Predicate Logic, Howard Pospesel, Prentice Hall, 2nd edition, 2003, chpt. 11, ex. 21

A line from the poem Desiderata: If you compare yourself with others, you may become vain and bitter; for always there will be greater and lesser persons than yourself, is spun out into an argument, which goes like this:

If one compares oneself with greater people, one will become bitter. And if one compares oneself with lesser people, one will become vain. If people compare themselves with others, they will compare themselves with some who are greater as well as some who are lesser. Consequently, if you compare yourself with others, you will become vain and bitter.

The domain (universe of discourse) is people. For simplicity's sake, I keep the symbolisation to the minimum. The aim is to prove the argument. Our glossary: Gxy - x is greater than y, Lxy - x is lesser than y, Bx - x is bitter, Vx - x is vain, Cxxy - x compares oneself to y.

1. (x)(y)[(Gyx • Cxxy) ⊃Bx]
2. (x)(y)[(Lyx • Cxxy) ⊃Vx]
3. (x)(y){Cxxy ⊃[(∃z)(Gzx • Cxxz) • (∃w)(Lwx • Cxxw)]}
4. ∴ (x)(y)[Cxxy ⊃(Vx • Bx)]
5. * Cxxy / ACP
6. * (y){Cxxy ⊃[(∃z)(Gzx • Cxxz) • (∃w)(Lwx • Cxxw)]} / 3UI x/x
7. * Cxxy ⊃[(∃z)(Gzx • Cxxz) • (∃w)(Lwx • Cxxw)] / 6UI y/y
8. * (∃z)(Gzx • Cxxz) • (∃w)(Lwx • Cxxw) / 5,7MP
9. * (∃z)(Gzx • Cxxz) / 8Simp.
10. * Gax • Cxxa / 9EI z/a
11. * (y)[(Gyx • Cxxy) ⊃Bx] / 1UI x/x
12. * (Gax • Cxxa) ⊃Bx / 11UI y/a
13. * Bx / 10,12MP
14. * (∃w)(Lwx • Cxxw) / 8Simp.
15. * Lmx • Cxxm / 14EI w/m
16. * (y)[(Lyx • Cxxy) ⊃Vx] / 2UI x/x
17. * (Lmx • Cxxm) ⊃Vx / 16UI y/m
18. * Vx / 15,17MP
19. * Bx • Vx / 13,18Conj.
20. Cxxy ⊃(Vx • Bx) / 5-19CP
21. (y)[Cxxy ⊃(Vx • Bx)] / 20UG
22. (x)(y)[Cxxy ⊃(Vx • Bx)] / 21UG