Final answer:
The question involves a Fitch-style proof in propositional logic, using given premises to deduce that E is true. The logical proof utilizes rules of inference to systematically determine that the premises entail the conclusion that E must be true.
Step-by-step explanation:
The question presented is a logical argument within the realm of philosophy known as a Fitch-style proof. This type of proof is used in propositional logic to determine the validity of a conclusion based on a given set of premises. We are asked to deduce that E is true given the premises: C ∨ E, A ∨ M, A → ¬C, and ¬S ∧ ¬M.
To approach this proof, we must use the rules of logical inference. Since we have ¬S ∧ ¬M, we know immediately that ¬M is true. Because A ∨ M is also true and ¬M is established, it must mean that A is true. With A → ¬C in our premises and establishing that A is true, we can infer that ¬C must also be true. Now, return to premise C ∨ E. Since we've found ¬C to be true, it must mean that E is the only remaining truth value in this disjunction. Therefore, E must be valid based on these premises, and we've completed the proof.