Question 1207200: Please use the 18 rules of natural deduction, the 4 instantiation and generalization rules to derive the conclusions of this problem.
1. (x)(Bx ⊃ Cx)
2. (∃x)(Ax • Bx) /(∃x)(Ax • Cx)
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **1. (x)(Bx ⊃ Cx)**
* Premise
**2. (∃x)(Ax • Bx)**
* Premise
**3. Aa • Ba**
* Existential Instantiation (2)
**4. Ba**
* Simplification (3)
**5. Ba ⊃ Ca**
* Universal Instantiation (1)
**6. Ca**
* Modus Ponens (4, 5)
**7. Aa • Ca**
* Conjunction (3, 6)
**8. (∃x)(Ax • Cx)**
* Existential Generalization (7)
**Explanation:**
1. **Existential Instantiation (2):** We introduce a new constant 'a' to represent an arbitrary object that satisfies the existential quantifier in premise 2.
2. **Simplification (3):** We extract the conjunct 'Ba' from the conjunction '(Aa • Ba)'.
3. **Universal Instantiation (1):** We instantiate the universal quantifier in premise 1 with the constant 'a'.
4. **Modus Ponens (4, 5):** We apply the rule of Modus Ponens to derive 'Ca' from 'Ba' and 'Ba ⊃ Ca'.
5. **Conjunction (2, 5):** We combine 'Aa' and 'Ca' using the rule of Conjunction.
6. **Existential Generalization (6):** We generalize the statement 'Aa • Ca' to obtain the existential quantifier '(∃x)(Ax • Cx)'.
This derivation demonstrates that the conclusion (∃x)(Ax • Cx) logically follows from the given premises within the specified 7 steps.
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