| Phi 270 Fall 2013 |
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Phi 270 F06 test 3
F06 test 3 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. Two sorts of questions are possible here corresponding to the sorts of analyses you have done in chs. 5 and 6: (i) analysis by truth-functional connectives only, with atomic sentences as the ultimate components (the focus would, of course, be on conditionals—i.e., on the symbolic representation of if, only if, and unless) and (ii) analysis using truth-functional connnectives and the ideas of predicates, individual terms, and functors.
In the case of the latter sort of analysis, you might be asked to represent pronouns using abstracts and variables. (You will not find questions of this sort in the old exams, but your homework on this topic and exercise 2 for 6.2 provide examples.)
Synthesis. Again this might take two forms, depending on whether the expressions abbreviated by letters were are complete sentences or were terms, predicates, and functors.
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. There will be some derivations where detachment and attachment rules may be used and where they will shorten the proof. But there will be others where you must rely on other rules, either because detachment and attachment rules do not apply or because I tell you not to use them. In particular be ready to use the rule RC (Rejecting a Conditional) from ch. 5.
Remember that, if a derivation includes forms involving predicates and functors, presenting a counterexample will require the description of a structure and not merely an assignment of truth values. You will be allowed to use either tables or diagrams to describe structures.
F06 test 3 questions
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Analyze the sentences below in as much detail as possible using only connectives; that is, the unanalyzed components should all be sentences (rather than individual terms, predicates, or functors). Present the result in both symbolic and English notation. Be sure that the unanalyzed components of your answer are complete and independent sentences; also try to respect any grouping in the English. |
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| 1. |
There was an audience if there was food. answer |
| 2. |
Sam went unless he had to work, but he enjoyed the ride only if the weather was good. answer |
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Use derivations to check whether each of the entailments below holds. You may use detachment and attachment rules. If an entailment fails, confirm a counterexample that lurks in an open gap. |
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| 3. |
C → (B → A), C → B ⊨ C → A answer |
| 4. |
A → B, C → D ⊨ C → (E → ¬ B) answer |
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Analyze the sentence below in as much detail as possible, giving a key to your abbreviations of unanalyzed expressions. In this case you should identify components that are individual terms, predicates, or functors; however, you do not need to present the result in English notation (i.e., symbolic notation is enough). Your analysis should be in reduced form (i.e., you should not use abstracts and variables), so be sure that the unanalyzed components of your answer are independent—in particular, that none contains a pronoun whose antecedent is in another. (Also be sure also that the individual terms you identify really are individual terms and are not quantifier phrases or general terms, like simple common nouns.) |
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| 5. |
Nancy phoned Oliver and told him about his promotion. answer |
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Analyze the sentence using abstracts and variables to represent pronominal cross reference (instead of replacing pronouns by their antecedents). That is, each individual term in your analysis should appear only as often as it appears in the original sentence. In other respects, your analysis should be as described for 5. |
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| 6. |
Spot finished chewing his bone, and he buried it in a flowerbed. answer |
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Use a derivation to show that the entailment below holds. You may use detachment and attachment rules. Be sure to indicate the alias sets whenever an equation is added to the resources. |
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| 7. |
Ra(fb) ∧ ¬ Rc(fd), fb = fc ⊨ ¬ (a = c ∧ b = d) answer |
F06 test 3 answers
| 1. |
There was an audience if there was food there was an audience ← there was food
A ← F
A: there was an audience; F: there was food |
| 3. |
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| 5. |
Nancy phoned Oliver and told him about his promotion Nancy phoned Oliver ∧ Nancy told Oliver about his promotion Nancy phoned Oliver ∧ Nancy told Oliver about his promotion [ _ phoned _ ] Nancy Oliver ∧ [ _ told _ about _ ] Nancy Oliver Oliver’s promotion Pno ∧ Tno([ _’s promotion] Oliver) Pno ∧ Tno(po) P: [ _ phoned _ ]; T: [ _ told _ about _ ]; n: Nancy; o: Oliver; p: [ _’s promotion] |
| 6. |
Spot finished chewing his bone, and he buried it in a flowerbed Spot is such that (he finished chewing his bone, and he buried it in a flowerbed) [x finished chewing x’s bone, and x buried it in a flowerbed]x Spot [x’s bone is such that (x finished chewing it, and x buried it in a flowerbed)]xs [ [x finished chewing y, and x buried y in a flowerbed]y x’s bone]xs [ [x finished chewing y ∧ x buried y in a flowerbed]y([ _’s bone] x) ]xs
[[Cxy ∧ Bxy]y(bx)]xs
B: [ _ buried _ in a flowerbed]; C: [ _ finished chewing _ ]; b: [ _’s bone]; s: Spot (Note: a flowerbed is not an individual term so it must remain unanalyzed as part of a predicate) |
| 7. |
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