phi_270_F97_test

Phi 270 F97 test 4
(questions 1-3 are from quiz 4 and 4-9 are from quiz 5 out of 6 quizzes—these two quizzes addressed the part of the course your test is designed to cover)

Identify individual terms and quantifier phrases in the following sentence and indicate links between pronouns and their antecedents. (You can do this by marking up an English sentence; you are not being asked to provide a symbolic analysis.)

Everyone who Carol lent the book to spoke to her at length about it.
answer

Analyze the following generalizations in as much detail as possible. Provide a key to the non-logical vocabulary (upper and lower case letters) appearing in your answer and restate the result using an unrestricted quantifier.
2.	`Bob called no one`. answer
3.	`Among contestants, only professionals were finalists`. answer

Analyze the following sentences in as much detail as possible, providing a key to the non-logical vocabulary (upper and lower case letters) appearing in your answer.
4.	`Bob doesn’t own any map showing Dafter`. answer
5.	`Nothing anyone said bothered Dave`. answer

Use derivations to establish the validity of the following arguments. You may use attachment rules.

∀x (Fx ∧ Gx)

∀x Fx

answer

∀x (Rxa → ∀y Rxy)

∀x (∀y Rxy → Rxb)

answer

Use a derivation to show that the following argument is not valid and describe a structure dividing one of the derivation’s open gaps. (You will not need the rules UG+ and ST of 7.8 that were designed to avoid unending derivations.)

∀x (Fx → Rax)

∀x (Fx → Rxa)

answer

You will receive credit for one of the following (but you may attempt both):

9a.

Synthesize an English sentence whose analysis would yield the following form.
(∀x: Dx) (Okx → (∀y: Dy) Oky)
D: [ _ is a door]; O: [ _ opens _ ]; k: the key
answer

9b.

Use derivations to establish the validity of the following argument. You may use attachment rules.

∀x ∀y (Rxy → ¬ Fy)

∀x (Fx → Rxx)

∀x ¬ Fx

answer

Phi 270 F97 test 4 answers

Everyone who

Carol

lent

the book

to

spoke to

her

at length about

it

2.	`Bob called no one` `no one is such that (Bob called him or her)` (∀x: x `is an person`) ¬ `Bob` `called` x (∀x: Px) ¬ Cbx ∀x (Px → ¬ Cbx) C: [ _ `called` _ ]; P: [ _ `is person`]; b: `Bob`

Among contestants, only professionals were finalists

Among contestants, only professionals are such that (they were finalists)

(∀x: x was a contestant ∧ ¬ x was a professional) ¬ x was a finalist

(∀x: Cx ∧ ¬ Px) ¬ Fx
∀x ((Cx ∧ ¬ Px) → ¬ Fx)

C: [ _ was a contestant]; F: [ _ was a finalist]; P: [ _ was a professional]

Bob doesn’t own any map showing Dafter

every map showing Dafter is such that (Bob doesn’t own it)

(∀x: x is a map showing Dafter) ¬ Bob owns x

(∀x: x is a map ∧ x shows Dafter) ¬ Obx

(∀x: Mx ∧ Sxd) ¬ Obx

M: [ _ is a map]; O: [ _ owns _ ]; S: [ _ shows _ ]; b: Bob; d: Dafter

Nothing anyone said bothered Dave

everyone is such that (nothing he or she said bothered Dave)

(∀x: x is a person) nothing x said bothered Dave

(∀x: Px) nothing x said is such that (it bothered Dave)

(∀x: Px) (∀y: y is a thing x said) ¬ y bothered Dave

(∀x: Px) (∀y: x said y) ¬ Byd

(∀x: Px) (∀y: Sxy) ¬ Byd

B: [ _ bothered _ ]; P: [ _ is a person]; S: [ _ said _ ]; d: Dave

	│∀x (Fx ∧ Gx)	a:2
	├─
	│ⓐ
2 UI	││Fa ∧ Ga	3
3 Ext	││Fa
3 Ext	││Ga	(4)
	││●
	│├─
4 QED	││Fa	1
	├─
1 UG	│∀x Fx

	│∀x (Rxa → ∀y Rxy)
	├─
	│ⓒ
	│││∀y Rcy	a:3
	││├─
3 UI	│││Rcb	(4)
	│││●
	││├─
4 QED	│││Rcb	2
	│├─
2 CP	││∀y Rcy → Rcb	1
	├─
1 UG	│∀x (∀y Rxy → Rxb)

	│∀x (Rxa → ∀y Ryx)	c:4
	├─
	│ⓒ
	│││∀y Rcy	a:3
	││├─
3 UI	│││Rca	(5)
4 UI	│││Rca → ∀y Ryc	5
5 MPP	│││∀y Ryc	b:6
6 UI	│││Rbc	(7)
	│││●
	││├─
7 QED	│││Rbc	2
	│├─
2 CP	││∀y Rcy → Rbc	1
	├─
1 UG	│∀x (∀y Rxy → Rbx)

[The first premise is never used in the derivation for this question (shown at the left). The fact that it was not needed was a slip on my part in making up the question; at the right is the sort of example I probably had in mind.]

	│∀x (Fx → Rax)	b:3, a:5
	├─
	│ⓑ
	│││Fb	(4)
	││├─
3 UI	│││Fb → Rab	4
4 MPP	│││Rab
5 UI	│││Fa → Raa	7
	│││
	││││¬ Rba
	│││├─
	││││││¬ Fa
	│││││├─
	││││││○	Fb, Rab, ¬ Rba, ¬ Fa ⇏ ⊥
	│││││├─
	││││││⊥	8
	││││├─
8 IP	│││││Fa	7
	││││
	│││││Raa
	││││├─
	│││││○	Fb, Rab, ¬ Rba, Raa ⇏ ⊥
	││││├─
	│││││⊥	7
	│││├─
7 RC	││││⊥	6
	││├─
6 IP	│││Rba	2
	│├─
2 CP	││Fb → Rba	1
	├─
1 UG	│∀x (Fx → Rxa)

The structure below divides both gaps; it will divide only the first gap if the arrow from 1 to itself is dropped.

9a.

(∀x: x is a door) (the key opens x → (∀y: y is a door) the key opens y)

(∀x: x is a door) (the key opens x → every door is such that (the key opens it))

(∀x: x is a door) (the key opens x → the key opens every door )

(∀x: x is a door) if the key opens x, then it opens every door

every door is such that (if the key opens it, then it opens every door)

If the key opens any door, then it opens every door

9b.

	│∀x ∀y (Rxy → ¬ Fy)	a:2
	│∀x (Fx → Rxx)	a:4
	├─
	│ⓐ
2 UI	││∀y (Ray → ¬ Fy)	a:6
	││
	│││Fa	(5), (8)
	││├─
4 UI	│││Fa → Raa	5
5 MPP	│││Raa	(7)
6 UI	│││Raa → ¬ Fa	7
7 MPP	│││¬ Fa	(8)
	│││●
	││├─
8 Nc	│││⊥	3
	│├─
3 RAA	││¬ Fa	1
	├─
1 UG	│∀x ¬ Fx