phi_270_F04_test

Phi 270 F04 test 4

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.
5.	∀x Fx ∀x ¬ Gx ∀x (Fx ∧ ¬ Gx) answer
6.	∀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.)
7.	∀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: [ _ `checked` _ ]; L: [ _ `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: [ _ answered _ ]; P: [ _ is a person]; N: [ _ was in _ ]; 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: [ _ is a person]; G: [ _ got _ ]; 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: [ _ is a bestseller]; L: [ _ is a list]; N: [ _ was on _ ]

Counterexample presented by a diagram

Counterexample presented by tables

range: 1, 2

a	b
1	2