| Phi 270 Fall 2013 |
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Phi 270 F06 test 4
F06 test 4 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.
Analysis. Be ready to handle any of the key issues discussed in class—for example, the proper analysis of every, no, and only (see §7.2.2), how to incorporate bounds and exceptions (see §7.2.3), ways of handling compound quantifier phrases (such as only cats and dogs, see §7.3.2), the distinction between every and any (see §§7.3.3 and 7.4.2), how to represent multiple quantifier phrases with overlapping scope (see §7.4.1). You should be able restate your analysis using unrestricted quantifiers (see §7.2.1), but you will not need to present it in English notation.
Synthesis. You may be given a symbolic form and an interpretation of its non-logical vocabulary and asked to express the sentence in English. Remember that the distinction between every and any can be important here, too.
Derivations. Be able to construct derivations to show that entailments hold and to show that they fail. 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 rules introduced in §7.8.1.
F06 test 4 questions
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Analyze the sentences below in as much detail as possible, providing a key to the non-logical vocabulary you use. State your analysis also in a form that expresses any generalizations using unrestricted quantifiers. |
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| 1. |
Every door was locked. answer |
| 2. |
Only people who had witnessed the event were able to follow the description of it. [It is possible for the scope of only to change with emphasis; although varying interpretations are less likely with this sentence than with others, you may choose whichever scope seems most plausible to you.] answer |
| 3. |
No key opened every door. [You should understand this sentence to leave open the possibility that some key opened some door.] answer |
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Synthesize an English sentence with the following logical form; that is, find a sentence that would have the following analysis: |
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| 4. |
(∀x: Px ∧ Nxa) (Dxm ∨ Axm) A: [ _ was acted on at _ ]; D: [ _ was discussed at _ ]; N: [ _ was on _ ]; P: [ _ was a proposal]; a: the agenda; m: the meeting answer |
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Use derivations to show that the following arguments are valid. You may use any rules. |
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| 5. |
∀x (Fx → (Gx → Hx))
answer
∀x Gx ∀x (Fx → Hx) |
6. |
∀x (Fx → ∀y Rxy)
answer
∀x Fx ∀x ∀y Ryx |
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Use a derivation to show that the following argument is not valid and present a counterexample by describing a structure that is a counterexample lurking an open gap. (You may describe the structure either by depicting it in a diagram, as answers in the text usually do, or by giving tables.) |
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| 7. |
∀x Rax
answer
∀x Rxb ∀x Rxx |
F06 test 4 answers
| 1. |
Every door was locked Every door is such that (it was locked) (∀x: x is a door) x was locked
(∀x: Dx) Lx
D: [ _ is a door]; L: [ _ was locked] |
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| 7. |
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Counterexample presented by a diagram |
Counterexample presented by tables
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