| Analyze the following sentences in as much detail as possible, providing a key to the non-logical vocabulary (upper and lower case letters) appearing in your answer. Notice the additional instructions given for each of the first two. | |
| 1. |
Sam mentioned someone Tina didn’t know. [Give this analysis also using an unrestricted quantifier.]
answer |
| 2. |
Every shoe fit someone. [This sentence is ambiguous. Analyze it in two different ways, and describe a situation in which the sentence is true on one of your interpretations and false on the other.]
answer |
| 3. |
Sam found at least two pieces.
answer |
| Analyze the sentence below using each of the two ways of analyzing definite descriptions. That is, analyze it using Russell’s analysis of definite descriptions as quantifier phrases and then analyze it again using the description operator. | |
| 4. |
The elephant standing on Sam sighed.
answer |
| [The following question was on a topic not covered in F08] Put the following sentence into prenex normal form (i.e., into a form which contains no restricted quantifiers and in which no quantifier is in the scope of a connective). Show each step where you move a quantifier past a connective separately. | |
| 5. |
¬ ∀x ((Px ∧ ∃y Rxy) → ∃z Sxz)
answer |
| Use derivations to show that the following argument is valid. You may use attachment rules (but not replacement by equivalence). | |
| 6. |
∀x ∀y (Rxy → (Ryx → Rxx))
answer
∃x ∃y (Rxy ∧ Ryx) ∃x Fxx |
| Use a derivation to show that the following argument is not valid and describe a structure dividing an open gap. | |
| 7. |
∃x Fx
answer
∃x (Gx ∧ Hx) ∃x (Fx ∧ Hx) |
| Complete the following to give a definition of entailment by a single sentence (i.e., implication) in terms of truth values and possible worlds: | |
| 8. |
A sentence φ entails a sentence ψ if and only if …
answer |
| Complete the following truth table by calculating the truth value of the sentence on the given assignment. Show the value of each component by writing it under the main connective of that component. |
| 9. |
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| Describe a structure (i.e., an assignment of extensions to the non-logical vocabulary) which makes the sentences below all true. (You may use either tables or a diagram.) | |
| 10. |
a = fb, fb = fc, fa = c, Pa, Pb, ¬ Pc, Rab, Rbc, Rc(fb)
answer
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second analysis:
Every shoe fit someone someone is such that (every shoe fit him or her) (∃x: x is a person) every shoe fit x (∃x: Px) every shoe is such that (it fit x) (∃x: Px) (∀y: y is a shoe) y fit x
(∃x: Px) (∀y: Sy) Fyx
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F: [ _ fit _ ]; P: [ _ is a person]; S: [ _ is a shoe]
The first is true and the second false if every shoe could be worn but not all by the same person |
| 7. |
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| 8. | A sentence φ entails a sentence ψ if and only if there is no possible world in which φ is true but ψ is false (or: if and only if ψ is true in every possible world in which φ is true) |
| 9. |
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| 10. |
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The diagram above provides a complete answer, as do the tables to its left. The tables below illustrate a way of finding this structure. |
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