Question

If E ⊃ (F • G) / F ⊃ (G ⊃ H) // E ⊃ H, what can be inferred?

A) E ⊃ (F • G)
B) G ⊃ H
C) F ⊃ (G ⊃ H)
D) E ⊃ F

Answer 1

Step-by-step explanation:

To determine what can be inferred from the given logical expression E ⊃ (F • G) / F ⊃ (G ⊃ H) // E ⊃ H, we need to simplify and analyze the expression.

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

E ⊃ (F • G) / F ⊃ (G ⊃ H) // E ⊃ H

E ⊃ (F • G) / F ⊃ (~G v H) // E ⊃ H [Using the material implication G ⊃ H ≡ ~G v H]

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

From the first part, E ⊃ (F • G), we can infer that E ⊃ (F • G) is a valid inference.

From the second part, F ⊃ (~G v H), we cannot directly infer any of the given options.

However, combining the first part (E ⊃ (F • G)) and the third part (E ⊃ H), we can infer that E ⊃ (F • G) AND E ⊃ H.

Therefore, the correct inference from the given logical expression E ⊃ (F • G) / F ⊃ (G ⊃ H) // E ⊃ H is:

A) E ⊃ (F • G)

Additionally, we can also infer:

D) E ⊃ F, since E ⊃ (F • G) implies E ⊃ F by simplifying the conjunction (F • G) to just F.

However, B) G ⊃ H and C) F ⊃ (G ⊃ H) cannot be directly inferred from the given expression.

Therefore, the correct inferences are:

A) E ⊃ (F • G)

D) E ⊃ F

Answer 2

Given the logical expressions, the correct inference is D) E ⊃ F, indicating that if E is true, then F is also true. This conclusion is reached by understanding that E implies both F and G together; hence E must imply F on its own.

Step-by-step explanation:

The student's question pertains to logical operations within propositional logic, a branch of mathematics. Given E ⊃ (F • G) (E implies F and G) and F ⊃ (G ⊃ H) (F implies G implies H), the student is asked which of the following can be inferred: A) E ⊃ (F • G), B) G ⊃ H, C) F ⊃ (G ⊃ H), D) E ⊃ F. The answer is D) E ⊃ F. This result is because we are given that E implies both F and G together (E ⊃ (F • G)), and therefore E must imply F alone as a logical consequence. We do not have enough information to infer that G implies H on its own (option B), reflect the first given statement (option A), or the second given statement (option C) as a direct inference from the information we have; these options do not directly answer what can be inferred from the provided premises regarding the implication of E leading to H.