Question 1208960: Use 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 ⊃ (Bx ≡ Cx)]
2.An • Am
3.Cn • ~Cm
Conclusion:
Bn • ~Bm
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! Here's the derivation of the conclusion using the eighteen rules of inference:
**1. (x) [Ax ⊃ (Bx ≡ Cx)]** (Given)
**2. An • Am** (Given)
**3. Cn • ~Cm** (Given)
**4. An** (Simplification from 2)
**5. Am** (Simplification from 2)
**6. A** (Universal Instantiation of 4 in 1) - Ax is implied for any x, so substituting A for x gives us A.
**7. Bx ≡ Cx** (Modus Ponens from 1 and 6) - Since we have A (from step 4) and the premise states Ax implies Bx≡Cx, we can infer Bx ≡ Cx.
**8. Cn** (Simplification from 3)
**9. ~(∃x) Bx** (Assuming for reductio ad absurdum) - We temporarily assume the opposite of what we want to conclude (Bn) to reach a contradiction.
**10. ~(∃x) Cx** (Since Bx ≡ Cx from step 7, if there's no Bx, there can't be Cx either)
**11. ~Bn** (Universal Instantiation of 9 in 7) - If there's no Bx for any x (from 9), then specifically there's no B for A (which we established as true in step 6).
**12. Cm** (Reductio ad Absurdum from 3, 11) - We reach a contradiction. Premise 3 states Cn AND ~Cm, but if ~Bn leads to ~Cm (step 11), then assuming ~Bn is false. Therefore, Bn must be true.
**13. Bn** (from Reductio ad Absurdum in 12) - We reject the initial assumption (~Bn in step 9) because it led to a contradiction. So, Bn must be true.
**14. ~Bm** (Conjunction Simplification from 3 after establishing Bn in 13) - Since we now know Bn is true (from 13), we can separate the conjunction (Cn • ~Cm) in premise 3 to get ~Cm.
**Therefore, the conclusion is Bn • ~Bm.**
This derivation uses the following rules of inference:
* Simplification
* Universal Instantiation
* Modus Ponens
* Reductio ad Absurdum (Assuming for reductio ad absurdum)
* Conjunction Simplification
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