| Phi 270 Fall 2013 |
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Phi 270 F12 test 5
F12 test 5 topics
The following are the topics to be covered. The proportion of the test covering each will approximate the proportion of the classes so far that have been devoted to that topic. Your homework and the collection of old tests will provide specific examples of the kinds of questions I might ask.
This test will have a few more questions than earlier ones (about 9 or 10 instead of about 7) and I will allow you as much of the 3 hour period as you want. The bulk of the questions (6 or 7 of the total) will be on ch. 8 but there will also be a few questions directed specifically towards earlier material (see below).
Analysis. This will represent the majority of the questions on ch. 8. The homework assignments give a good sample of the kinds of issues that might arise but you should, of course, consider examples and exercises in the text as well. In particular, pay attention to the variety of special issues that show up (e.g., how to handle there is or else).
Synthesis. You may be given a symbolic form and an interpretation of its non-logical vocabulary and asked to express the sentence in English. (This sort of question is less likely to appear than a question about analysis, and there would certainly be substantially fewer such questions.)
Derivations. Be able to construct derivations to show that entailments hold and to show that they fail (derivations that hold are more likely). I may tell you in advance whether an entailment holds or leave it to you to check that using derivations. If a derivation fails, you may be asked to present a counterexample, which will involve describing a structure. You will not be responsible for the rule for the description operator introduced in §8.6 or for the supplemented rules (i.e., PCh+, etc.) used to find finite counterexamples.
Earlier material. These questions will concern the following topics.
Basic concepts. You may be asked for a definition of a concept or asked questions about the concept that can be answered on the basis of its definition. You are responsible for: You will be responsible for entailment, tautologousness and absurdity, and the relations between pairs of sentences (i.e., implication, equivalence, exclusiveness, joint exhaustiveness, and contradictoriness). These are the concepts you were responsible for on the first test. You can find sample definitions in Appendix A.1.
Calculations of truth values. You should be able to complete a row of a truth table for a sentence formed using truth-functional connectives. (That is, you should be able to carry out the sort of calculation used to complete the confirmation of a counterexample in chs. 2-5.)
Using abstracts to analyze sentences involving pronouns. You might be asked to represent pronouns using abstracts and variables in order to avoid repeating individual terms (i.e., the sort of question on 6.2 you were responsible for on the third test).
F12 test 5 questions
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Analyze the following sentences in as much detail as possible, providing a key to the items of non-logical vocabulary (upper and lower case letters apart from variables) that appear in your answer. Notice the special instructions given for each of 1, 2, and 3. |
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| 1. |
Al wrapped a package. [Give an analysis using a restricted quantifier, and restate it using an unrestricted quantifier. Remember that a name like Al is not a quantifier phrase and should not be analyzed using a quantifier.] answer |
| 2. |
No one in the house heard the doorbell. [Do not use ∀ in your analysis of this; that is, use ∃ in your analysis of any quantifier phrases. Analyze the definition descriptions in this sentence as you would have before §8.4; that is, do not use Russell’s analysis or the description operator.] answer |
| 3. |
Everyone sang a song to someone. [On one way of understanding this sentence, it would be false if there is no one song that is sung by everyone but could still be true if no one person was sung to by everyone. Analyze it according to that interpretation.] answer |
| 4. |
Al nominated himself and someone else, too. answer |
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Analyze the sentence below twice, in each case using one the two ways of analyzing the definite description. That is, give an analysis that uses Russell’s analysis of definite descriptions as quantifier phrases as well as one that uses the description operator to analyze the definite description. |
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| 5. |
Sally rode in the red sleigh. answer |
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Use a derivation to show that the following entailment holds. You may use any rules. |
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| 6. |
∃x (Fx ∧ ¬ Gx) ⊨ ¬ ∀x Gx answer |
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Use a derivation to show that the following claim of entailment fails, and use either a diagram or tables to present a counterexample that lurks in an open gap of your derivation. |
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| 7. |
∃x ¬ ∀y Rxy ⊨ ∃x ¬ Rxx answer |
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Complete the following to give a definition of equivalence in terms of truth values and possible worlds: |
| 8. |
A pair of sentences φ and ψ are mutually exclusive (i.e., φ ▵ ψ) if and only if ... answer |
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Complete the following truth table row. Indicate the value of each non-atomic component of the sentence on the right by writing the value under the main connective of that component (so in your answer every connective must have a truth value directly under it); also circle the value that is under the main connective of the whole sentence (and circle no other value). |
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F12 test 5 answers
| 1. |
Al wrapped a package Some package is such that (Al wrapped it) (∃x: x is a package) Al wrapped x
P: [ _ is a package]; S: [ _ wrapped _ ]; a: Al |
| 4. |
Al nominated himself and someone else, too Al nominated himself ∧ Al nominated someone else
Al nominated Al Naa ∧ (∃x: x is a person other than Al) Al nominated x Naa ∧ (∃x: x is a person ∧ ¬ x = Al) Nax Naa ∧ (∃x: Px ∧ ¬ x = a) Nax N: [ _ nominated _ ]; P: [ _ is a person]; a: Al If unrestricted quantifiers were used (they weren’t required), the answer might be stated as: Naa ∧ ∃x ((Px ∧ ¬ x = a) ∧ Nax) |
| 5. |
Using Russell’s analysis: Sally rode in the red sleigh Some red sleigh that alone was a red sleigh was such that (Sally rode in it) (∃x: x was a red sleigh ∧ only x was a red sleigh) Sally rode in x (∃x: x was a red sleigh ∧ (∀y: ¬ y = x) ¬ y was a red sleigh) Osx (∃x: (x was a sleigh ∧ x was red) ∧ (∀y: ¬ y = x) ¬ (y was a sleigh ∧ y was red)) Osx (∃x: (Sx ∧ Rx) ∧ (∀y: ¬ y = x) ¬ (Sy ∧ Ry)) Osx
or: (∃x: (Sx ∧ Rx) ∧ ¬ (∃y: ¬ y = x) (Sy ∧ Ry)) Osx
Using the description operator: Sally rode in the red sleigh [ _ rode in _ ] Sally the red sleigh Os(Ix x was a red sleigh) Os(Ix (x was a sleigh ∧ x was red)) Os(Ix (Sx ∧ Rx)) O: [ _ rode in _ ]; R: [ _ was red]; S: [ _ was a sleigh]; s: Sally |
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or |
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| 8. |
A pair of sentences φ and ψ are mutually exclusive if and only if there is no possible world in which φ and ψ are both true or A pair of sentences φ and ψ are mutually exclusive if and only if, in each possible world, at least one of φ and ψ is false |
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