Question

Given the following premises:

1) ∼W
2) C ∨ W
3) R ⊃ ∼(C ∨ W)
Group of answer choices
A) R ⊃ (∼C • ∼W) 3, DM
B) ∼R 2, 3, MT
C) C 1, 2, DS
D) (C ∨ W) ⊃ ∼R 3, Trans
E) ∼C ⊃ W 2, Impl

Answer 1

The correct conclusion based on propositional logic and the given premises is B) ¬R. This utilizes the modus tollens rule of inference, stating that if R leads to a contradiction of the premises, then R cannot be true.

Step-by-step explanation:

The question you're asking relates to logic and reasoning, which is an essential part of mathematics, specifically within the realm of propositional logic. The premises given are:

• ¬W (not W)
• C ∨ W (C or W)
• R ⊃ ¬(C ∨ W) (If R, then not (C or W))

From these premises, we need to determine the correct conclusion. By applying rules like disjunctive syllogism, modus tollens, and modus ponens, we can infer the validity of certain conclusions. We are given several conclusions, but only one can be correct based on the rules of inference:

1. R ⊃ (¬C • ¬W)
2. ¬R (not R)
3. C (C is true)
4. (C ∨ W) ⊃ ¬R (If C or W, then not R)
5. ¬C ⊃ W (If not C, then W)

In this case, by applying the modus tollens to the third premise with the first one, we can infer B) ¬R. This is because if R implies not (C or W), and we are told that C or W is true, then R cannot be true.