Translate the following arguments into symbolic form and use the rules of inference to derive the conclusion of each: A. If topaz is harder than quartz, then it will scratch quartz and also feldspar. Topaz is harder than quartz and it is also harder than calcite. Therefore, either topaz will scratch quartz or it will scratch corundum. (T, Q, F, C, O) B. If clear-cutting continues in primary forests and the Endangered Species Act is not repealed, then either the Endangered Species Act will be repealed or thousands of animals species will become extinct. Clear-cutting continues in primary forests. The Endangered Species Act will not be repealed. Therefore, thousands of animal species will become extinct. (C, E, T) C. Either animals are mere mechanisms or they feel pain. If either animals feel pain or they have souls, then they have a right not to be subjected to needless pain and humans have a duty not to inflict needless pain on them. It is not the case that animals are mere mechanisms. Therefore, animals have a right not to be subjected to needless pain. (M, P, S, R, D)

(A)

T: Topaz is harder then quartz

Q: Topaz will scratch quartz

F: Topaz will scratch feldspar

C: Topaz is harder than calcite

O: Topaz will scratch corundum

1. T→ (Q • F)
2. T • C // Q v O
3. T Simp, 2
4. Q • F MP, 1,3
5. Q Simp, 4
6. Q v O Add, 5

(B)

C: Clear-cutting continues in primary forests

E: the Endangered Species is repealed

T: Thousands of animals species will become extinct

1. (C • ~E) → (E v T)
2. C
3. ~E //T
4. C • ~E Conj, 2,3
5. E v T MP, 1,4
6. T DS, 3,5

(C)

M: Animals are mere mechanisms

P: Animals feel pain

S: Animals have soul

R: Animals have a right not to be subjected to needless pain

D: Humans have a duty not to inflict needless pain on animals

1. M v P
2. (P v S) → (R • D)
3. ~M // R
4. P DS, 1,3
5. P v S Add, 4
6. R • D MP, 2,5
7. R Simp, 6

Step-by-step explanation

A.

• Premises:

1. If topaz is harder then quartz, then it will scratch quartz and also feldspar. [T→ (Q • F)]
2. Topaz is harder then quartz and it is also harder than calcite. [T • C]

• Conclusion:
• Either topaz will scratch quartz or it will scratch corundum. [ Q v O]

B.

• Premises:

1. If clear-cutting continues in primary forests and the Endangered Species Act is not repealed, then either the Endangered Species Act will be repealed or thousands of animal species will become extinct. [(C • ~E) → (E v T)]
2. Clear-cutting continues in primary forests. [C]
3. The Endangered Species Act will not be repealed [~E]

• Conclusion:

1. Thousands of animal species will become extinct [T]

(C)

• Premises

1. Either animals are mere mechanisms or they feel pain. [M v P]
2. If either animals feel pain or they have souls, then they have a right not to be subjected to needless pain and humans have a duty not to inflict needless pain on them. [(P v S) → (R • D)]
3. It is not the case that animals are mere mechanisms. [~M]

• Conclusion

1. Therefore, animals have a right not to be subjected to needless pain. [R]