phi_270_F09_test

Phi 270 F09 test 4

Analyze the sentences below in as much detail as possible, providing a key to the non-logical vocabulary you use. Also restate your analyses using unrestricted quantifiers.
1.	`Everyone saw the eclipse.` answer
2.	`Al didn’t find any book that he was looking for.` answer
3.	`No one ate only potato chips.` answer

Synthesize an English sentence that has the following logical form; that is, devise a sentence that would have the following analysis:

(∀x: ¬ Sbx) Sax

S: [ _ saw _ ]; a: Al; b: Bill

Use derivations to show that the following arguments are valid. You may use any rules.
5.	∀x (Gx → Hx) ∀x (Fx ∧ Gx) ∀x Hx answer
6.	∀y ∀x (Px → ¬ Fxy) ∀x ∀y (Fyx → ¬ Py) answer

Use a derivation to show that the following argument is not valid and present a counterexample that divides an open gap.
7.	∀x Rxa ∀x Rxx answer

Phi 270 F09 test 4 answers

1.	`everyone saw the eclipse` `everyone is such that (he or she saw the eclipse)` (∀x: x `is a person`) x `saw the eclipse` (∀x: Px) Sxe ∀x (Px→ Sxe) P: [ _ `is a person`]; S: [ _ `saw` _ ]; e: `the eclipse`

Al didn’t find any book that he was looking for

every book that Al was looking for is such that (he didn’t find it)

(∀x: x is a book that Al was looking for) Al didn’t find x

(∀x: x is a book ∧ Al was looking for x) ¬ Al found x

(∀x: Bx ∧ Lax) ¬ Fax
∀x ((Bx ∧ Lax) → ¬ Fax)

B: [ _ is a book]; F: [ _ found _ ]; L: [ _ was looking for _ ]; a: Al

no one ate only potato chips

no one is such that (he or she ate only potato chips)

(∀x: x is a person) ¬ x ate only potato chips

(∀x: Px) ¬ only potato chips are such that (x ate them)

(∀x: Px) ¬ (∀y: ¬ y is a potato chip) ¬ x ate y

(∀x: Px) ¬ (∀y: ¬ Cy) ¬ Axy
∀x (Px → ¬ ∀y (¬ Cy → ¬ Axy))

A: [ _ ate _ ]; C: [ _ is a potato chip]; P: [ _ is a person]

4.	(∀x: ¬ `Bill saw` x) `Al saw` x (∀x: `Bill didn’t see` x) `Al saw` x `everything that Bill didn’t see is such that (Al saw it)` `Al saw everything that Bill didn’t see`

Counterexample presented by a diagram