SOLUTION: 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

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Question 1208963: 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.
Premise:
1.(∃x) Ax ⊃ ~(∃x) (Bx • Ax)
2.~(x) Bx ⊃ ~(∃x) (Ex • ~Bx)
3.An
Conclusion:
~(x) Ex
Answer by textot(100) About Me  (Show Source):
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**1. (∃x) Ax ⊃ ~(∃x) (Bx • Ax)**
Given
**2. ~(x) Bx ⊃ ~(∃x) (Ex • ~Bx)**
Given
**3. An**
Given
**4. (∃x) Ax**
Existential Introduction (from 3)
**5. ~(∃x) (Bx • Ax)**
Modus Ponens (1, 4)
**6. ~(x) (Bx • Ax)**
Quantifier Negation (5)
**7. ~(x) Bx v ~(x) Ax**
De Morgan's Law 1 (6)
**8. ~(x) Bx**
Modus Ponens (2, 7)
**9. ~(∃x) (Ex • ~Bx)**
Modus Ponens (2, 8)
**10. ~(x) (Ex • ~Bx)**
Quantifier Negation (9)
**11. ~(x) Ex v ~(x) ~Bx**
De Morgan's Law 1 (10)
**12. ~(x) Ex v (x) Bx**
Double Negation (11)
**13. (x) Bx**
Modus Tollens (8, 12)
**14. ~(∃x) Ex**
Modus Ponens (2, 13)
**15. (x) ~Ex**
Quantifier Negation (14)
**Therefore, ~(x) Ex**
This derivation demonstrates that the conclusion ~(x) Ex can be derived from the given premises using the specified rules of inference and quantifier negation.
**1. (∃x) Ax ⊃ ~(∃x) (Bx • Ax)**
Given
**2. An**
Given
**3. (∃x) Ax**
Existential Introduction (from 2)
**4. ~(∃x) (Bx • Ax)**
Modus Ponens (1, 3)
**5. ~(x) (Bx v ~Ax)**
Quantifier Negation (4)
**6. ~(Bx v ~An)**
Universal Instantiation (5)
**7. ~Bx • ~~An**
De Morgan's Law 1 (6)
**8. ~~An**
Simplification (7)
**9. An**
Double Negation (8)
**10. ~Bx**
Simplification (7)
**11. (x) ~Bx**
Universal Generalization (10)
**12. ~(∃x) Bx**
Quantifier Negation (11)
**13. ~(∃x) (Ex • ~Bx)**
Modus Ponens (2, 12)
**14. (x) ~(Ex • ~Bx)**
Quantifier Negation (13)
**15. (x) (~Ex v ~~Bx)**
De Morgan's Law 1 (14)
**16. (x) (~Ex v Bx)**
Double Negation (15)
**17. ~(∃x) (Ex • ~Bx)**
Quantifier Negation (16)
**18. (x) ~(Ex • ~Bx)**
Repetition (17)
**19. (x) (~Ex v Bx)**
De Morgan's Law 1 (18)
**20. (x) ~Ex**
Simplification (19)
**Therefore, ~(x) Ex is derived.**
This derivation demonstrates the use of the quantifier negation rule and other rules of inference to derive the desired conclusion from the given premises.