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Analyze each sentence below in as much detail as possible, presenting the result in both symbols and English notation (i.e., using ∧, etc. and also both ... and, etc.). Be sure that the unanalyzed components of your answer are complete and independent sentences, and try to respect any grouping in the English. |
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| 1. |
Neither Ann nor Bill got the joke, but Carol did. answer |
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| 2. |
Either Ann didn’t reach Bill, or he wasn’t both free and able to help her. answer |
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Synthesize an English sentence (the more idiomatic the better) that has the following analysis: |
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| 3. |
¬ (B ∨ M) (B: Sam had heard of the book; M: Sam had heard of the movie) answer |
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Use derivations to check whether each of the entailments below holds. If one fails, present a counterexample by providing a table in which you calculate the truth values of the premises and conclusion on an extensional interpretation (i.e., an assignment of truth values) that divides an open gap. |
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Do not use attachment or detachment rules in 4-6. That is, do not use Adj or the rules MTP, MPT, and Wk of §4.3; instead use only the basic rules for exploiting resources, planning for goals, and closing gaps. |
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| 4. |
A ∧ ¬ B ⊨ ¬ (C ∧ B) ∧ A
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| 5. |
¬ (A ∧ B), A ⊨ ¬ B
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| 6. |
B ∨ A ⊨ C ∨ B
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For 7 you should show the first stage of each of the possible ways of beginning the derivation with the basic rules (i.e., the rules allowed in 4-6); and you should complete one of these derivations. In completing it, you may use attachment and detachment rules (and their use can simplify the derivation). |
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| 7. |
B ∨ A ⊨ A ∨ B
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| 4. |
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| 6. |
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