Question

Prove the following argument using hypothetical reasoning.

a)A ∧ ¬D
b)B → (C → D)
c)(A → B) → ¬C
The answer should be the format below :
1. p ⤑ q Premise
2. q ⤑ r Premise
3. |p Hypothesis
4. |q M.P. (1, 3)
5. | r M.P. (2, 4)
6. p ⤑ r. C.P. (3–5)

Answer 1

The argument uses hypothetical reasoning to derive a conclusion from three premises. By introducing hypothesis, applying modus ponens, and modus tollens, it is determined that A implies ¬C, which supports the given premises.

Step-by-step explanation:

To prove the argument using hypothetical reasoning, we follow a step-by-step process and apply logical rules such as modus ponens (M.P.) and conditional proof (C.P.). The argument provided consists of three premises, and we are to derive a valid conclusion from these.

1. A ∧ ¬D — Premise
2. B → (C → D) — Premise
3. (A → B) → ¬C — Premise
4. | A — Hypothesis
5. | ¬D — Simplification (1)
6. | A → B — Hypothesis
7. | B — M.P. (4, 6)
8. | C → D — M.P. (2, 7)
9. | ¬C — Modus Tollens (5, 8)
10. A → ¬C — C.P. (4–9)

The argument proceeds by introducing the first premise A and by negating D, then it proceeds to derive B through a hypothetical assumption of A → B. B then allows for the derivation of C → D due to the second premise. Given that D is false, C must also be false by modus tollens; thus, A implies ¬C. This successfully concludes the hypothetical reasoning process.