Question

Use the first four rules of inference (MP, MT, HS, DS) to derive the conclusion of the following symbolized argument.

1. PREMISE: (B ⊃ ∼M) ⊃ (T ⊃ ∼S)

2. PREMISE: B ⊃ K

3. PREMISE: K ⊃ ∼M

4. PREMISE: ∼S ⊃ N

CONCLUSION: T ⊃ N

Answer 1

To derive T ⊃ N from the given premises using rules of inference, we first use Hypothetical Syllogism to infer B ⊃ ∼M, then Modus Ponens to derive T ⊃ ∼S from Premise 1 and the inferred B ⊃ ∼M. Lastly, we apply Hypothetical Syllogism again with T ⊃ ∼S and ∼S ⊃ N to conclude T ⊃ N.

Step-by-step explanation:

To derive the conclusion T ⊃ N using the first four rules of inference (Modus Ponens, Modus Tollens, Hypothetical Syllogism, and Disjunctive Syllogism), we proceed as follows:

• From Premise 2 (B ⊃ K) and Premise 3 (K ⊃ ∼M), we can use Hypothetical Syllogism (HS) to infer B ⊃ ∼M.
• With the inferred B ⊃ ∼M and Premise 1 ((B ⊃ ∼M) ⊃ (T ⊃ ∼S)), we apply Modus Ponens (MP) to derive T ⊃ ∼S.
• Having T ⊃ ∼S and Premise 4 (∼S ⊃ N), we again apply Hypothetical Syllogism (HS) to infer the conclusion T ⊃ N.

Each step follows a deductive reasoning pattern that ensures the validity of the argument; if the premises are true, then the conclusion must also be true.