Question

Question: Select the conclusion that follows in a single step from the given premises. Given the following premises: 1. ∼P 2. L ⊃ (P ∨ M) 3. (P • M) ⊃ (∼R ∨ ∼R) a. (L ⊃ P) ∨ (L ⊃ M) 2, Dist b. (P • M) ⊃ ∼R 3,

Select the conclusion that follows in a single step from the given premises.
Given the following premises:
1. ∼P
2. L ⊃ (P ∨ M)
3. (P • M) ⊃ (∼R ∨ ∼R)
a. (L ⊃ P) ∨ (L ⊃ M) 2, Dist
b. (P • M) ⊃ ∼R 3, Taut
c. L ⊃ (∼R ∨ ∼R) 2, 3, HS
d. P 3, Simp
e. M 1, 2, DS

Answer 1

Using the principles of logical form and deductive reasoning, the correct conclusion that follows in a single step from the given premises is option b: \( (P \cdot M) \supset \sim R \), which uses tautology to reduce the premise to a simpler form.

Step-by-step explanation:

The question involves using logical form and deductive reasoning to select a conclusion that follows in a single step from the given premises. Here are the premises:

1. \( \sim P \)
2. \( L \supset (P \lor M) \)
3. \( (P \cdot M) \supset (\sim R \lor \sim R) \)

We are asked to select from the options a conclusion that follows in a single step. When we apply valid deductive inferences such as disjunctive syllogism, modus ponens, and modus tollens, it becomes clear that the correct answer is:

\( (P \cdot M) \supset \sim R \) (3, Taut)

This uses tautology, where \( \sim R \lor \sim R \) simplifies to \( \sim R \), as the presence of the same term in a disjunction is redundant. Thus, the conclusion is that if both P and M are true, then R must not be true, following directly from premise 3.