There are at most two scientists in the laboratory. At least two scientists in the laboratory are Russians. No Russians are Chinese. Therefore, if Norene is a Chinese scientist, then she is not in the laboratory. (Sx: x is a scientist; Lx: x is in the laboratory; Rx: x is Russian; Cx: x is Chinese; n: Norene) / All accounting is shown below the proof due to blogger formatting constraints.
1. (x)(y)(z)[(Sx • Lx • Sy • Ly • Sz • Lz) ⊃ (x = y v x = z v y = z)]
2. (∃x)(∃y)(Sx • Lx • Rx • Sy • Ly • Ry • x ≠ y)
3. (x)(Rx ⊃ ~ Cx)
∴ (Sn • Cn) ⊃ ~ Ln
4. Sn • Cn 5. Rn ⊃ ~ Cn 6. Cn • Sn 7. Cn 8. ~ Rn 9. (∃y)(Sm • Lm • Rm • Sy • Ly • Ry • m ≠ y) 10. Sm • Lm • Rm • Sr • Lr • Rr • m ≠ r 11. (y)(z)[(Sm • Lm • Sy • Ly • Sz • Lz) ⊃ (m = y v m = z v y = z) 12. (z)[(Sm • Lm • Sr • Lr • Sz • Lz) ⊃ (m = r v m = z v r = z) 13. (Sm • Lm • Sr • Lr • Sn • Ln) ⊃ (m = r v m = n v r = n) 14. Rr ⊃ ~ Cr 15. Rr • Sm • Lm • Rm • Sr • Lr • m ≠ r 16. Rr 17. r ≠ n 18. Rm ⊃ ~ Cm 19. Rm • Sm • Lm • Sr • Lr • Rr • m ≠ r 20. Rm 21. ~ Cm 22. m ≠ n 23. m ≠ r • Sm • Lm • Rm • Sr • Lr • Rr 24. m ≠ r 25. m ≠ r • m ≠ n • r ≠ n 26. ~ (m = r v m = n v r = n) 27. ~ (Sm • Lm • Sr • Lr • Sn • Ln) 28. ~ (Sm • Lm • Sr • Lr • Sn) v ~ Ln 29. Sm • Lm • Sr • Lr • Rm • Rr • m ≠ r 30. Sm • Lm • Sr • Lr 31. Sn 32. Sm • Lm • Sr • Lr • Sn 33. ~ Ln |
34. (Sn • Cn) ⊃ ~ Ln
4. ACP; 5. 3 UI; 6. 1 Com; 7. 6 Simp; 8. 5,7 MT; 9. 2 EI; 10. 9 EI; 11. 1 UI; 12. 11 UI; 13. 12 UI; 14. 3 UI; 15. 10 Com; 16. 15 Simp; 17. 8,16 Id; 18. 3 UI; 19. 10 Com; 20. 19 Simp; 21. 18,20 MP; 22. 7,21 Id; 23. 10 Com; 24. 23 Simp; 25. 17,22,24 Conj; 26. 25 DM; 27. 13,26 MT; 28. 27 DM; 29. 10 Com; 30. 29 Simp; 31. 4 Simp; 32. 30,31 Conj; 33. 28,32 DS; 34. 4-33 CP
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