5.1.x. Exercise questions
| 1. | Analyze each of the following sentences in as much detail as possible. | |
| a. | If it was raining, the roads were slippery. | |
| b. | He was home if the light was on. | |
| c. | Ann and Bill helped if Carol was away | |
| d. | Sam will help—and Tom will, too, if we ask him. | |
| e. | If it was warm, they ate outside provided it didn’t rain. | |
| f. | If the new project was approved, Carol started work on it and so did Dave if he was finished with the last one. | |
| g. | If he found the instructions, Tom set up the new machine; otherwise, he packed up the old one. | |
| 2. | Restate each of the following forms, putting English notation into symbols and vice versa and indicating the scope of connectives in the result by underlining: | |||
| a. | A ∧ (B → C) | c. |
if A then both B and if C then D
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| b. | (A ∧ B) → C | d. |
both if A then B and if not A then not B
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| 3. | Synthesize idiomatic English sentences that express the propositions that are associated with the logical forms below by the intensional interpretations that follow them. | |
| a. |
¬ S → ¬ B S: I’ll see it; B: I’ll believe it |
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| b. |
S → ¬ (R ∨ N) S: it was sunny; R: it rained; N: it snowed |
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| c. |
¬ W ← ¬ (P ∧ ¬ B) W: the set works; P: the set is plugged in; B: the set is broken |
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| d. |
¬ (A ∨ B) → (G ← ¬ (C ∨ D)) A: Adams will back out; B: Brown will back out; G: the deal will go through; C: Collins will have trouble with financing; D: Davis will have trouble with financing |
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| 4. | Calculate truth values for all components of the forms below on each possible extensional interpretation. Since the first two each have two unanalyzed components, there will be 4 interpretations and your table will have 4 rows of values; with three components, as in the third and fourth, there will be 8 interpretations giving 8 rows of values. | |
| a. | (A → B) ∧ (B → A) | |
| b. | ¬ (A ∧ B) → (¬ B ∨ A) | |
| c. | (A → C) ∧ (B → ¬ C) | |
| d. | ¬ (A → C) ∧ (¬ B → C) | |