| │∃x Rax | ||
| ├─ | ||
| │ⓑ | ||
| ││Rab | ||
| │├─ | ||
| │││∀x ¬ Rxa | a:3,b:4 | |
| ││├─ | ||
| 3 UI | │││¬ Raa | |
| 4 UI | │││¬ Rba | |
| │││○ | Rab, ¬Raa, ¬Rba ⇏ ⊥ | |
| ││├─ | ||
| │││⊥ | 2 | |
| │├─ | ||
| 2 NcP | ││∃x Rxa | 1 |
| ├─ | ||
| 1 PCh | │∃x Rxa |
| 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. | |
| 1. |
Tom phoned someone who had left a message for him. [Give this analysis also using an unrestricted quantifier.]
answer |
| 2. |
Santa said something to each child. [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. |
Ron asked Santa for at least two things.
answer |
| 4. |
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.
Bill lent the book Ann gave him to Carol
answer
|
| 5. |
Use derivations to show that the following argument is valid. You may use any rules.
∃x ∃y (Rxy ∧ Sxy)
answer
∃y ∃x (Sxy ∧ Rxy) |
| 6. |
Use a derivation to show that the following argument is not valid and describe a structure dividing an open gap.
∃x Rax
answer
∃x Rxa |
| 7. |
Complete the following to give a definition of inconsistency in terms of truth values and possible worlds:
A set Γ is inconsistent if and only if … answer |
| 8. | Describe a structure (i.e., an assignment of extensions to the non-logical vocabulary) which makes the list of 5 sentences below all true and use it to calculate a truth value for the sentence that follows them. (You may present the structure using either tables or a diagram.) | |
| make these true: | b = ga, fa = f(ga), Rab, R(fa)a, ¬ R(fb)b | |
| calculate the value: | (b = gb ∨ Ra(ga)) → (R(fa)(ga) ∧ f(gb) = g(fb)) | |
| answer | ||
| 9. |
Give two different restatements of the sentence below in expanded form as a complex predicate (i.e., an abstract) applied to a term.
∃y Rayb
answer
|
|
second analysis:
something is such that (Santa said it to each child) ∃x Santa said x to each child ∃x each child is such that (Santa said x to him or her) ∃x (∀y: y is a child) Santa said x to y
∃x (∀y: Cy) Dsxy
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||
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C: [ _ is a child]; D: [ _ said _ to _ ]; s: Santa
The first is true and the second false if Santa spoke to each child but said different things to different children |
||
| 3. |
Ron asked Santa for at least two things ∃x (∃y: ¬ y = x) (Ron asked Santa for x ∧ Ron asked Santa for y)
∃x (∃ y: ¬ y = x) (Arsx ∧ Arsy)
A: [ _ asked _ for _ ]; r: Ron; s: Santa
|
|
using the description operator:
Bill lent the book Ann gave him to Carol Lb(the book Ann gave Bill)c Lb(Ix x is a book Ann gave Bill)c Lb(Ix (x is a book ∧ Ann gave Bill x))c
Lb(Ix (Bx ∧ Gabx))c
|
||
| B: [ _ is a book]; G: [ _ gave _ _ ]; L: [ _ lent _ to _ ]; a: Ann; b: Bill; c: Carol | ||
| 6. |
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| 7. | A set Γ is inconsistent if and only if there is no possible world in which every member of Γ is true. |
| 8. |
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Your values for some of the compound terms and equations may differ from those shown here in gray, but your values for other predications and for truth-functional compounds should be the same as those shown. The diagram above provides a complete answer, and so do the tables to its left. The tables below show a way of arriving at these answers. |
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| 9. |
The following are 3 possibilities (up to choice of the variable) from which your two might be chosen; in the last, τ may be any term:
[∃y Rxyb]xa, [∃y Rayx]xb, [∃y Rayb]xτ
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