∃x)~ (~ Hx v ~ Ozx) • (y)~ (Ay • Ozy)] 10. (∃z)[(∃x)(Hx • Ozx) • (y)(~ Ay v ~ Ozy)] 11. (∃x)(Hx • Omx) • (y)(~ Ay v ~ Omy) 12. (∃x)(Hx • Omx) 13. Hn • Omn 14. Hn ⊃ An 15. H 16. An 17. (y)(~ Ay v ~ Omy) • (∃x)(Hx • Omx) 18. (y)(~ Ay v ~ Omy) 19. ~ An v ~ Omn 20. ~ Omn 21. Omn • Hn 22. Omn 23. Omn • ~ Omn 24. ~ ~ (z)[(∃x)(Hx • Ozx) ⊃ (∃y)(Ay • Ozy)] 25. (z)[(∃x)(Hx • Ozx) ⊃ (∃y)(Ay • Ozy)]

AIP 2 Impl 3 CQ 4 CQ 5 DM 6 CQ 7 CQ 8 Impl 9 DM 10 EI 11 Simp 12 EI 1 UI 13 Simp 14,15 MP 11 Com 17 Simp 18 UI 16,19 DS 13 Com 21 Simp 20-23 Conj 2-23 IP 24 DN