Phi 270 F04 test 4

Phi 270 F04 test 4 in pdf format

Analyze the sentences below in as much detail as possible, providing a key to the non-logical vocabulary you use. Restate 2 using an unrestricted quantifier.
1.	`Sam checked every lock` [answer]
2.	`No one who was in the office answered the call` [Remember to restate your answer in 2 using an unrestricted quantifier.] [answer]
3.	`Ralph got the joke if anyone did` [answer]
4.	`Only bestsellers were on every list` [answer]

Use derivations to show that the following arguments are valid. You may use any rules.

∀x Fx

∀x ¬ Gx

∀x (Fx ∧ ¬ Gx)

[answer]

(∀x: Rxa) ∀y Txy

∀x (∀y: Rya) Tyx

[answer]

Use a derivation to show that the following argument is not valid and present a counterexample by describing a structure that divides an open gap. (You may describe the structure either by depicting it in a diagram, as answers in the text usually do, or by giving tables.)

∀x Rax

(∀x: Rxa) Rxx

[answer]

Phi 270 F04 test 4 answers

1.	`Sam checked every lock` `Every lock is such that (Sam checked it)` (∀x: x `is a lock`) Sam `checked` x (∀x: Lx) Csx [C: λxy (x `checked` y); L: λx (x `is a lock`); s: `Sam`]

No one who was in the office answered the call

No one who was in the office is such that (he or she answered the call)

(∀x: x is a person who was in the office) ¬ x answered the call

(∀x: x is a person ∧ x was in the office) ¬ Axc

(∀x: Px ∧ Nxo) ¬ Axc
∀x ((Px ∧ Nxo) → ¬ Axc)

[A: λxy (x answered y); P: λx (x is a person); N: λxy (x was in y); c: the call; o: the office]

Ralph got the joke if anyone did

Everyone is such that (Ralph got the joke if he or she did)

(∀x: x is a person) Ralph got the joke if x did

(∀x: Px) (Ralph got the joke ← x got the joke)

(∀x: Px) (Grj ← Gxj)
(∀x: Px) (Gxj → Grj)

[P: λx (x is a person); G: λxy (x got y); j: the joke]

Only bestsellers were on every list

Only bestsellers are such that (they were on every list)

(∀x: ¬ x is a bestseller) ¬ x was on every list

(∀x: ¬ Bx) ¬ every list is such that (x was on it)

(∀x: ¬ Bx) ¬ (∀y: y is a list) x was on y

(∀x: ¬ Bx) ¬ (∀y: Ly) Nxy

[B: λx (x is a bestseller); L: λx (x is a list); N: λxy (x was on y)]

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

	│(∀x: Rxa) ∀y Txy	c:3
	├─
	│ⓑ
	││ⓒ
	│││Rca	(3)
	││├─
3 SB	│││∀y Tcy	b: 4
4 UI	│││Tcb	(5)
	│││●
	││├─
5 QED	│││Tcb	2
	│├─
2 RUG	││(∀y: Rya) Tyb	1
	├─
1 UG	│∀x (∀y: Rya) Tyx

	│∀x Rax	a:3, b:4
	├─
	│ⓑ
	││Rba
	│├─
	│││¬ Rbb
	││├─
3 UI	│││Raa
4 UI	│││Rab
	│││○	Rba, ¬Rbb, Raa, Rab ⇏⊥
	││├─
	│││⊥	2
	│├─
2 IP	││Rbb	1
	├─
1 RUG	│(∀x: Rxa) Rxx

Counterexample presented by a diagram

Counterexample presented by tables

range: 1, 2

a	b
1	2

R	1	2
1	T	T
2	T	F