Question

Given ~(K ≣ S) / S ⊃ ~(R v K) // R v ~S, what follows?

A) K ⊃ S
B) R v ~S
C) S ⊃ (R v K)
D) ~(R v K)

Answer 1

Step-by-step explanation:

To determine what follows from the given logical expression ~(K ≣ S) / S ⊃ ~(R v K) // R v ~S, we need to simplify and analyze the expression.

1. Let's simplify the expression step by step:

~(K ≣ S) / S ⊃ ~(R v K) // R v ~S

~(K ≣ S) / S ⊃ ~(R v K) // ~(R v K) v ~S [Using the material implication A ⊃ B ≡ ~A v B]

~(K ≣ S) / S ⊃ ~(R v K) // (~R • ~K) v ~S [Using De Morgan's law ~(A v B) ≡ ~A • ~B]

2. Now, let's analyze the simplified expression:

From the first part, ~(K ≣ S), we cannot directly infer any of the given options.

From the second part, S ⊃ ~(R v K), we cannot directly infer any of the given options.

However, combining the first part (~(K ≣ S)) and the third part ((~R • ~K) v ~S), we can infer ~(K ≣ S) AND ((~R • ~K) v ~S).

Simplifying further, we can rewrite ~(K ≣ S) AND ((~R • ~K) v ~S) as ~K • (S v ~S) v ~R.

From this expression, we can infer:

B) R v ~S, since ~R is part of the expression.

Additionally, we can also infer:

D) ~(R v K), since ~K is part of the expression.

However, we cannot directly infer:

A) K ⊃ S, since the expression does not provide enough information to determine the relationship between K and S.

C) S ⊃ (R v K), since the expression does not provide enough information to determine the relationship between S and (R v K).

Therefore, the correct inferences are:

B) R v ~S

D) ~(R v K)

Answer 2

To determine what follows from the given premises, we apply propositional logic rules and derive the conclusion R v ~K.

Step-by-step explanation:

To determine what follows from the given premises, we can apply the rules of propositional logic.

1. We are given ~(K ≣ S) which can be simplified to K ⊃ ~S and ~S ⊃ K using the equivalence rule.
2. From S ⊃ ~(R v K), we can apply the rule of Modus Tollens to get ~S ⊃ ~(R v K) which can be further simplified to S ⊃ (R v ~K) using De Morgan's law.
3. Combining K ⊃ ~S with S ⊃ (R v ~K) using the transitive property, we get K ⊃ (R v ~K).
4. Applying the rule of Simplification to K ⊃ (R v ~K), we can derive R v ~K as the final result.