phi_270_F08_test

Phi 270 F08 test 4

Analyze the sentences below in as much detail as possible, providing a key to the non-logical vocabulary you use. State your analysis also in a form that expresses any generalizations using unrestricted quantifiers.
1.	`No cover fit the container.` answer
2.	`Everyone who Sam spoke to had seen the movie.` answer
3.	`Only dogs chewed every bone.` answer
4.	`No one who everyone knew bought anything.` answer

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

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

Phi 270 F08 test 4 answers

1.	`no cover fit the container` `no cover is such that (it fit the container)` (∀x: x `is a cover`) ¬ x `fit the container` (∀x: Cx) ¬ Fxc ∀x (Cx→ ¬ Fxc) C: [ _ `is a cover`]; F: [ _ `fit` _ ]; c: `the container`

everyone who Sam spoke to had seen the movie

everyone who Sam spoke to is such that (he or she had seen the movie)

(∀x: x is a person who Sam spoke to) x had seen the movie

(∀x: x is a person ∧ Sam spoke to x)) Sxm

(∀x: Px ∧ Ksx) Sxm
∀x ((Px ∧ Ksx) → Sxm)

K: [ _ spoke to _ ]; P: [ _ is a person]; S: [ _ had seen _ ]; m: the movie; s: Sam

only dogs chewed every bone

only dogs are such that (they chewed every bone)

(∀x: ¬ x is a dog) ¬ x chewed every bone

(∀x: ¬ Dx) ¬ every bone is such that (x chewed it)

(∀x: ¬ Dx) ¬ (∀y: y is a bone) x chewed y

(∀x: ¬ Dx) ¬ (∀y: By) Cxy
∀x (¬ Dx → ¬ ∀y (By → Cxy))

B: [ _ is a bone]; C: [ _ chewed _ ]; D: [ _ is a dog]

No one who everyone knew bought anything

everything is such that (no one who everyone knew bought it)

∀x no one who everyone knew bought x

∀x no one who everyone knew is such that (he or she bought x)

∀x (∀y: y is a person who everyone knew) ¬ y bought x

∀x (∀y: y is a person ∧ everyone knew y) ¬ Byx

∀x (∀y: Py ∧ everyone is such that (he or she knew y)) ¬ Byx

∀x (∀y: Py ∧ (∀z: z is a person) z knew y) ¬ Byx

∀x (∀y: Py ∧ (∀z: Pz) Kzy) ¬ Byx
∀x ∀y ((Py ∧ ∀z (Pz → Kzy)) → ¬ Byx)

B: [ _ bought _ ]; K: [ _ knew _ ]; P: [ _ is person]

Counterexample presented by a diagram