7.6.x. Exercise questions
| 1. |
Use the system of derivations to establish the following. You may use detachment and attachment rules. |
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| a. | ∀x ∀y (Rxy → ¬ Ryx) ⊨ ∀x ∀y (¬ x = y → ¬ (Rxy ∧ Ryx)) | |
| b. |
∀x ∀y (¬ x = y → ¬ (Rxy ∧ Ryx)) ∀x ¬ Rxx ∀x ∀y (Rxy → ¬ Ryx) |
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| 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 |
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| 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 |
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| 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 in such way that the resulting argument is valid. Then analyze the premises and conclusion and construct a derivation to confirm its validity. You may use detachment and attachment rules. |
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| 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 |
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