phi_270_F03_test

Phi 270 F03 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.	`No one called the new number` answer
2.	`Sam asked everyone he could think of` [Remember to restate this one using an unrestricted quantifier.] answer
3.	`If any door was opened, the alarm sounded` answer
4.	`Only people who’d read everything the author had written were asked to review the book` answer

Use derivations to show that the following arguments are valid. You may use any rules.
5.	∀x (Fx ∧ Gx) ∀x Gx answer
6.	∀x (Fx → Gx) ∀x ∀y (Gy → Rxy) ∀x ∀y (Fy → Rxy) answer

Use a derivation to show that the following argument is not valid and describe a structure (by using either a diagram or tables) that divides an open gap.
7.	∀x (Fx → Rxa) Fa → ∀x Rxx answer

Phi 270 F03 test 4 answers

1.	`No one called the new number` `No one is such that (he or she called the new number)` (∀x: x `is a person`) ¬ x `called the new number`) (∀x: Px) ¬ Cxn C: [ _ `called` _ ]; P: [ _ `is a person`]; n: `the new number`

Sam asked everyone he could think of

everyone Sam could think of is such that (Sam asked him or her)

(∀x: x is a person Sam could think of) Sam asked x

(∀x: x is a person ∧ Sam could think of x) Asx

( ∀x: Px ∧ Tsx) Asx
∀x ((Px ∧ Tsx) → Asx)

A: [ _ asked _ ]; P: [ _ is a person]; T: [ _ could think of _ ]; s: Sam

If any door was opened, the alarm sounded

every door is such that (if it was opened, the alarm sounded)

(∀x: x is a door) if x was opened, the alarm sounded

(∀x: Dx) (x was opened → the alarm sounded)

( ∀x: Dx) (Ox → Sa)

D: [ _ is a door]; O: [ _ was opened]; S: [ _ sounded]; a: the alarm

Only people who’d read everything the author had written were asked to review the book

Only people who’d read everything the author had written are such that (they were asked to review the book)

(∀x: ¬ x is a person who’d read everything the author had written) ¬ x was asked to review the book

(∀x: ¬ (x is a person ∧ x had read everything the author had written)) ¬ Axb

(∀x: ¬ (x is a person ∧ everything the author had written is such that (x had read it))) ¬ Axb

(∀x: ¬ (Px ∧ (∀y: y is a thing the author had written) x had read y)) ¬ Axb

(∀x: ¬ (Px ∧ (∀y: the author had written y) Rxy)) ¬ Axb

(∀x: ¬ (Px ∧ (∀y: Way) Rxy)) ¬ Axb

A: [ _ was asked to review _ ]; P: [ _ is a person]; R: [ _ had read _ ]; R: [ _ had written _ ]; a: the author; b: the book

Counterexample presented by tables

Counterexample presented by a diagram

range: 1, 2

a	b
1	2

(This interpretation divides both gaps; the value of F2 is needed only for the 1st and the value of R21 only for the 2nd.)