Phi 270 F96 Quiz 4-5

Phi 270 F96 part of quiz 4 and all of quiz 5 (of 6) in pdf format
(questions from these two tests addressed the part of the course your test is designed to cover)

4-4.	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.)
	`Al called everyone who left him a message concerning the accident and told them he had seen 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.
4-5.	`Every employee received the letter.` [answer]
4-6.	`Among bystanders, Sam interviewed only soldiers.` [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.
5-1.	`If anyone guessed the number, the prize was awarded`. [answer]
5-2.	`Everyone who worked on any part of the project was honored`. [answer]

Synthesize an English sentence whose analysis would yield the following form.
5-3.	(∀x: Px) ¬ ∀y Axy [A: λxy (x `ate` y); P: λx (x `is a person`)] [answer]

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

5-4.

∀x Fx
∀x Gx

∀x (Fx ∧ Gx)

[answer]

5-5.

(∀x: Fx) Rxa
(∀x: Rxa) ∀y Ryx

∀x (∀y: Fy) Rxy

[answer]

5-6.

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+, RUG+, and ST introduced in §7.8 that are designed to avoid unending gaps.)

∀x Rxx

Rab → ∀x Rxa

[answer]

Phi 270 F96 Answers to part of quiz 4 and all of quiz 5

4-4.

Al

called

everyone who left

him

a message

concerning

the accident

and told

them

he

had seen

it

[it could instead have a message concerning the accident as its antecedent]

4-5.	`Every employee received the letter` `Every employee is such that (he or she received the letter)` (∀x: x `is an employee`) x `received` `the letter` (∀x: Ex) Rxl ∀x (Ex → Rxl) [E: λx (x `is an employee`); R: λxy (x `received` y); l: `the letter`]

4-6.

Among bystanders, Sam interviewed only soldiers
Among bystanders, only soldiers are such that (Sam interviewed them)
(∀x: x was a bystander ∧ ¬ x was a soldier) ¬ Sam interviewed x

(∀x: Bx ∧ ¬ Sx) ¬ Isx
∀x ((Bx ∧ ¬ Sx) → ¬ Isx)

[B: λx (x was a bystander); I: λxy (x interviewed y); S: λx (x was a soldier); s: Sam]

5-1.

If anyone guessed the number, the prize was awarded
Everyone is such that (if he or she guessed the number, the prize was awarded)
(∀x: x is a person) (if x guessed the number, the prize was awarded)
(∀x: Px) (x guessed the number → the prize was awarded)

(∀x: Px) (Gxn → Ap)

[P: λx (x is a person); G: λxy (x guessed y); n: the number]

5-2.

Everyone who worked on any part of the project was honored
Every part of the project is such that (everyone who worked on it was honored)
(∀x: x is a part of the project) everyone who worked on x was honored
(∀x: Rxj) (∀y: y is a person who worked on x) y was honored
(∀x: Rxj) (∀y: y is a person ∧ y worked on x) Hy

(∀x: Rxj) (∀y: Py ∧ Wyx) Hy

[H: λx (x was honored); P: λx (x is a person); R: λxy (x is a part of y); W: λxy (x worked on y); j: the project]

5-3.	(∀x: x `is a person`) ¬ ∀y x `ate` y (∀x: x `is a person`) ¬ x `ate everything` `No one is such that (he or she ate everything)` `No one ate everything`

5-4.

	│∀x Fx	a:2
	│∀x Gx	a:3
	├─
	│ⓐ
2 UI	││Fa	(5)
3 UI	││Ga	(6)
	││
	│││●
	││├─
5 QED	│││Fa	4
	││
	│││●
	││├─
6 QED	│││Ga	4
	│├─
4 Cnj	││Fa ∧ Ga	1
	├─
1 UG	│∀x (Fx ∧ Gx) 1

5-5.

	│(∀x: Fx) Rxa	c:3
	│(∀x: Rxa) ∀y Ryx	c:4
	├─
	│ⓑ
	││ⓒ
	│││ Fc	(3)
	││├─
3 SB	│││Rca	(4)
4 SB	│││∀y Ryc	b:5
5 UI	│││Rbc	(6)
	│││●
	││├─
6 QED	│││Rbc	2
	│├─
2 RUG	││(∀y: Fy) Rby	1
	├─
1 UG	│∀x (∀y: Fy) Rxy

5-6.

	│∀x Rxx	a:1,b:2,c:5
	├─
1 UI	│Raa
2 UI	│Rbb
	│
	││Rab
	│├─
	││ⓒ
5 UI	│││Rcc
	│││
	││││ ¬ Rca
	│││├─
	││││ ○	Raa,Rab,Rbb,Rcc,¬Rca ⇏ ⊥
	│││├─
	││││⊥	6
	││├─
6 IP	│││Rca	4
	│├─
4 UG	││∀x Rxa	3
	├─
3 CP	│Rab → ∀x Rxa