Question 1208961: Premise:
1.(∃x) (Ax • Bx) ⊃ (x) (Cx • Dx)
2.(∃x) Ax ⊃ (x) (Bx • Cx)
3.Ae
Conclusion:
(∃x) (Dx • Bx)
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! Here's the derivation for the conclusion using the eighteen rules of inference and quantifier negation:
**1. (∃x) (Ax • Bx) ⊃ (x) (Cx • Dx)** (Given)
**2. (∃x) Ax ⊃ (x) (Bx • Cx)** (Given)
**3. Ae** (Given)
**4. (∃x) Ax** (Existential Introduction from 3)
**5. (x) (Bx • Cx)** (Modus Ponens from 2, 4)
**6. Be** (Universal Instantiation from 5)
**7. (∃x) (Ax • Bx)** (Existential Introduction from 3 and 6)
**8. (x) (Cx • Dx)** (Modus Ponens from 1, 7)
**9. (∃x) Dx** (Existential Introduction from 8)
**10. (∃x) (Dx • Bx)** (Commutative Property - order doesn't matter within disjunction)
**Therefore, (∃x) (Dx • Bx)** is the conclusion derived from the premises.
**Explanation:**
1. We start with the given premises.
2. We use existential introduction based on premise 3 (Ae) to infer that there exists an x such that Ax is true.
3. We use modus ponens with premise 2 and the inferred (∃x) Ax to conclude that for all x, if Ax is true, then (Bx • Cx) must also be true.
4. Since we know Ae is true (premise 3), we can infer Be using universal instantiation from the conclusion in step 5 (which states for all x...). This means Bx is also true.
5. We use existential introduction again, this time with the information from premise 3 (Ae) and the newly inferred Be, to conclude that there exists an x such that both Ax and Bx are true (Ax • Bx).
6. We use modus ponens again, but this time with premise 1 and the inferred (∃x) (Ax • Bx). This tells us that for all x, if (∃x) (Ax • Bx) is true, then (x) (Cx • Dx) must also be true.
7. Since we know (∃x) (Ax • Bx) is true (from step 5), we can infer the existence of Dx using existential introduction from the conclusion in step 8 (which states for all x...).
8. Finally, we use the commutative property of disjunction to rearrange the order of Dx and Bx within the disjunction (Dx • Bx). This is a valid transformation because the order doesn't matter when elements are connected by OR.
This derivation shows that the conclusion (∃x) (Dx • Bx) logically follows from the given premises.
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