phi_270_F11_test

F11 test 2 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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Analysis. Be able to analyze the logical form of a sentence as fully as possible using conjunction, negation, and disjunction and present the form in both symbolic and English notation.

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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 a derivation where attachment rules (Adj and Wk) and detachment rules (MTP and MPT) may be used and where they will shorten the proof; but there will be other derivations where you must rely on the basic rules, either because detachment and attachment rules do not apply or because I tell you not to use them.

F11 test 2 answers

the club’s president wasn’t available, but either its first or its second vice president was

the club’s president wasn’t available ∧ either the club’s first or its second vice president was available

¬ the club’s president was available ∧ (the club’s first vice president was available ∨ the club’s second vice president was available)

¬ P ∧ (F ∨ S)
both not P and either F or S

F: the club’s first vice president was available; P: the club’s president was available; S: the club’s second vice president was available

Ann and Bill weren’t both happy with the resort, and neither visited it again

Ann and Bill weren’t both happy with the resort ∧ neither Ann nor Bill visited the resort again

¬ Ann and Bill were both happy with the resort ∧ ¬ either Ann or Bill visited the resort again

¬ (Ann was happy with the resort ∧ Bill was happy with the resort) ∧ ¬ (Ann visited the resort again ∨ Bill visited the resort again)

¬ (H ∧ P) ∧ ¬ (V ∨ S)
both not both H and P and not either V or S

H: Ann was happy with the resort; P: Bill was happy with the resort; S: Bill visited the resort again; V: Ann visited the resort again

[¬ V ∧ ¬ S is also correct as an analysis of the second conjunct, and ¬ H ∨ ¬ P is a correct (though less likely) analysis of the first; however, ¬ H ∧ ¬ P and ¬ V ∨ ¬ S are not correct.]

(M ∧ N) ∨ ¬ (M ∨ N) with (M: Mike finished the game; N: Nick finished the game)

(Mike finished the game ∧ Nick finished the game) ∨ ¬ (Mike finished the game ∨ Nick finished the game)

Mike and Nick finished the game ∨ ¬ Mike or Nick finished the game

Mike and Nick finished the game ∨ neither Mike nor Nick finished the game

either Mike and Nick both finished the game or neither did

	│¬ S	(3)
	├─
	││S ∧ ¬ T	2
	│├─
2 Ext	││S	(3)
2 Ext	││¬ T
	││●
	│├─
3 Nc	││⊥	1
	├─
1 RAA	│¬ (S ∧ ¬ T)

	│B ∧ ¬ (A ∧ B)	1
	├─
1 Ext	│B	(7), (8)
1 Ext	│¬ (A ∧ B)	4
	│
	│││A	(6)
	││├─
	│││││●
	││││├─
6 QED	│││││A	5
	││││
	│││││●
	││││├─
7 QED	│││││B	5
	│││├─
5 Cnj	││││A ∧ B	4
	││├─
4 CR	│││⊥	3
	│├─
3 RAA	││¬ A	2
	│
	││●
	│├─
8 QED	││B	2
	├─
2 Cnj	│¬ A ∧ B

	│A ∨ B	1
	├─
	││A	(3)
	│├─
	│││●
	││├─
3 QED	│││A	2
	││
	││││¬ B
	│││├─
	││││○	¬ B, A ⊭ ⊥
	│││├─
	││││⊥	4
	││├─
4 IP	│││B	2
	│├─
2 Cnj	││A ∧ B	1
	│
	││B	(7)
	│├─
	││││¬ A
	│││├─
	││││○	¬ A, B ⊭ ⊥
	│││├─
	││││⊥	6
	││├─
6 IP	│││A	5
	││
	│││●
	││├─
7 QED	│││B	5
	│├─
5 Cnj	││A ∧ B	1
	├─
1 PC	│A ∧ B

A	B	A	∨	B	/	A	∧	B
T	F		Ⓣ				Ⓕ		lurks in the first gap
F	T		Ⓣ				Ⓕ		lurks in the second gap

It is OK to end your derivation when you have reached a dead-end gap, so, in this case, stages 5-7 were not needed

The following are two possible answers, but there are others:

	│C ∧ D	1
	├─
1 Ext	│C
1 Ext	│D	(3)
	│
	││¬ E
	│├─
	││●
	│├─
3 QED	││D	2
	├─
2 PE	│D ∨ E

	│C ∧ D	1
	├─
1 Ext	│C
1 Ext	│D	(4)
	│
	││¬ D	(4)
	│├─
	│││¬ E
	││├─
	│││●
	││├─
4 Nc	│││⊥	3
	│├─
3 IP	││E	2
	├─
2 PE	│D ∨ E

Analyze each sentence below in as much detail as possible, presenting the result in symbols and in English notation. Provide a key to your abbreviations of unanalyzed components, and be sure that these components are complete and independent sentences. Try to respect any grouping in the English.
1.	`The club’s president wasn’t available, but either its first or its second vice president was.` answer
2.	`Ann and Bill weren’t both happy with the resort, and neither visited it again.` answer

Use derivations to check whether each of the claims of entailment below holds. If a derivation fails, confirm a counterexample by providing a table in which you calculate the truth values of the premises and conclusion on an extensional interpretation (i.e., an assignment of truth values) that lurks 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, and you should be sure to mark the final values you have calculated for the premises and the conclusion (e.g., by circling them). *Do not* use attachment or detachment rules in these derivations. That is, do not use Adj or the rules MTP, MPT, and Wk of §4.3; instead use only the basic rules for exploiting resources, planning for goals, and closing gaps.
4.	¬ S ⊨ ¬ (S ∧ ¬ T) answer
5.	B ∧ ¬ (A ∧ B) ⊨ ¬ A ∧ B answer
6.	A ∨ B ⊨ A ∧ B answer
7.	C ∧ D ⊨ D ∨ E answer

Synthesize an English sentence (the more idiomatic the better) that has the following analysis:
3.	(M ∧ N) ∨ ¬ (M ∨ N) M: `Mike finished the game`; N: `Nick finished the game` answer