phi_270_F99_test

Phi 270 F99 test 4

F99 test 4 questions

Analyze the following sentences in as much detail as possible, providing a key to the non-logical vocabulary (upper and lower case letters) appearing in your answer.
1.	`Sam invited every vertebrate to the party, but only people accepted his invitation` answer
2.	`Tom didn’t send anything to the printer` answer
3.	`No game that every child liked was complete` answer
Synthesize an English sentence whose analysis would yield the following form.
4.	(∀x: Px) (∀y: Ry ∧ Txy) Sy P: [ _ `is a person`]; R: [ _ `is a room`]; S: [ _ `was reserved`]; T: [ _ `thought of` _ ] answer

Use derivations to establish the validity of the following arguments. You may use attachment rules.
5.	∀x (Fx → Gx) ∀x Fx → ∀x Gx answer
6.	∀x ∀y (Fyx → ¬ Py) ∀x (Px → ∀y ¬ Fxy) 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 is a counterexample lurking one of the derivation’s open gaps.
7.	∀x ∀y (Fy → ¬ Rxy) ∀x Rxx ∀x ∀y ¬ Rxy answer

F99 test 4 answers

Sam invited every vertebrate to the party, but only people accepted his invitation

Sam invited every vertebrate to the party ∧ only people accepted Sam’s invitation

every vertebrate is such that (Sam invited it to the party) ∧ only people are such that (they accepted Sam’s invitation)

(∀x: x is a vertebrate) Sam invited x to the party ∧ (∀x:¬ x is a person) ¬ x accepted Sam’s invitation

(∀x: Vx) Isxp ∧ (∀x:¬ Px) ¬ Ax(Sam’s invitation)

(∀x: Vx) Isxp ∧ (∀x: ¬ Px) ¬ Ax(is)

A: [ _ accepted _ ]; I: [ _ invited _ to _ ]; P: [ _ is a person]; V: [ _ is a vertebrate]; i: [ _ ’s invitation]; p: the party; s: Sam

2.	`Tom didn’t send anything to the printer` `everything is such that (Tom didn’t send it to the printer)` ∀x `Tom didn’t send` x `to the printer` ∀x ¬ `Tom` `sent` x `to` `the printer` ∀x ¬ Stxp S: [ _ `sent` _ `to` _ ]; p: `the printer`; t: `Tom`

No game that every child liked was complete

No game that every child liked is such that (it was complete)

(∀x: x was a game that every child liked) ¬ x was complete

(∀x: x was a game ∧ every child liked x) ¬ Cx

(∀x: x was a game ∧ every child is such that (he or she liked x)) ¬ Cx

(∀x: Gx ∧ (∀y: y was a child) y liked x) ¬ Cx

(∀x: Gx ∧ (∀y: Dy) Lyx) ¬ Cx

C: [ _ was complete]; D: [ _ was a child]; G: [ _ was a game]; L: [ _ liked _ ]

(∀x: x is a person) (∀y: y is a room ∧ x thought of y) y was reserved

(∀x: x is a person) (∀y: y is a room x thought of) y was reserved

(∀x: x is a person) every room x thought of was such that (it was reserved)

(∀x: x is a person) every room x thought of was reserved

everyone is such that (every room he or she thought of was reserved)

every room anyone thought of was reserved

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

	│∀x ∀y (Fyx → ¬ Py)	b:5
	├─
	│ⓐ
	│││Pa	(8)
	││├─
	│││ⓑ
	│││││Fab	(7)
	││││├─
5 UI	│││││∀y (Fyb → ¬ Py)	a:6
6 UI	│││││Fab → ¬ Pa	7
7 MPP	│││││¬ Pa	(8)
	│││││●
	││││├─
8 Nc	│││││⊥	4
	│││├─
4 RAA	││││¬ Fab	3
	││├─
3 UG	│││∀y ¬ Fay	2
	│├─
2 CP	││Pa → ∀y ¬ Fay	1
	├─
1 UG	│∀x (Px → ∀y ¬ Fxy)

	│∀x ∀y (Fy → ¬ Rxy)	a:4,b:5
	│∀x Rxx	a:6,b:7
	├─
	│ⓐ
	││ⓑ
	││││Rab	(11)
	│││├─
4 UI	││││∀y (Fy → ¬ Ray)	a:8, b:9
5 UI	││││∀y (Fy → ¬ Rby)	a:12, b:13
6 UI	││││Raa	(10)
7 UI	││││Rbb	(14)
8 UI	││││Fa → ¬ Raa	10
9 UI	││││Fb → ¬ Rab	11
10 MTT	││││¬ Fa
11 MTT	││││¬ Fb
12 UI	││││Fa → ¬ Rba	15
13 UI	││││Fb → ¬ Rbb	14
14 MTT	││││¬ Fb
	││││
	││││││¬ Fa
	│││││├─
	││││││○	¬Fa,¬Fb,Rab,Raa,Rbb ⊭ ⊥
	│││││├─
	││││││⊥	16
	││││├─
16 IP	│││││Fa	15
	││││
	│││││¬ Rba
	││││├─
	│││││○	¬Fa,¬Fb,Rab,
	│││││	Raa,Rbb,¬Rba ⊭ ⊥
	││││├─
	│││││⊥	15
	│││├─
15 RC	││││⊥	3
	││├─
3 RAA	│││¬ Rab	2
	│├─
2 UI	││∀y ¬ Ray	1
	├─
1 UI	│∀x ∀y ¬ Rxy

The counterexample below lurks in both gaps: