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Complete the following to give a definition in terms of truth values and possible worlds. |
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| 1. |
φ and ψ are mutually exclusive (i.e., φ, ψ ⊨ ⊥) if and only if …
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Analyze each sentence below in as much detail as possible, presenting the result using both in symbols and using English notation (i.e., both … and, etc.). Be sure that the unanalyzed components of your answer are complete and independent sentences; also try to respect any grouping in the English. |
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| 2. |
The job didn’t have both good pay and flexible hours, and Sam didn’t apply for it.
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| 3. |
Although neither Luke nor Mary saw the movie, either Nancy or Oscar did.
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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. |
¬ B ⊨ ¬ (A ∧ (B ∧ C))
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| 5. |
¬ (A ∧ B) ⊨ ¬ A
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| 6. |
(A ∧ B) ∨ C ⊨ C ∨ B
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In 7 you may use attachment and detachment rules (and their use can simplify the derivation). |
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| 7. |
A ∨ B, ¬ (B ∧ C), C ⊨ A
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| 1. |
φ and ψ are mutually exclusive if and only if there is no possible world in which both are true (or: … if and only if, in every possible world, at least one is false) |
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| 6. |
It is also possible to begin with PE; if that’s done, IP and Nc will be needed to close one of the gaps. |
| 7. |
The first answer below uses detachment rules while the second shows one way to construct a derivation without them.
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