phi_270_F10_test

Phi 270 F10 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.	`No one was disappointed.` answer
2.	`If any part was missing, the set wasn't assembled.` answer
3.	`Only cartoons appealed to everyone.` answer

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

¬ (∀x: Jx ∧ ¬ Sx) ¬ Fx

F: [ _ was finished]; J: [ _ is a job]; S: [ _ is small]

answer

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

Use a derivation to show that the following argument is not valid and present a counterexample that divides an open gap. (You may present the counterexample either by a diagram or by tables.)
7.	Rab ∀x Rxa ∀x Rxb answer

Phi 270 F09 test 4 answers

1.	`no one was disappointed.` `no one is such that (he or she was disappointed)` (∀x: x `is a person`) ¬ x `was disappointed` (∀x: Px) ¬ Dx ∀x (Px→ ¬ Dx) D: [ _ `was disappointed`]; P: [ _ `is a person`]

if any part was missing, the set wasn't assembled

every part is such that (if it was missing, the set wasn't assembled)

(∀x: x is a part) if x was missing, the set wasn't assembled

(∀x: Px) (x was missing → the set wasn't assembled)

(∀x: Px) (Mx → ¬ the set was assembled)

(∀x: Px) (Mx → ¬ As)
∀x (Px → (Mx → ¬ As))

A: [ _ was assembled]; M: [ _ was missing]; P: [ _ is a part]; s: the set

only cartoons appealed to everyone

only cartoons were such that (they appealed to everyone)

(∀x: ¬ x is a cartoon) ¬ x appealed to everyone

(∀x: ¬ Cx) ¬ everyone is such that (x appealed to him or her)

(∀x: ¬ Cx) ¬ (∀y: y is a person) x appealed to y

(∀x: ¬ Cx) ¬ (∀y: Py) Axy
∀x (¬ Cx → ¬ ∀y (Py → Axy))

A: [ _ appealed to _ ]; C: [ _ is cartoon]; P: [ _ is a person]

¬ (∀x: x is a job ∧ ¬ x is small) ¬ x was finished

¬ (∀x: x is a job ∧ x isn’t small) x was unfinished

¬ (∀x: x is a job that isn’t small) x was unfinished

¬ every job that isn’t small it such that (it was unfinished)

¬ every job that isn’t small was unfinished

not every job that isn’t small was unfinished
or: among jobs not only small ones were finished
or: not only small jobs were finished
or: it’s false that no jobs that are not small were finished

	│∀x Mx	a:2
	│∀x (Mx → Qx)	a:3
	├─
	│ⓐ
2 UI	││Ma	(4)
3 UI	││Ma → Qa	4
4 MPP	││Qa	(5)
	││●
	│├─
5 QED	││Qa	1
	├─
1 UG	│∀x Qx

	│∀x ∀y (Fx → Gy)	a:3
	├─
	││Fa	(5)
	│├─
	││ⓑ
3 UI	│││∀y (Fa → Gy)	b:4
4 UI	│││Fa → Gb	5
5 MPP	│││Gb	(6)
	│││●
	││├─
6 QED	│││Gb	2
	│├─
2 UG	││∀x Gx	1
	├─
1 CP	│Fa → ∀x Gx

	│Rab
	│∀x Rxa	a:2, b:3, c:4
	├─
	│ⓒ
2 UI	││Raa
3 UI	││Rba
4 UI	││Rca
	││
	│││¬ Rcb
	││├─
	│││○	¬ Rcb, Rca, Rba, Raa, Rab ⊭ ⊥
	││├─
	│││⊥	5
	│├─
5 IP	││Rcb	1
	├─
1 UG	│∀x Rxb

Counterexample presented by a diagram

Counterexample presented by tables

a	b	c
1	2	3

R	1	2	3
1	T	T	F
2	T	F	F
3	T	F	F