| Use the basic system of derivations to check each of the claims below; if a derivation indicates that a claim fails, present a counterexample (that is, give an interpretation that divides an open gap and calculate truth values for the premises and conclusion from it—as is done in the example in 2.3.3): | |
| 1. | A ⇒ A ∧ B |
| 2. | A ∧ B ⇒ A ∧ (B ∧ A) |
| 3. | B ∧ E, C ∧ ⊤ ⇒ (A ∧ B) ∧ (C ∧ D) |
| 4. | A ∧ B, B ∧ C, C ∧ D ⇒ A ∧ D |
| 5. | A, B ∧ A, D ⇒ B ∧ ((C ∧ A) ∧ D) |