| Phi 270 Fall 2013 |
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Phi 270 F12 test 3
F12 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, in addition to truth-functional connectives, 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 find questions of this sort only in the last 5 years or so of exams, 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 may 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.
F12 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 (i) symbolic notation, (ii) symbolic notation with conditionals written so that all arrows point left to right (if that’s different), and (iii) 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. |
The project went ahead if either Bill or Carol supported it. answer |
| 2. |
The project will go ahead only if both financing and staffing are approved. 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. (Your truth-table for a counterexample should show the truth-value of each compound component of sentence under the main connective of that component, and it should indicate the final truth-value of each sentence.) |
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| 3. |
A → (B → C), A → B ⊨ A → C answer |
| 4. |
A → (B → C) ⊨ C → (A → 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—and that they are completely specified. In particular, the expressions that letters stand for should be complete sentences or complete individual terms—except, perhaps, for blanks marking the places of predicates and functors (so, in particular, no letter should stand for a bare common noun like dog—as opposed to the dog, [ _ is a dog], or [ _’s dog]—since dog by itself is not a complete individual term). |
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| 5. |
Al will send Carol’s book to her unless she picks it up. 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. |
Ann spoke to Bill, and he spoke to her mother. 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. |
g(fa) = c, R(gb)(gc) ⊨ gc = a → (¬ Rca → ¬ fa = b) answer |
F12 test 3 answers
| 3. |
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| 5. |
Al will send Carol’s book to her unless she picks it up Al will send Carol’s book to her ← ¬ Carol will pick up her book [ _ will send _ to _ ] Al Carol’s book Carol ← ¬ [ _ will pick up _ ] Carol Carol’s book S a ([ _’s book] Carol) c ← ¬ P c ([ _’s book] Carol)
Sa(bc)c ← ¬ Pc(bc) S: [ _ will send _ to _ ]; P: [ _ will pick up _ ]; b: [ _'s book]; a: Al; c: Carol |
| 6. |
Ann spoke to Bill, and he spoke to her mother Ann and Bill are such that (she spoke to him, and he spoke to her mother) [x spoke to y, and y spoke to x’s mother]xy Ann Bill [x spoke to y ∧ y spoke to x’s mother]xy Ann Bill [ [ _ spoke to _ ] x y ∧ [ _ spoke to _ ] y x ’s mother ]xyab [ [ _ spoke to _ ] x y ∧ [ _ spoke to _ ] y ([ _ ’s mother] x) ]xyab [ Sxy ∧ Sy(mx) ]xyab S: [ _ spoke to _ ]; m: [ _ ’s mother]; a: Ann; b: Bill |
| 7. |
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