Phi 270 F10 test 1

F10 test 1 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.

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Basic concepts of deductive logic. 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). You should be able to define any of these ideas in terms of truth values and possible worlds (see appendix A.1 and 1.2.8 for samples of such definitions), and you should be ready to answer questions about these concepts and explain your answers in a way that uses the definitions.

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Implicature. Be able to define it and distinguish it from implication. Be able to give examples and explain them. Be ready to answer questions about it, justifying your answer in a way that uses the definition.

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Analysis. Be able to analyze the logical form of a sentence as fully as possible using conjunction and present the form in both symbolic and English notation (that is, with the logical-and symbol ∧ and with the both … and … way of expressing forms).

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Synthesis. Be able to synthesize an English sentence that has a given logical form.

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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 may be some derivations where the rule Adj introduced in 2.4 would be convenient to use; but it is never necessary. You should be ready to use EFQ and ENV (the rules for ⊤ and ⊥) in addition to Ext, Cnj, and QED; but derivations requiring EFQ and ENV are only a possibility while you are certain to run into derivations requiring the latter three rules.

F10 test 1 questions

1.

Define entailment by completing the following with a definition in terms of truth values and possible worlds:

a set Γ of sentences entails a sentence φ (i.e., Γ ⊨ φ) if and only if …

answer

2.

Suppose that φ and ψ are contradictory (i.e., that φ ⋈ ψ) and also that ψ implies χ (i.e., that ψ ⊨ χ). Can you conclude anything about the deductive relations holding between φ and χ? That is, does the information given allow you to rule out one or more of the four conceivable patterns of truth values for two sentences (i.e., TT, TF, FT, FF) in the case of φ and χ, or is it consistent with what you are told that φ and χ be logically independent? You should justify your answer in a way that shows you know the definitions of contradictoriness and implication (but you need not provide the name of the relation between φ and χ if you conclude that they are related).

answer

3.

(i) Present a sentence that, when used in a certain context, has an implicature that suggests something beyond what the sentence says literally, and (ii) briefly explain why the sentence has that implicature in the context you describe. In addressing part (i), be sure to show that the implicature is not part of what the sentence says by describing a way that the implicature could be false while what the sentence says literally is true.

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4.

Analyze the sentence below in as much detail as possible, presenting the result using symbolic notation and (and present the same analysis also using English notation—i.e., using both … and … to indicate conjunction). Be sure that the unanalyzed components of your answer are complete and independent sentences, and give a key to the abbreviations you use for them; also try to respect any grouping in the English.

Ann posed the problem, and Bill and Carol each solved it

answer

Synthesize an English sentence that has the analysis below. Choose a simple and natural sentence whose organization reflects the grouping of the logical form.

5.

(F ∧ O) ∧ (C ∧ W)

C: Tom gathered up the contents of the package; F: Sam found the package; O: Sam opened the package; W: Tom gathered up the wrapping of the package

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Use derivations to check whether each of the claims of entailment below holds. If an entailment fails, confirm a counterexample by providing a table in which you calculate the truth values of the premises and conclusion on an assignment of truth values that is a counterexample lurking in an open gap. (Your table should indicate the value of any compound component by writing this value under the main connective of the component.) Do not use the rule Adj.

6.

E ∧ (A ∧ K) ⊨ K ∧ E

answer

7.

D ∧ E, R ∧ S ⊨ R ∧ (D ∧ T)

answer

F10 test 1 answers

1.

a set Γ of sentences entails a sentence φ (i.e., Γ ⊨ φ) if and only if there is no possible world in which every member of Γ is true and φ is false (or: if and only if φ is true in each possible world in which every member of Γ is true)

2.

We can conclude that φ and χ cannot be both false. Since φ and ψ are contradictory, we know that they cannot be both true or both false—i.e., their truth values must be different—so ψ must be true if φ is false. And, since ψ implies χ, we know that χ cannot be false when ψ is true. Therefore, χ cannot be false when φ is false.

[Although it is not part of what you were asked, note that we cannot conclude anything else about the truth values of φ and χ. In some cases they might exhibit the patterns TF and FT because φ and ψ might exhibit both, and χ might in fact be equivalent to ψ. And they might exhibit the pattern TT since the given information is consistent with this pattern in a case where ψ is false; for example, suppose φ says The glass is not empty, ψ says The glass is empty, and χ says The glass is not full, and consider a case where the glass is half full.]

3.

Here’s a sample answer. (i) If I say, “I have a lot of work to do” in answer to the question “Are you going to the movie?,” the person asking would in most circumstances conclude that I am not going; but it would be perfectly possible for me to have a lot work to do (so what I have said is true) and nevertheless go to the movie (in which case what I implicated would be false). (ii) The implicature arises because my response to the question is appropriate only if it is taken as an answer; and, unless I am known to be someone makes a point of going to the movies when I have a lot of work to do, the answer made most likely by my response is that I am not going.

4.

Ann posed the problem, and Bill and Carol each solved it

Ann posed the problem ∧ Bill and Carol each solved the problem

Ann posed the problem ∧ (Bill solved the problem ∧ Carol solved the problem)

A ∧ (B ∧ C)
both A and both B and C

A: Ann posed the problem; B: Bill solved the problem; C: Carol solved the problem

[The function of each in this sentence is to pointedly leave open the possibility that Bill and Carol worked independently; a sentence that said instead that they worked together could not be analyzed as a conjunction of the components B and C above.]

5.

(Sam found the package ∧ Sam opened the package) ∧ (Tom gathered up the contents of the package ∧ Tom gathered up the wrapping of the package)

Sam found the package and opened it ∧ Tom gathered up both its contents and its wrapping

Sam found the package and opened it, and Tom gathered up both its contents and its wrapping

[If the switch from Sam to Tom was unexpected, you’d probably express the main conjunction by using but (or a synonym of it) instead of and.]

6.
│E ∧ (A ∧ K)1
├─
1 Ext│E(5)
1 Ext│A ∧ K2
2 Ext│A
2 Ext│K(4)
│
││●
│├─
4 QED││K3
│
││●
│├─
5 QED││E3
├─
3 Cnj│K ∧ E
7.
│D ∧ E1
│R ∧ S2
├─
1 Ext│D(6)
1 Ext│E
2 Ext│R(4)
2 Ext│S
│
││●
│├─
4 QED││R3
│
│││●
││├─
6 QED│││D5
││
│││○D, E, R, S ⊭ T
││├─
│││T5
│├─
5 Cnj││D ∧ T3
├─
3 Cnj│R ∧ (D ∧ T)
  D   E   R   S   T     D ∧ E ,   R ∧ S   /   R ∧ (D ∧ T)   
  T   T   T   T   F      Ⓣ      Ⓣ        Ⓕ F