| 1. | Give the instances of each of the following for the terms a, b, and c (remembering that you will drop the main quantifier, and only the main one, when giving an instance): | |
| a. | ∀x Fx | |
| b. | ∀y Fy | |
| c. | ∀x Rxa | |
| d. | ∀x Saxb | |
| e. | ∀x ∀y Rxy | |
| f. | ∀x (Fx → Gx) | |
| g. | ∀x (Fx → Gd) | |
| h. | ∀x (Fx → ∀y Rxy) | |
| i. | ∀x (Fx → ∀x Rxx) | |
| 2. | Use the system of derivations to establish each of the following. You may use detachment and attachment rules. | |
| a. | ∀x Fx, ∀x (Fx → Gx) ⇒ Ga | |
| b. | ∀x (Fx ∧ Gx) ⇒ Fa ∧ Gb | |
| c. | ∀x Rxa, ∀x (Rbx → Gx) ⇒ Ga | |
| d. | ∀x Fx, ∀x (Fx → Gx) ⇒ ∀x Gx | |
| e. | ∀x (Fx ∧ Gx) ⇔ ∀x Fx ∧ ∀x Gx | |
| f. | ∀x ∀y Rxy ⇒ (Rab ∧ Rbb) ∧ Rca | |
| g. | ∀x ∀y Rxy ⇒ ∀y Rya | |
| h. | ∀x ∀y (Rxy → ¬ Ryx) ⇒ ∀x ¬ Rxx | |
| i. | ∀x ∀y ∀z ((Rxy ∧ Ryz) → Rxz), ∀x ¬ Rxx ⇒ ∀x ∀y (Rxy → ¬ Ryx) | |