phi270F12tst1

Phi 270 F12 test 1

F12 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. 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 that require EFQ or ENV are much less likely than ones that require only Ext, Cnj, and QED.

F12 test 1 questions

1.	Define the idea of sentences φ and ψ being equivalent by completing the following with a definition in terms of truth values and possible worlds: φ and ψ are (logically) equivalent (i.e., φ ≃ ψ) if and only if … answer

Which of the patterns of truth values for three sentences φ, ψ, and χ (i.e., which of the eight ways of assigning truth values to the three sentences) are ruled out if you know that

(i): φ, ψ ⊨ χ, and
(ii): χ ⊨ ψ

Use the definition of entailment to explain why each of the patterns you cite is ruled out.

answer

Give an example of a sentence that has an implicature in one sort of circumstance but not in another. Use the definition of implicature in explaining your example—i.e., explaining why the sentence has the implicature in the one sort of circumstance but not the other. (The nature of the “sort of circumstance” can be quite varied: it might be, for example, a matter of who utters the sentence or of the fact that it is uttered in response to a certain question.)

answer

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 yelled to Bill, and Carol and Dave each yelled to her.

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. (L ∧ P) ∧ S L: `The apartment was clean`; P: `The apartment was cheap`; S: `The apartment was up 5 flights of stairs` 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.	(A ∧ B) ∧ C ⊨ A ∧ C answer
	7.	A ∧ C ⊨ (A ∧ B) ∧ C answer

F12 test 1 answers

1.	φ and ψ are equivalent (φ ≃ ψ) if and only if there is no possible world in which φ and ψ have different truth values (or: if and only if, in each possible world, φ has the same truth value different as ψ).

Three patterns are ruled out: (i) rules out the pattern in which its premises φ and ψ are true and its conclusion χ is false, and (ii) rules out the two patterns in which its premise χ is true and its conclusion ψ is false (with φ true in one of these patterns and false in the other). The other five patterns are compatible with both entailments.

A sample answer: The sentence “There’s a cooler in the trunk” implicates that there is beer in the cooler when said in response to the question “Is there any beer?” (because it would not be an appropriate answer otherwise if there was no beer in the cooler), but it does not have this implicature when said in response to the question “Where should I put the beer I brought?” (because simple information about the location of a cooler is appropriate under those circumstances).

Ann yelled to Bill, and Carol and Dave each yelled to her

Ann yelled to Bill ∧ Carol and Dave each yelled to Ann

Ann yelled to Bill ∧ (Carol yelled to Ann ∧ Dave yelled to Ann)

A ∧ (C ∧ D)
both A and both C and D

A: Ann yelled to Bill; C: Carol yelled to Ann; D: Dave yelled to Ann

(the apartment was clean ∧ the apartment was cheap) ∧ the apartment was up 5 flights of stairs

the apartment was clean and cheap ∧ the apartment was up 5 flights of stairs

The apartment was clean and cheap, but it was up 5 flights of stairs

Of course, there are other possible syntheses (especially if you like exercise).

	│(A ∧ B) ∧ C	1
	├─
1 Ext	│A ∧ B	2
1 Ext	│C	(5)
2 Ext	│A	(4)
2 Ext	│B
	│
	││●
	│├─
4 QED	││A	3
	│
	││●
	│├─
5 QED	││C	3
	├─
3 Cnj	│A ∧ C

	│A ∧ C	1
	├─
1 Ext	│A	(4)
1 Ext	│C	(5)
	│
	│││●
	││├─
4 QED	│││A	3
	││
	│││○	A, C ⊭ B
	││├─
	│││B	3
	│├─
3 Cnj	││A ∧ B	2
	│
	││●
	│├─
5 QED	││C	2
	├─
2 Cnj	│(A ∧ B) ∧ C

A	B	C	A	∧	C	/	(A	∧	B)	∧	C
T	F	T		Ⓣ				F		Ⓕ

Since a dead end is reached after stage 3 in the derivation, using QED to close gaps at stages 4 and 5 is not required

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