1. G > H
2. A > (G • E)
3. A / A • (H v B)
4.
5.
6.
7.
8.
To solve this logical argument, we will use a method called proof by contradiction. The goal is to assume that the statement is false and then derive a contradiction, which will prove that the statement must be true.
1. Given: G > H (Premise)
2. Given: A > (G • E) (Premise)
3. A (Assumption)
To prove: A • (H v B)
Assume the opposite of what we're trying to prove (Proof by contradiction):
4. ¬(A • (H v B)) (Assumption)
We can simplify ¬(A • (H v B)) to ¬A v ¬(H v B) using De Morgan's Law.
5. ¬A v (¬H • ¬B) (De Morgan's Law)
From premise 2 (A > (G • E)), we can use Modus Ponens to infer that ¬A > ¬(G • E).
6. ¬A > ¬(G • E)
Combining ¬A v ¬(H v B) from step 5 with premise 1 (G > H), we can use Disjunctive Syllogism to infer that ¬A > H.
7. ¬A > H (Disjunctive Syllogism)
From step 6 (¬A > ¬(G • E)) and premise 2 (A > (G • E)), we can use Transitive Property to derive that ¬A > (G • E).
8. ¬A > (G • E) (Transitive Property)
Now, we have ¬A > H (from step 7) and ¬A > (G • E) (from step 8). By Combining these two implications, we can use Constructive Dilemma to derive H v (G • E).
9. H v (G • E) (Constructive Dilemma)
Using step 3 (A) and step 9 (H v (G • E)), we can use Simplification to infer A • (H v B).
10. A • (H v B) (Simplification)
Therefore, we have proven that A • (H v B) holds true.