Use derivations to check each of the claims below; if a derivation indicates that a claim fails, describe a structure that divides an open gap.
| 1. | Fa ⇒ ∀x Fx |
| 2. | ∀x Rxx ⇒ ∀x Rxa |
| 3. | ∀x ¬ Fx ⇔ ¬ ∀x Fx |
| 4. |
No widget is a gadget No gizmo is a widget No gizmo is a gadget |
| 5. | No widget is a gadget ⇔ Not every widget is a gadget |
| 6. |
Everything is either finished or unstarted Either everything is finished or everything is unstarted |
| 7. | ¬ ∀x ¬ ∀y Rxy ⇒ ∀x ¬ ∀y ¬ Rxy |