Question 1210176: Use the quantifier negation rule together with the eighteen rules of inference to derive the conclusion of the following symbolized argument. Do not use either conditional proof or indirect proof. (Complete the proof in the logic tool. See the Getting Started text for further instructions. Select the Submit button to grade your response.)
Step Argument Justification
1. (∃x)Ax ⊃ [(∃x)Bx ∨ (x)Cx]
2. (∃x)(Ax ⦁ ~Cx)
3. ~(x)Cx ⊃ [(x)Fx ⊃ (x)~Bx] / (∃x)~Fx
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! ```
1. (∃x)Ax ⊃ [(∃x)Bx ∨ (x)Cx] Premise
2. (∃x)(Ax ⦁ ~Cx) Premise
3. ~(x)Cx ⊃ [(x)Fx ⊃ (x)~Bx] Premise
4. Ax ⦁ ~Cx EI 2
5. Ax Simp 4
6. ~Cx Simp 4
7. (∃x)Ax EG 5
8. (∃x)Bx ∨ (x)Cx MP 1, 7
9. ~(x)Cx DN 6
10. (∃x)~Cx QN 9
11. (x)Fx ⊃ (x)~Bx MP 3, 9
12. ~(x)~Bx ⊃ ~(x)Fx Contra 11
13. (∃x)Bx ∨ (∃x)~Bx Tautology
14. (x)Cx ∨ ~(x)Cx Tautology
15. (∃x)Bx ∨ ~(x)Cx Add 8
16. (∃x)Bx ∨ (∃x)~Cx Replace 10, 15
17. (∃x)~Bx ∨ (∃x)Bx Comm 16
18. (∃x)~Bx DS 10, 16
19. ~(x)Fx MP 12, 18
20. (∃x)~Fx QN 19
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