1. |
Define the idea of two sentences being mutually exclusive by completing the following with a definition in terms of truth values and possible worlds:
φ and ψ are mutually exclusive if and only if …
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2. |
Suppose you are told about some sentences φ and ψ and the tautology ⊤ that φ ⇒ ⊤ and ⊤ ⇒ ψ. (i) What does this tell you about the possible truth values of φ? And (ii) what does it tell you about the possible truth values of ψ? In each case, explain your answer by reference to the definitions of a tautology and of implication (i.e., entailment). answer |
3. |
Give an example of three sentences where the first implies (i.e., entails) the second, the second implicates the third (i.e., has the third as an implicature), but the first does not implicate the third. (It may be easiest to choose the second and third sentences as an example of an implicature—and it doesn’t have to be a new example—and then look for a sentence that implies one without implicating the other.) Be sure to say enough about the context of your sentences for me to be able to see that what you claim about them is so. answer |
4. |
Analyze the sentence below in as much detail as possible, presenting the result using symbolic notation and also English notation (i.e., using |
Although Al took the first turn, he missed the second; but he found his way to the meeting. answer |
5. |
Synthesize an English sentence that has the analysis below. Choose a simple and natural sentence whose organization reflects the grouping of the logical form.
B ∧ (C ∧ D)
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B: Al wrote to Bob; C: Al spoke to Carol; D: Al spoke to Dave |
Use derivations to check whether each of the claims of entailment below holds. If an entailment fails, present a counterexample by providing a table in which you calculate the truth values of the premises and conclusion on an assignment of truth values that divides an open gap. (Your table should indicate the value of any compound component by writing this value under the main connective of the component.) Do not use the rule Adj in the first derivation, but you may use it in the second. |
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6. |
A ∧ (B ∧ C), D ⇒ C ∧ D
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7. |
A ∧ B, B ∧ D ⇒ A ∧ (C ∧ D)
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1. |
φ and ψ are mutually exclusive if and only if there is no possible world in which φ and ψ are both true (or: if and only if, in each possible world, at least one of φ and ψ is false) |
6. |
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