A Concise Introduction to Logic, Patrick J. Hurley, Wadsworth, 2006, 9th ed., 8.6, III, 8, p. 436

Translate and derive the conclusion.

If there are any policemen, then if there are any robbers, then they will arrest them. If any robbers are arrested by policemen, they will go to jail. There are some policemen, and Macky is a robber. Therefore, Macky will go to jail. (Px: x is a policeman; Rx: x is a robber; Axy: x will arrest y; Jx: x will go to jail)

The first premise (x)[Px ⊃ (y)(Ry ⊃ Axy)] simplifies to (x)(y)[(Px • Ry) ⊃ Axy], while the second premise (x){[Px ⊃ (y)[R ⊃ (Axy ⊃ Jy)]} simplifies to (x)(y)[(Px • Ry • Axy) ⊃ Jy].

 

1.     (x)(y)[(Px • Ry) ⊃ Axy] 2.     (x)(y)[(Px • Ry • Axy) ⊃ Jy] 3.     (∃x)Px • Rm ∴ Jm 4.     ~ Jm 5.     (∃x)Px 6.     Pr 7.     (y)[(Pr • Ry • Ary) ⊃ Jy] 8.     (Pr • Rm • Arm) ⊃ Jm 9.     ~ (Pr • Rm • Arm) 10. ~ Pr  v ~ Rm v ~ Arm 11. ~ Rm v ~ Arm 12. Rm • (∃x)Px 13. Rm 14. ~ Arm 15. (y)[(Pr • Ry) ⊃ Ary] 16. (Pr • Rm) ⊃ Arm 17. ~ (Pr • Rm) 18. ~ Pr v ~ Rm 19. ~ Rm 20. ~ Rm • Rm 21. ~ ~ Jm 22. Jm

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