Question

Prove that the following arguments are invalid. 1. (∃x) (Ax * ～Bx) 2. (∃x) (Ax *...

Prove that the following arguments are invalid.

1. (∃x) (Ax * ～Bx)

2. (∃x) (Ax * ～Cx)

3. (∃x) ( ～Bx * Dx)

/ (∃x) [Ax *(～Bx * Dx)]

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Answer #1

Answer :

1. (∃x) (Ax * ～Bx) Premise

2. (∃x) (Ax * ～Cx)   Premise

3. (∃x) ( ～Bx * Dx) Premise

4. Aa * ～Ba (1) , Existential Instantiation

5. Aa (4) , Simplification

6. ～Ba    (4) , Simplification

7. Aa * ～Ca (2) , Existential Instantiation

8. Aa (7) , Simplification

9. ～Ca (7) , Simplification

10.  ～Ba * Da (3) ,Existential Instantiation

11. Aa *(～Ba * Da) (10), Simplification

12. (∃x) [Ax *(～Bx * Dx)] (11), Existential Generalization

Thus , the conclusion  (∃x) [Ax *(～Bx * Dx)] is derived from the given premises and hence the argument is valid

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