Question

Given the following premises:

1. S ⊃ (∼∼T • ∼∼C)
2. (S • Q) ∨ C
3. ∼C
Group of answer choices
a) S ⊃ (T • ∼∼C) 1;DN
b) S ⊃ ∼∼ T 1,Simp
c) S ⊃ (T • ∼C) 1,DN
d) S 2,Simp
e) ~(S • Q) 2,3,DS

Answer 1

The question deals with logical inference using premises that include logical connectives. By applying the rules of double negation and disjunctive syllogism, we derive that S implies T, where answer choice (b) S ⊃ ∼∼ T is correct by simplification and double negation.

Step-by-step explanation:

The student's question involves applying rules of logical inference to the given premises to derive a conclusion. The goal is to determine which conclusion is valid based on the initial premises provided. The logical connectives used in the premises include the conditional (⊃), conjunction (•), disjunction (∨), and negation (∼). Double negation (∼∼) simply removes the negation, turning ∼∼T into T, and ∼∼C into C.

Detailed Solution

1. Start with Premise 1: S ⊃ (∼∼T • ∼∼C), which simplifies to S ⊃ (T • C) by double negation.
2. From Premise 2: (S • Q) ∨ C, and Premise 3: ∼C, we apply the rule of disjunctive syllogism which tells us that if ∼C is true, then (S • Q) must be the case because C cannot be the case (due to ∼C).
3. Using the simplification rule (simp), we can conclude that from (S • Q), S is true.
4. The answer choice that matches our derived conclusion is S ⊃ (T • C) but this is not listed in the answer choices provided.

Hence, if we look for an option that correctly uses the premises, answer choice (b) S ⊃ ∼∼ T is correct by simplifying Premise 1 directly (ignoring C) using simplification and double negation.