Phi 270 F08 test 3
F08 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 preserve pronouns, representing them using abstracts and variables. (You will not find questions of this sort in the exams before 2006, but your homework on this topic and exercise 2 for 6.2 provide further 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—i.e., depending on whether the question is directed at ch. 5 or ch. 6.
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.
In the case of a derivation that includes forms involving predicates and functors, you won’t be asked to present a counterexample if the derivation fails (though you will still need to be able to recognize that such a derivation has failed). In short, the test won’t cover the new material introduced in 6.4.
F08 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. |
If John was invited, then he attended if he was free. answer |
| 2. |
Unless we find the key, we’ll get in only if we break the lock. 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, present a counterexample that divides an open gap. |
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| 3. |
B → C ⊨ (A ∧ B) → C answer |
| 4. |
¬ (C → D) → (A → B) ⊨ A → D 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. |
Sam wrote to Linda, and she sent his book to him. answer |
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Analyze the sentence below using abstracts and variables to represent pronominal cross reference (instead of replacing pronouns by their antecedents). That is, use expanded form to the extent necessary so that each individual term in your analysis appears 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. |
The rock hit the road, but it didn’t hit Oscar. 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), fa = gb ⊨ a = b → (Rb(ga) ∧ fb = gb) answer |
F08 test 3 answers
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| 5. |
Sam wrote to Linda, and she sent his book to him Sam wrote to Linda ∧ Linda sent Sam’s book to him Sam wrote to Linda ∧ Linda sent Sam’s book to Sam [ _ wrote to _ ] Sam Linda ∧ [ _ sent _ to _ ] Linda Sam’s book Sam Wsl ∧ Sl([ _’s book] Sam)s Wsl ∧ Sl(bs)s S: [ _ sent _ to _ ]; W: [ _ wrote to _ ]; b: [ _’s book]; l: Linda; s: Sam |
| 6. |
The rock hit the road, but it didn’t hit Oscar The rock is such that (it hit the road, but it didn’t hit Oscar) [x hit the road, but x didn’t hit Oscar]x the rock [x hit the road ∧ x didn’t hit Oscar]x the rock [x hit the road ∧ ¬ x hit Oscar]x the rock [Hxr ∧ ¬ Hxo]xk H: [ _ hit _ ]; k: the rock; o: Oscar; r: the road |
| 7. |
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