| 1. |
Define (logical) relative inconsistency by completing the following: φ is inconsistent with Γ (i.e., Γ, φ ⇒ ) if and only if ... . (Your answer need not replicate the wording of the text's definitions, but it should define equivalence in terms of truth values and possible worlds.)
[answer] |
| Analyze the sentences below in as much detail as possible, presenting the result in both symbolic and English notation. Be sure that the unanalyzed components of your answer are complete and independent sentences; also try to respect any grouping in the English. | ||
| 2. |
Sam didn't eat his cake and keep it, too, but he wasn't disappointed.
[answer] |
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| 3. |
Either the intruder woke neither the cat nor the dog or it was someone they both knew. [answer] |
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| Use derivations to check whether each of the entailments below holds. You may use detachment and attachment rules. If an entailment fails, present a counterexample that divides an open gap. | ||
| 4. |
C ∧ D ⇒ ¬ (B ∧ ¬ C)
[answer] |
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| 5. |
(A ∧ C) ∨ (B ∧ D) ⇒ B ∨ C
[answer] |
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| 6. |
¬ (A ∨ B), A ∨ D, ¬ (C ∧ D) ⇒ C
[answer] |
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| 7. | [This question was on a topic not covered in F04]
Use replacement principles to put the following sentence into disjunctive normal form (in which there are no negated compounds and no conjunction has a disjunction as a component):
A ∧ ¬ (B ∧ ¬ C)
[answer] |
| 1. | Γ, φ ⇒ if and only if there is no possible world in which φ is true along with and all members of Γ. |
| 4. |
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| 6. | This answer illustrates the use of detachment rules; other, longer, derivations are possible without them. IP is used at the first stage in order to make it possible to exploit the first premise by CR, the only rule available for exploiting negated disjunctions.
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| 7. | [This question was on a topic not covered in F04]
A ∧ ¬ (B ∧ ¬ C)
⇔ A ∧ (¬ B ∨ C) ⇔ (A ∧ ¬ B) ∨ (A ∧ C) |