Question

Select the Conclusion that follows in a single step from the given premises. Given the following premises:

1. ∼R ∨ ∼R
2. R ∨ (∼J • ∼H)
3.∼R ⊃ (H • B)
a. ∼J • ∼H 1, 2, DS
b. R 1, DN
c. R ∨ ∼(J ∨ H) 2, DM
d. (R ∨ ∼J) • ∼H 2, Assoc
e. H• B 1, 3, MP

Answer 1

Using the rules of inference and the given premises, option (e) H• B is the correct conclusion that follows in a single step from premises 1 and 3 using modus ponens.

Step-by-step explanation:

To determine which conclusion follows in a single step from the given premises, we must evaluate each option based on rules of inference: disjunctive syllogism (DS), DeMorgan's Theorem (DM), and modus ponens (MP).

First, premise 1 (∼R ∨ ∼R) simplifies to ∼R via the rule of double negation (DN), supporting option (b). However, this doesn't follow in one step directly from the premises given but requires a simplification step. This means we must look for another option that does follow in a single step.

Premise 2 (R ∨ (∼J • ∼H)) presents a disjunction, and premise 3 (∼R ⊃ (H • B)) is a conditional statement. Since we know from premise 1 that ∼R is true, by modus ponendo ponens (another term for MP), we can derive H • B from premise 3. Therefore, option (e) is the correct conclusion and follows in a single step from premises 1 and 3.