Phi 270
Fall 2013
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Phi 270 F13 test 1

F13 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 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 logical form that I specify (as in the second part of the homework on 2.1).

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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. You can expect to be asked to confirm a counterexample for any entailment that fails. Derivations will focus on the rules Ext, Cnj, and QED, but I won’t rule out ones requiring EFQ and ENV (the rules for ⊤ and ⊥). There may be some derivations where the rule Adj introduced in 2.4 would be convenient to use, but it is never necessary.

F13 test 1 questions

1.

Define the idea of sentences φ and ψ being mutually exclusive by completing the following with a definition in terms of truth values and possible worlds:

φ and ψ are mutually exclusive (i.e., φ ▵ ψ) if and only if …
answer
2.

Suppose you know that φ ⊨ ψ but that φ and ψ are not equivalent. What truth values are possible for φ and ψ and why? (Your answer should show that you understand the definitions of entailment and equivalence, and showing that you understand them will earn you a significant part of the credit for this question.)

answer
3.

Consider the following dialogue:

Al: Was it raining when you walked over?
Bob: The road was wet.

Does Bob’s answer implicate that it was raining? Say why or why not and explain your answer in a way that shows you understand the definition of implicature. (I can imagine either a yes or a no answer being supported, so the key here is the way you justify whichever answer you give.)

answer
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.

Al found the clue and started after the treasure, but Bill got to it before he did.

answer
5.

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

W ∧ (S ∧ C)

W: Al went to Paris; S: Al saw Paris; C: Al was conquered by Paris

answer

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 from §2.4.

  6. K ∧ (L ∧ M) ⊨ L ∧ K
answer
  7. D ∧ E, F ∧ G ⊨ D ∧ (F ∧ H)
answer

F13 test 1 answers

1.

φ and ψ are mutually exclusive (i.e., φ ▵ ψ) if and only if there is no possible world in which both φ and ψ are true (or: if and only if, in each possible world, at least one of φ and ψ is false).

2.

Knowing that φ ⊨ ψ tells you that it is not possible to have φ true and ψ false; and knowing that they are not equivalent tells you that it is possible for them to have different truth values one way or the other, so it must be possible to have φ false and ψ true. This much—i.e., the impossibility of separating φ from ψ and the possibility of separating ψ from φ—is enough to insure that φ implies ψ but they aren't equivalent, so you don’t know anything more about possibilities for truth values. In particular, you don’t know whether or not it is possible for them to be both true or both false.

3.

I’d guess that yes is the most likely answer. It could be justified by noting what Bob says ought to suggest something about whether it is raining if it is to answer Al’s question and that it is raining is the natural suggestion. But then what he says is not appropriate if it is not raining, and that’s what it means for it to implicate that it is raining.

The answer no might be justified by noting that Bob really only has to supply relevant information (and not necessarily a complete answer) to respond to Al’s question. And saying that the roads were wet does this, so there need be no further suggestion. But this means that it is not necessary for it to actually be raining for his response to be appropriate.

4.

Al found the clue and started after the treasure, but Bill got to it before he did

Al found the clue and started after the treasure ∧ Bill got to the treasure before Al did

(Al found the clue ∧ Al started after the treasure) ∧ Bill got to the treasure before Al did

(F ∧ S) ∧ G
both both F and S and G

F: Al found the clue; G: Al started after the treasure; S: Bill got to the treasure before Al did

5.

Al went to Paris ∧ (Al saw Paris ∧ Al was conquered by Paris)

Al went to Paris ∧ Al saw and was conquered by Paris

Al went to Paris, and he saw and was conquered by it

6.
│K ∧ (L ∧ M)1
├─
1 Ext│K(5)
1 Ext│L ∧ M2
2 Ext│L(4)
2 Ext│M
│
││●
│├─
4 QED││L3
│
││●
│├─
5 QED││K3
├─
3 Cnj│L ∧ K
7.
│D ∧ E1
│F ∧ G2
├─
1 Ext│D(4)
1 Ext│E
2 Ext│F(6)
2 Ext│G
│
││●
│├─
4 QED││D3
│
│││●
││├─
6 QED│││F5
││
│││○ F, G, D, E ⊭ H
││├─
│││H5
│├─
5 Cnj││F ∧ H3
├─
3 Cnj│D ∧ (F ∧ H)
DEFGH D∧E,F∧G/D∧(F∧H)
TTTTF ⓉⓉⒻF