Phi 270 text 7.2.xa

7.2.xa. Exercise answers

1.	a.	∀x (Fx → Gx) `Everything`, x, `is` `such` `that` `if` Fx [or: F `fits` x] `then` Gx
	b.	(∀x: Fx) ¬ Gx `Everything`, x, `such` `that` F x `is` `such` `that` `not` Gx
	c.	∀x ((Fx ∧ ¬ Gx) → Hx) `Everything`, x, `is` `such` `that` `if` `both` Fx `and` `not` Gx `then` Hx
	d.	(∀x: Px ∧ ¬ Rxx) Rxa `Everything`, x, `such` `that` `both` Px `and` `not` Rxx `is` `such` `that` Rxa
	e.	∀x ((Rxa ∧ ¬ Rbx) → ¬ (Fx ∨ Gx)) `Everything`, x, `is` `such` `that` `if` `both` Rxa `and` `not` Rbx `then` `not` `either` Fx `or` Gx
	f.	(∀x: Fx ∨ Gx) (Hx ∧ ¬ Kx) `Everything`, x, `such` `that` `either` Fx `or` Gx `is` `such` `that` `both` Hx `and` `not` Kx

Everyone had heard about the accident
Everyone is such that (he or she had heard about the accident)
(∀x: x is a person) x had heard about the accident

(∀x: Px) Hxa
∀x (Px → Hxa)

[H: λxy (x had heard about y); P: λx (x is a person); a: the accident]

Every relative of Sam agreed with him about the issue

Every relative of Sam is such that (he or she agreed with Sam about the issue)

(∀x: x is a relative of Sam) x agreed with Sam about the issue

(∀x: Rxs) Axsi
∀x (Rxs → Axsi)

[A: λxyz (x agreed with y about z); R: λxy (x is a relative of y); i: the issue; s: Sam]

	c.	`Edna took pleasure in none of her possessions` `No possession of Edna is such that (Edna took pleasure in it)` (∀x: x `is a possession of` `Edna`) ¬ `Edna` `took pleasure in` x (∀x: Pxe) ¬ Lex ∀x (Pxe → ¬ Lex) [L: λxy (x `took pleasure in` y); P: λxy (x `is a possession of` y); e: `Edna`]
	d.	`Tom found only empty boxes` `Among boxes, only empty ones are such that (Tom found them)` (∀x: x `is a box` ∧ ¬ x `is empty`) ¬ `Tom` `found` x (∀x: Bx ∧ ¬ Ex) ¬ Ftx ∀x ((Bx ∧ ¬ Ex) → ¬ Ftx) [B: λx (x `is a box`); E: λx (x `is empty`); F: λx (x `found` y); t: `Tom`]
	e.	`The survey was sent to all members of the organization except its officers` `All members of the organization except the organization’s officers are such that (the survey was sent to them)` (∀x: x `is a member of` `the organization` ∧ ¬ x `is an officer of` `the organization`) `the survey` `was sent to` x (∀x: Mxo ∧ ¬ Oxo) Ssx ∀x ((Mxo ∧ ¬ Oxo) → Ssx) [M: λxy (x `is a member of` y); O: λxy (x `is an officer of` y); S: λxy (x `was sent to` y); o: `the organization`; s: `the survey`]
	f.	`Only countries bordering the Pacific will prosper` `Among countries, only those bordering the Pacific are such that (they will prosper)` (∀x: x `is a country` ∧ ¬ x `borders` `the Pacific`) ¬ x `will prosper` (∀x: Cx ∧ ¬ Bxp) ¬ Px ∀x ((Cx ∧ ¬ Bxp) → ¬ Px) [B: λxy (x `borders` y); C: λx (x `is a country`); P: λx (x `will prosper`); p: `the Pacific`]

3.	a.	(∀x: x `is a dog`) x `chases cats` `Every dog is such that (it chases cats)` `Every dog chases cats` (or: `All dogs chase cats`; or: `Dogs chase cats`)
	b.	(∀x: x `is a hole`) `Holly patched` x `Every hole is such that (Holly patched it)` `Holly patched every hole` (or: `Holly patched each hole`)
	c.	(∀x: x `is a person`) ¬ x `volunteered` `No one is such that (he or she volunteered)` `No one volunteered`.
	d.	(∀x: ¬ x `is a cockroach`) ¬ x `will survive` `Only cockroaches are such that (they will survive)` `Only cockroaches will survive`
	e.	∀x ¬ x `seemed right` `Nothing is such that (it seemed right)` `Nothing seemed right`
	f.	(∀x: x `was a reviewer` ∧ ¬ x `was a friend of the director`) x `panned the movie` `All reviewers except friends of the director are such that (they panned the movie)` `All reviewers except friends of the director panned the movie` (or: `Among reviewers, all but friends of the director panned the movie`)
	g.	(∀x: x `is a bird` ∧ ¬ x `is an early bird`) ¬ x `gets a worm` `Among birds, only early ones are such that (they get worms)` `Only early birds get worms` (or: `No birds except early ones get worms`)
	h.	(∀x: ¬ x `is a small child`) `the movie bored` x `All things except small children are such that (the movie bored them)` `The movie bored all but small children`

Glen Helman 29 Oct 2004