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1. (x) Ax > ( ∃ x) Bx 2. (x) ~Bx / ( ∃ x) ~Ax CQ of conclusion: ~(x) Ax CQ of line 2: ~ ( ∃ x) Bx 3. ~( ∃ x) Bx 4. ~(x) Ax

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1. ( ∃ x) ~Ax v ( ∃ x) ~Bx 2. (x) Bx / ~(x) Ax 3. ~( ∃ x) ~BxCQ 2 4. ( ∃ x) ~Axcm, ds 1,3 5. ~ (x) Ax CQ 4

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5) If all philosophers are either ethicists or metaphysicians, then there are no logicians. But Russell’s a logician, so some philosophers are not metaphysicians. 1. (x) (Px > (Ex v Mx)) > ~( ∃ x ) Lx 2. L r / ( ∃ x) (Px. ~Mx) 3. ( ∃ x) Lx EG 2 4. ~~( ∃ x) Lx DN 3 5. ~(x)(Px > (Ex v Mx))MT 4,1 6. ( ∃ x)~(Px > (Ex v Mx))CQ 5 7. ~(Pq > (Eq v Mq))EI 6 8. ~(~Pq v (Eq v Mq)IMP 7 9. Pq. ~(Eq v Mq)DM 8 10. Pq. (~Eq. ~Mq) DM 9 11. Pq. ~ MqCM, AS,SM 10 12. ( ∃ x) (Px. ~Mx)EG 11

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7) All utilitarians are ethicists and all idealists are metaphysicians. Therefore, since it is not true that some ethicists are metaphysicians, it is not the case that some utilitarians are idealists. 1.(x) (Ux > Ex). (x) (Ix > Mx) 2. ~( ∃ x) (Ex. Mx)/ ~( ∃ x) (Ux. Ix) 3. (x) ~(Ex. Mx)CQ 2 4. ~(Ex. Mx) UI 3 5. Ux > ExSM, UI 1 6. Ix > MxCM, SM, UI 1 7. ~Ex v ~ MxDM 4 8. Ex > ~Mx IMP 7 9. Ux > ~Mx HS 5, 8 13. (x) ~ (Ux. Ix)CM, DM, UG 9 14. ~( ∃ x) (Ux. Ix)CQ 13 10. Mx > ~Ux TRAN 9 11. Ix > ~Ux HS 10, 6 12. ~Ix v ~UxIMP 11

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