7.6.x. Exercise questions
| 1. |
Use the system of derivations to establish the following. You may use detachment and attachment rules. |
|
| a. | ∀x ∀y (Rxy → ¬ Ryx) ⇒ ∀x ∀y (¬ x = y → ¬ (Rxy ∧ Ryx)) | |
| b. |
∀x ∀y (¬ x = y → ¬ (Rxy ∧ Ryx)) ∀x ¬ Rxx ∀x ∀y (Rxy → ¬ Ryx) |
|
| c. | ∀x ∀y ∀z ((Rxy ∧ Ryz) → Rxz), ∀x ¬ Rxx ⇒ ∀x ∀y (Rxy → ¬ Ryx) | |
| d. |
Everyone loves everyone who loves anyone If anyone loves anyone, then everyone loves everyone |
|
| e. | ∀x ∀y Rxy, ∀x (∀y Ryx → (Fx → Gx)) ⇒ ∀x (Fx → Gx) | |
| f. |
Al said everything he remembered Al is a person who said nothing Anyone who remembered nothing forgot everything Al forgot everything |
|
| 2. |
Choose one of each alternative pair of premises (enclosed in square brackets) and one of each alternative pair of words or phrases in the conclusion so as to make a valid argument. Then analyze the premises and conclusion and construct a derivation to show that the argument is valid. You may use detachment and attachment rules. |
|
| a. |
Everyone watched every snake [Every cobra is a snake | Every snake is a reptile] Everyone watched every [cobra | reptile] |
| b. |
No one watched every snake [Every cobra is a snake | Every snake is a reptile] No one watched every [cobra | reptile] |
| c. |
No one watched any snake [Every cobra is a snake | Every snake is a reptile] No one watched any [cobra | reptile] |
| d. |
Everyone who likes every snake was pleased [Every cobra is a snake | Every snake is a reptile] Everyone who likes every [cobra | reptile] was pleased |
| e. |
Everyone who likes a snake was pleased [Every cobra is a snake | Every snake is a reptile] Everyone who likes a [cobra | reptile] was pleased |